Theory of strain percolation in metals: mean field and strong boundary universality class

Physica A 283 (2000) 307–327

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Theory of strain percolation in metals: mean eld and strong boundary universality class a Materials

Robb Thomsona , L.E. Levinea; ∗ , D. Stauerb

Science and Engineering Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA b Institute for Theoretical Physics, Cologne University, D-50923 K oln, Germany Received 24 January 2000

Abstract For the percolation model of strain in a deforming metal proposed earlier, we develop sum rule and mean eld approximations which predict a critical point. The numerical work is restricted to the simpler of two cases proposed in the earlier work, in which the cell walls are “strong”, and unzipping of the dislocation entities which lock the walls into the lattice is not permitted. For this case, we nd that strain percolation is a new form of correlated percolation, but that it c 2000 Published by Elsevier Science is in the same universality class as standard percolation. B.V. All rights reserved. PACS: 62.20.Fe; 64.60.Ak; 81.40.Ef Keywords: Dislocation percolation; Metal deformation

1. Introduction In this paper, we extend two aspects of an earlier letter [1] (called I hereafter) in which we introduced a percolation model for strain in a deforming metal. In the present paper, we will develop two analytic approximations (a sum rule theorem and mean eld version) of the model in 2D (the relevant case for metal deformation), and compute the critical exponents for the case where strain is transmitted through dislocation walls only by the activation of dislocation sources. In the language of our earlier paper this case corresponds to “strong walls with unzippable locks”. When a metal deforms, large numbers of line defects, called dislocations, are produced which, by their motion under stress, give rise to the irreversible plastic ow of ∗

Corresponding author. E-mail address: [email protected] (L.E. Levine).

c 2000 Published by Elsevier Science B.V. All rights reserved. 0378-4371/00/$ - see front matter PII: S 0 3 7 8 - 4 3 7 1 ( 0 0 ) 0 0 0 9 7 - 2

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Fig. 1. Micrograph of Cu single-crystal deformed plastically by uniaxial tension along a [100]. The dark regions are the walls of high dislocation density surrounding regions of very low dislocation density. Micrograph courtesy of H. Mughrabi.

the metal. One of the most striking characteristics of these dislocations is that they interact with each other with long-range forces, which in 2D are angle dependent 1=r forces. In addition to their elastic interactions, however, dislocations in crystals are also constrained to glide only on certain crystallographic planes (though movement o the plane is possible by diusive “recovery” processes), and when their atomic cores overlap, a number of distinctive atomic scale interactions can occur which induce dislocation reactions (i.e., creation of new dislocation types), some of which can eectively lock a dislocation and its local surrounding structure into a speci c region of the lattice [2]. Given the long-range interactions, the ability of the dislocations to contort in 3D, and the “zoo” of atomic scale interactions, it is not surprising that the dislocation con gurations developed in deforming metals are exceedingly complex. But the most striking feature of the dislocation structure which develops is that it is partially ordered. It is conventional to speak of several “stages” in the deformation process, beginning with an initial stage I dominated by deformation on only a single-slip plane, followed by the multiple-slip stage II in which more than one slip plane of the allowed crystallographic-type operates, and followed again by a stage III in which thermally activated processes such as dislocation climb or diusion play a role. Dislocation ordering plays a role in each of these stages, but between stages II and III, a typical three-dimensional cellular structure develops, such as that shown in the micrograph of Fig. 1 [3] with walls of high dislocation density surrounding 3D regions where the dislocation density is at least an order of magnitude lower. It is this 3D cellular structure

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on which this paper is focused. The reader will nd detailed reviews of all aspects of the problem of dislocation structure evolution in deforming metals, and the “work hardening” which it causes in any of several recent reviews [4 –7]. The classic problem in metal deformation is to predict the stress–strain law for a deforming metal in terms of the physics of the generation and motion of the underlying dislocations, including the problem of the development of ordered patterns. But a solution to this problem in its entirety has eluded theorists for three quarters of a century, and we will not attempt its resolution here. Rather, our point of departure is to note that the total problem can be usefully and logically separated into a part which deals with the development of the dislocation patterns, and a part which deals with the transport of mobile dislocations through this partially ordered structure. Our focus in this paper is on the transport part of the problem. The transport of dislocations under stress is in a way the most important part of the problem, because it leads to insight into, and predictions of, the ultimate stress–strain response of the metal, which is the primary goal of any theory of work hardening. Of course, any transport theory will require information about how the structure evolves with deformation. We will simply assume that information on the structure evolution can be obtained either from experimental measurement, or ultimately, from an adequate theory of the evolution. It goes without saying, of course, that an ultimate and fully satisfactory theory of the work hardening problem can only follow from a properly uni ed view of both the ordering and transport parts of the problem. But until such time as such a theory is available, our separation of the problem is a very useful strategy. Our approach is to view the transport problem as a version of percolation theory, in which the walls of the ordered cells act as both sources of mobile dislocations, as well as barriers to their propagation. The goal of this paper is to make the desired connections between the percolation model of dislocation transport and the already highly developed eld of percolation theory [8]. The idea that strain in a deforming metal is a percolation problem goes back to work by Kocks [9], in which he generalized an earlier model of the penetration of mobile dislocations through a random distribution of pinning points [10]. Following his suggestion, others have explored the problem of dislocations bowing around randomly distributed pinning points with computer simulations [11–13]. Although Kocks has also proposed that the dislocation percolation problem is similar or equivalent to invasion percolation with trapping [14,15], a serious invocation of percolation theory for the work hardening problem has not been pursued. Nevertheless, his is an appealing picture, in that the observed external strain is a natural outcome in the model, and he also realistically proposes that the system hardens because parts of the mobile population become trapped in the postulated “hard spots”. But his model envisions essentially zero-dimensional hard spots, with an open structure through which individual dislocations can penetrate by Orowan bowing [2]. There is no place in such a model for the 3D dislocation cells which are observed, and, indeed, Kocks has expressed the view that the closed network structures are the result of relaxations which occur only after the driving stress is turned o [9].

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Fig. 2. Square network of walls on a regular lattice. The cell at the center initiates the strain with a dislocation pileup on the slip plane which impinges on the surrounding walls. Some of these walls allow new dislocation pileups to be formed in the neighboring unstrained cells, and the strained cluster grows. Strain growth takes place only on the cluster surface. The strained cluster thus formed has a fractal character at the critical limit.

In contrast, for us, the 3D cell structure is a central feature of the deforming system, with the walls of the cells acting as both barriers to the percolation of dislocations through the structure, as well as sources (and sinks) for the mobile percolating dislocations. In particular, we assume that the dislocations trapped in the walls can be induced to act as sources of new dislocations in adjacent cells, for example by means of the Frank-Read bowing mechanism [2]. It is also possible that the dislocations of the walls can be unlocked or “unzipped” [1], in which case, a portion of the wall can be broken down, and a large multiplication of dislocations takes place, but this second mechanism is not considered in this paper. The incipient sources in the walls are activated by the stress induced at the head of a dislocation pileup, and because of the highly random character of the wall dislocation content, we will treat the source action as a stochastic process. Any strain thus generated propagates from cell to cell until it runs out of steam or propagates to the surface where a slip step is formed. Such a strain propagation process is clearly determined by statistics, with probability distributions governing the cell size, the source density in the walls, and the propagation, itself. In our modeling, however, we will simplify the picture somewhat in order to make the numerical problem tractible, and in particular, we will assume that the 2D section through the cell structure on which the slip propagates is a regular square lattice. The implications of this assumption will be explored in the next section. Fig. 2 illustrates the model we use, showing a central initiating site propagating dislocation pileups into neighboring lattice cells in a stochastic manner. By invoking the full panoply of statistical percolation theory, we are able to show that our model of the deforming metal describes a self-organizing critical system [16],

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with the evolution of the system taking place on a critical surface in the space of the input parameters of the system [1]. In this paper, we extend the results reported in I in two principal ways. First, we develop a sum rule and a mean eld approximation for strain percolation in 2D, which leads to a useful physical picture for the model. Second, we investigate the numerical model for the simple case of strong cell boundaries (no “unzippable” locks [1]), compare it with the mean eld and sum rule results, and investigate the universality class of this correlated percolation problem. More precise details of the model are presented in Section 2. In Section 2, the sum rule and the mean eld approximation in 2D are developed. In Section 4, we brie y summarize the aspects of percolation theory we apply to the strain problem, and in the following section, present the numerical results for the case where the locks in the cell boundaries cannot be unzipped. In an appendix, we present a variation of the strain problem which is closer to standard percolation, and discuss results for this related model.

2. The model The model sketched in the Introduction will now be more completely de ned. We begin with a physical system which has been brought to a state of strain wherein a 3D partially ordered dislocation structure has been formed. In particular, we envision the 3D cell structure which occurs in the rst part of stage III deformation in fcc metals shown in Fig. 1. In stage I deformation, the ne slip lines on the metal surface are long and uniformly distributed. But by stage III, the slip lines bunch together, forming distinct bands (see Fig. 3 [17]). Each slip line corresponds to strain which occurs on a single-slip plane in the interior of the crystal, and exits the surface with the production of the observed slip step. Because the slip bands are so far apart, it will be presumed that the interaction between separate slip bands is small. But the observed thickening of a slip band which occurs as deformation proceeds implies that once a slip event has taken place on a single plane, subsequent slip events will be clustered on nearby planes, a topic to which we will return in a later paper. It is observed that the creation of a single slip step is a discontinuous event, and it is this discontinuous planar slip process which the percolation model is speci cally designed to describe. In the remainder of the paper, when we refer to slip within a slip “line”, we will be referring to the slip within the plane that intersects the surface, rather than the slip line itself. With the system brought to its initial strain state, the external stress is then presumed to be incremented by . We propose that the system responds with a burst of strain, s0 , in the weakest cell of a new slip line. In the percolation model, we follow the evolution of the strain, s, as it is transmitted from cell to cell throughout the slip plane of the slip line. For simplicity, we take the variable, s, to be the number of new slip dislocations in a cell. The initiating burst is equivalent to a set of dislocations piled up against the walls of the cell. As noted in the Introduction, the transmission of strain to a neighboring cell

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Fig. 3. Slip lines and bands. This early gure shows the slip lines and slip bands as they appear on the surface of a single crystal of Copper. The sample was rst pre-strained by 60% polished, and then strained an additional 10%. After Seeger [17].

will be driven by the stress concentration from the pileup acting on the intervening wall between a strained and an unstrained cell. According to dislocation theory, the force on the head dislocation in a pileup, and therefore the driving force for strain transmission through the wall, is proportional to the number of dislocations in the pileup [2]. Thus, we write s∗ = as ;

(1)

where s∗ is the number of dislocations transmitted into the unstrained cell, s is the number in the cell containing the pileup, and a is a stochastic variable which is a measure of the ability of a wall to transmit strain from a strained cell to an unstrained one. The linear character of this propagation equation is a crucial feature of the overall problem, and leads to a variety of correlated percolation theory [18–21] unlike that of either standard random percolation [8], or of invasion percolation [14]. It is a correlated percolation problem, because the probability that a site is strained depends on the strain of its neighbors. In standard random percolation theory, the occupation of a site is a random uncorrelated event. But the correlation induced in the cluster geometry by (1) is unlike that of any of the previously studied cases, and thus we cannot draw directly on the experience with those models to gain insight into the present case.

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A rule which follows immediately from the physics of the strain problem is that s ¿1, because the dislocations are discrete, and unless at least one dislocation is produced in the newly strained cell, s∗ = 0. Because strain is propagated by dislocation glide, the growing strain cluster will keep (more or less) to a single glide plane of the metal. Indeed, it is generally observed that even in a system exhibiting overall multiple slip, the slip in a given region will usually consist primarily of glide on a single plane. Thus, the appropriate percolation model is 2D. As the strain proceeds from cell to cell, a connected strain cluster grows as depicted in Fig. 2. An essential and heuristic assumption in all that follows is that the growth of the cluster takes place only on the cluster periphery, and that once an unstrained cluster becomes strained, it maintains its state of strain thereafter without increase. A growing cluster will ordinarily enclose islands of unstrained cells, so that part of the periphery will consist of the internal perimeter of these unstrained islands. This peripheral growth assumption is consistent with a well-posed percolation problem, and is quite easy to program on a computer, but it does not follow from a proper stress analysis of the problem. That is, a true solution of the dislocation transport problem would proceed from a complete analysis of the internal stresses in all the cells of the system, with a prediction of which cells contain dislocation sources with critical stresses acting on them, whether or not the cells are on a periphery. But specifying the full internal stress distribution (even randomized) in the system would be tantamount to a rst principles solution of the dislocation problem. Instead, the growth assumption is a heuristic statement based on the physical fact that after a cell becomes strained, some of the mobile dislocations activated will be absorbed by the walls. As noted by Kocks [9], the walls are thereby hardened, so that further dislocation activity in an interior cell is inhibited until the external stress is increased. A further rule must be adopted to cover the situation on a periphery where more than one strained cell is adjacent to an unstrained cell. In the numerical work, we will assume that the cell producing the largest value of s∗ is the active one. There is no strong physical reason for this rule, but unless some such rule is adopted, the model is not well de ned. We have explored other assumptions, and nd that this rule does not lead to results signi cantly dierent from other rules, and it is easy to implement in the numerical work. In the mean eld approximation, however, such a maximum choice rule is dicult to implement, and we will instead adopt a rule that the transmitted strain is the sum of the s∗ ’s, calculated from some or all of the strained neighbors of an unstrained cell. Finally, we must provide a method for obtaining the stochastic variable, a, in Eq. (1). We proceed from our physical view sumarized in the Introduction of the processes by which strain is transferred from a strained to an unstrained cell. As noted there, we envision two distinct mechanisms. In the rst, we suppose the walls will contain a distribution of dislocation sources which can be activated by the stress eld of a pileup within a strained cell on the periphery. Thus, we write the probability that such a ∗

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wall source will be activated as a = P1  ; P1 ; (2) 2 where 0661 is a random number, as might be obtained from a computer random number generator, and P1 is a parameter which re ects the properties of the wall. P1 will be of order unity at the percolation limit. The average value of the ampli cation factor, a,  will be important in the following. In passing, we note the possibility that dislocations of a pileup may actually pass through a wall by Orowan bowing [2] past the immobile dislocations of the wall is also a mechanism which can be incorporated in P1 . Thus, Orowan bowing can be considered to be a subclass of dislocation sources in the wall. As noted in the Introduction, we believe a second important mechanism for strain transfer is associated with the unzipping of the locks which attach the wall to the lattice. An exploration of this mechanism was begun in I, but the more complex results of that case will be presented in a separate paper. As noted in the Introduction, in the simulations, we replace the actual irregular geometry with a regular square lattice (Fig. 2). In standard site percolation, changing the lattice from square to hexagonal changes the percolation limit from 0.59 to 12 , but does not change the universality class. The fact that deformation cells are more like a random network should cause a comparable variation. Thus, the geometrical consequencies due purely to the connectivity of the square lattice approximation should not be serious. A more important consideration is the cell size distribution, which is known to have fractal character [22]. In our simulations, since the cells are completely uniform, with no size variation, several corrections must be considered. The rst is that the parameter, P1 , will depend on the cell size in a rather complex fashion, because it depends on the details of the dislocation structure in the wall, which in turn will depend on the cell size. Also, the number of dislocations transmitted into a new cell will depend on the size of the cell, because the back stress from the pileup on the sources acting in the wall will depend on the cell size. In this paper, we merely note these approximations, and return to their impact on the form and content of the mechanical constitutive laws in a later paper. For the purposes of this paper, then, all these considerations will simply be subsumed in the value of P1 . The strain percolation model is now mathematically de ned, and can be programmed by methods similar to those developed by Leath [23] for a percolation problem initiated with a value s = s0 at the origin. a =

3. Sum rule and mean ÿeld approximation A mean eld approach to the strain problem is useful because analytic approximations can be obtained thereby, which lead to very useful physical insights. In 2D, we

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will write the propagation equation, Eq. (1), on the periphery as a1 (m; n)s(m − 1; n) + a2 (m; n)s(m; n − 1) = s(m; n) ;

(3)

where m and n are integers that designate the cell position within the square lattice, and a1 and a2 are stochastic variables which are obtained (independently) for each direction from the assumed form, Eq. (2). In writing this equation, the sum rule for calculating the transmitted strain is adopted, as explained in the previous section. We have also made the initial assumption that every unstrained site on the periphery has exactly two strained neighbors. This assumption will be relaxed later on, and a more general solution will be presented. From (3), one can write the sum on the periphery as X X [a1 (m; n)s(m − 1; n) + a2 (m; n)s(m; n − 1)] = s(m; n) ; (4) p

p

where the sum on the left is over the previously strained cells at the periphery, and the one on the right is over the cells subsequently strained. By the mean value theorem, P  1 , etc., so we can write these equations as a1 s(m − 1; n) = a1 S  1 + a2 S  2 ) = Np S  : Np (a1 S

(5)

Because of the random character of the wall distribution from cell to cell, it is natural to assume isotrophy in the strain evolution on average. Then X X 0 ; s(m − 1; n) = s(m; n − 1) = S p

p

0=S  : (a1 + a2 )S

(6)

 and the last equation becomes Also, if the system is isotropic, then a1 = a2 = a,  : 0=S 2aS

(7)

To proceed further, it is necessary to explore the nature of the solution we except. When the ampli cation factor, a,  which controls the transmission of strain from cell to cell is below its percolation limit, the strain will be localized near the origin, in the way of all percolation systems, and the total strain in the local strained cluster will be nite. But above the critical limit, the strain grows in magnitude from cell to cell, and diverges at in nity. In this case, not only is the total integrated strain of the (in nite) strained cluster in nite, but the strain in the cells far from the initial strained cell at the origin grows without limit. This situation corresponds to an avalanche of strain. The percolation limit for the system divides these two regimes of behavior, and at the percolation limit, the strain does not grow, on average, from cell to cell. Thus, on the periphery, at the percolation limit, the average of the strain in the strained cells in the   0 = S. previous equation must exactly equal the average of the transmitted strain, S

Then, from the last equation, we get the condition for the percolation limit, ac =

1 2

:

(8)

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The derivation of the percolation limit above is valid only for the simpli ed form of the propagation equation, Eq. (4). But for this derivation, one can relax the two neighbor assumption, and carry out the summation over the actual distribution of neighbors of an unstrained cell on the periphery, giving the expression 0=S  ; (p1 + 2p2 + 3p3 + 4p4 )aS

(9)

where pn is the probability of nding n strained neighbors. The simplest approach for the pn is to assume that the distribution of probabilities is entirely random. In that case, p1 is the combination of four neighbor sites (because the lattice is square) taken one at a time, p2 is the combination of four sites taken two at a time, etc. After 4 6 4 1 normalizing the p’s, we nd p1 = 15 ; p2 = 15 ; p3 = 15 , and p4 = 15 . Substituting into (9) gives the critical percolation limit ac =

15 32

;

(10)

which is very close to Eq. (8). Although Eq. (9) is expected to be exact, the above probabilities p1 , etc., are not rigorously derived. Thus, the predicted value of ac = 15 32 will not be exact. This percolation limit derived from the sum rule can be compared to the more approximate mean eld limit by taking the continuum limit of the propagation equation, Eq. (3). Then, a1 (x; y)

@s(x; y) @s(x; y) + a2 (x; y) = (1 − a1 (x; y) − a2 (x; y))s(x; y) : @x @y

(11)

This equation can be solved by the standard separation of variables, with the assumptions a1 (x; y) = a1 (x); a2 (x; y) = a2 (y), and s(x; y) = s1 (x)s2 (y). Then     a2 (y) ds2 (y) a1 (x) ds1 (x) + a1 (x) + + a2 (y) = 1 : (12) s1 (x) dx s2 (y) dy There are now two ordinary dierential equations ds1 (x) b1 − a1 (x) = s1 (x) ; dx a1 (x) ds2 (y) b2 − a2 (y) = s2 (y) ; dy a2 (y) b1 + b2 = 1 :

(13) 1 2.

From symmetry, b1 = b2 = We expand the functions, a, about their average values, such that a1 = (1 + 1 )a1 , and then expand 1=a1 and 1=a2 as series, and write the quantity, s1 (x)(1 + 21 + · · ·) as a new stochastic variable, 1 (x), etc. Then, in lowest order, (13) can be written as the two independent Langevin equations   1 ds1 (x) = − 1 s1 (x) + 1 (x) ; dx 2a1   1 ds2 (y) − 1 s2 (y) + 2 (y) : (14) = dy 2a2

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When the stochastic noise variables, , are uncorrelated in the sense that h1 (0)1 (x)i= C(x), etc., then the correlation functions of the strains satisfy C −1 |x| e ; hs1 (x)s1 (0)i = 21 C −2 |y| e ; hs2 (y)s2 (0)i = 22 1 − 2ai : i = (15) 2ai When both the right-hand sides of these equations are divergent, that is when a1 = a2 = ac =

1 2

;

(16)

the system has an in nite correlation length in both dimensions, which is the de nition of a critical system. It is satisfying that predictions (16) and (8) are the same. In order to make this comparison, of course, it was necessary to make the restrictive two-neighbor assumption in the sum rule. But the more accurate prediction of the critical point given in Eq. (10) is very close. Of course, the mean eld limit results required severe approximations. We believe the most restrictive assumption is that the stochastic variable, a1 (x; y) is a function only of x, and similarly for a2 . Thus, in determining the stochastic variables for each point in the 2D system, there are not N 2 independence choices for a sample of side N , but only N for each cell boundary at a cell. But the fact that the critical point is identical to that predicted by the more rigorous sum rule gives one hope that the approximations have not changed the essential physics. The value of these mean eld results is in making the connection between the percolation limit of the discrete system and the true critical point of the mean eld system. Further, though we will not pursue it here, it also demonstrates the existence of the noise terms, , which have been used in a dierent connection by Hahner et al. [22] to obtain very interesting results about the cell size distributions. 4. Strain percolation In previous sections, the strain percolation model has been de ned, and the mean eld approximation has shown that a critical point exists (at least in mean eld). This leaves the problem of showing that the model does, indeed, lead to a percolation limit in the traditional sense of the term, and to explore the universality class for this model. In the Hoshen–Kopelman [24] version of standard percolation theory [8], an occupation probability is de ned for each site (or bond) in the system, and attention is then focused on the clusters of adjacent occupied sites. That is, if t is the number of adjacent occupied sites forming an occupied cluster, then nt will denote the number of such clusters (per lattice site) in the in nite system. (We will use the notation conventions of Ref. [20] where possible, but since we have already used s to denote the

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strain variable, we must choose a dierent letter for the number of sites in a cluster.) In our strain model, however, instead of an a priori occupation probability, we de ne the propagation of the strain, s, from a strained to an unstrained cell, as a linear function, Eq. (1). Second, our strain problem is akin to the Leath [23] version of standard percolation theory. In that version, the origin is a site known to the occupied, and a cluster is generated from this site with the prede ned occupation probabilities for every site in the lattice. That is, starting with the initial “cluster” at the origin, the sites neighboring the periphery of the occupied cluster are tested sequentially for addition to the occupied cluster. When no neighbors of the occupied periphery are occupied (and thus cannot be added to the occupied cluster) according to the preassigned probabilities for the lattice sites, then the cluster is mature. If no mature cluster is generated by this rule before the cluster spans the nite system, then the cluster is called a spanning or “in nite” cluster. In the Leath version of standard percolation, the probability that a cluster of size, t, will be generated in this way is given by wt = tnt , and a correspondence is thus established between the Leath method for generating a cluster distribution, and that computed by the Hoshen–Kopelman methodology. In our strain problem, one again starts with a known strained site at the origin, and propagates the strain from this site with the propagation law, Eq. (1). But in the strain problem, the amount of strain generated in a given cell is determined by the strain transmitted from an adjacent cell. Thus, the strain problem is a correlated percolation problem [18,20,21], and the size and shape of the strain cluster is path dependent. Moreover, the way a cluster of strained sites is generated in the strain problem is by transmission through a “bond” of the lattice, so the strain problem is a curious mixture of “site” and “bond” percolation in the terminology of the standard theory. That is, the physical entity to be studied in strain percolation is the cluster of strained lattice sites, but the sites are populated by a bond transmission algorithm. The path dependence along with the subtle correlated mixing of the two kinds of percolation means that the strain problem is a unique version of percolation theory, and no a priori presumption should be made about the expected percolation thresholds or the universality class. Further, the strain problem has no corresponding Hoshen–Kopelman version, and the occupation numbers, nt , de ned from the Leath transformation between wt and nt , have no physical signi cance. (In Appendix A, we introduce a variation of the strain problem to which a modi ed Hoshen–Kopelman algorithm can be applied, but it diers from the model of Section 2 in important ways). In our strain problem, there is a parameter, a,  which determines the percolation threshold in the mean eld limit, and it is reasonable to assume that this parameter plays the same role in the strain theory that p and pc play in the standard theory. With this assumption, and the standard scaling assumptions for any percolation problem, one can derive the appropriate equations for strain percolation theory in terms of a,  in a way analogous to that in standard percolation theory (see Ref. [20], for example). In summary form, we list the equations used. If wt is the probability of generating a non-spanning cluster consisting of t sites, the rst equation relates the mean cluster

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size, S, to the deviation from the threshold, for an in nite system, X twt (a)  ; S= t

 − : S ˙ |ac − a|

(17)

Again for an in nite system with a ¿ ac , the probability, w∞ , of generating an in nite spanning cluster obeys the scaling law, w∞ ˙ (a − ac ) :

(18)

The correlation length, , for a system is de ned from the radius of gyration for a cluster, P |ri − r0 |2 2 (19) Rt = i t for a given value of cluster size, t, with P 2 t tRt wt : (20) 2 = 2 P t twt For in nite systems, the correlation length scales as  − :  ˙ |ac − a|

(21)

For large, but nite systems,  is a strongly peaked function with a peak at a slightly shifted value of ac , exhibiting the power law in the tail region. For large, but nite, systems, S also exhibits the same behavior. As a function of system size, however, S(L; ) satis es the power law S(L) ˙ L− =

(22)

at the threshold value of a (or for nite systems with L). In the same way, at the percolation limit, it follows that w∞ (L) obeys the power law w∞ (L) ˙ L− = :

(23)

The fractal dimension of the spanning cluster is given by D = d − = :

(24)

This equation is independent of the previous equation, even though the exponents are the same, because here the actual spanning cluster size is plotted as a function of L. In standard percolation theory, w∞ is simply related to the spanning cluster size, but in the strain problem, this equivalence is not necessarily valid because of the special character of the site at the origin. In standard percolation theory, the hyperscaling relation

+ 2 = d connects the ratios, = and =, to give D as

 1 d+ : D= 2 

(25)

(26)

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D will be the slope of the average mass of the spanning clusters as a function of the system size. We note that this analysis refers to the purely geometrical character of the strained clusters, and the strain, itself, does not appear directly, i.e., the strain correlation length one can obtain from the correlation functions introduced in Section 2 is not the same as the geometrical correlation length, , discussed here. For further discussion of this point see the Conclusion section.

5. Numerical modeling In the numerical work, we start a simulation with an assumed value of s0 ¿ 1 at the origin, and employ one of two versions of the Leath algorithm [23]. The codes used for the simulations discussed in the appendix may be obtained from stau[email protected]. The codes used for the simulations presented in this section may be obtained from [email protected]. In our implementation of the Leath algorithm, each site gets the strain information from all those (1– 4) neighbors which were interrogated previously in the algorithm. We keep a list of growth sites, which when exhausted, means that a mature cluster is produced. In the simulation algorithm, a test is made to ascertain if an interrogated unstrained cell at the periphery is outside the simulation box (i.e., the cluster spans the system), and if it is, then all growth in that direction is terminated. But even if portions of the periphery have touched the box edge, other segments of periphery may be situated in growing regions of the box, and the simulation proceeds until all growth at the active periphery halts and the cluster is mature. Five quantities were extracted from the computer simulations: the in nite system critical threshold, ac , the values of the exponents and , the fractal dimension, D, of the spanning clusters at the percolation limit, and the ratio =. The critical threshold and the exponents and  were computed directly for large system sizes (up to 41 million cells). These values were obtained using a non-linear curve routine with Eqs. (17) and (21). According to the scaling theory for nite systems [8] these power laws would normally be shifted to an eective percolation threshold, ae . But in these simulations, the system sizes were large enough that no spanning clusters appeared in  the simulation set. That is, suciently below the in nite system threshold, w∞ (L; a) could be made small enough so that no spanning clusters developed for the number of simulations that were made. Thus, the computed eective threshold is actually the estimated in nite system value. Figs. 4 and 5 display the average cluster size and the correlation length, respectively, as a function of a.  From these data we nd = 2:35 ± 0:07 and  = 1:36 ± 0:03. All of the errors quoted in this paper are one-sigma errors. With these errors, the determined values for and  are consistent with the values expected from standard percolation theory ( = 2:39;  = 1:33). The in nite-system percolation threshold was extracted from these same simulations. We found P1c =2 = ac = 0:653 ± 0:001 as compared with ac = 0:665 as determined by the Stauer implementation. We have found that small

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Fig. 4. Plot of the mean cluster size, S(L; a),  as a function of a.  The smooth curve is a nonlinear, three-parameter t of Eq. (17) to these data. The resulting value for is 2:35 ± 0:07 (s0 = 2:2).

Fig. 5. Plot of (L; a)  as a function of a.  The extracted value of  is 1:36±0:03. See Eq. (21). The combined data from Figs. 2 and 3 yields ac = 0:653 ± 0:001 (s0 = 2:2).

algorithm changes can shift ac slightly but do not change the universality class of problem. All of the simulations that required a preset value of the critical threshold used the Levine=Thomson programs with ac = 0:653. The fractal dimension of the spanning clusters at the percolation limit was determined by plotting the mean size of the spanning clusters as a function of L (see Fig. 6) and tting a power law to the data. The resulting exponent was found to be 1:89 ± 0:02 which is the fractal dimension. This result is also consistent with the standard percolation value of 1.895. The ratio = was obtained from D using Eq. (24), giving = = 0:11 ± 0:02. An independent determination of = can be made using Eq. (23) by plotting the w∞ at the critical threshold as a function of L. This plot is shown in

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Fig. 6. Average spanning cluster size as a function of system size, L. The slope of the tted power law gives the fractal dimension for the spanning clusters, D = 2 − = = 1:89 ± 0:02. See Eq. (24) (s0 = 2:2).

Fig. 7. Spanning cluster probability, w∞ , plotted as a function of system size, L. There is a deviation from a straight line for smaller values of L, and the curve is tted only for L¿100. The tted slope is = = 0:12 ± 0:01. See Eq. (23). The deviation at the lower values of L is believed to be a local eect at the origin caused by the strain initiation (s0 = 2:2).

Fig. 7 along with the best- t power law. The slope of this t gives = = 0:12 ± 0:01. Given the size of the one-sigma errors, both values of = are consistent with the standard percolation value of 0.104. Because of the hyperscaling relation, Eq. (25), only two of the three scaling exponents, ; and  are independent, so the four independent exponent measurements provide considerable redundancy to cross check the theory. All of the above results are shown in Table 1, along with the one-sigma errors.

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Table 1 Critical threshold and exponents. One-sigma errors are listed. Calculated values were obtained using combinations of measured parameters and the hyperscaling relation Critical value

Measured

Calculated

Standard

ac

 = D

0:653 ± 0:001 2:35 ± 0:07 1:36 ± 0:03 0:12 ± 0:01 1:89 ± 0:02

— — — 0:11 ± 0:02 1:88 ± 0:01

— 2.39 1.33 0.104 1.895

Fig. 8. Slope of the average non-spanning cluster size, S, plotted as a function of system size, L. See Eq. (22). The smooth curve is a guide to the eye and is not model-based. After an s0 induced dip at small L, the exponent increases monotonically with increasing system size. The exponent is expected to level o at the standard percolation value of 1.79 but the necessary system sizes were larger than those we were able to explore (s0 = 2:2).

We also explored the system size dependence of S which, by Eq. (22), should yield the ratio = for large system sizes. The slope of these data did not approach a constant value for the system sizes available to us. Averaging over large numbers of simulations (400 000 for the two smallest values of L and 150 000 for the remaining L values) produced points on the S(L) curve with maximum percentage errors of 0.7%. Fitting a power law to successive points produced a sequence of powers that should approach the standard value of = = 1:792 for large system sizes. As shown in Fig. 8, the extracted powers display a slight decrease at the smallest system sizes and then increase steadily toward the expected large-system value. Since system sizes much larger than L = 1000 are impractical for us, we were unable to determine a useful value for =. We nd that such curvature in log plots of S(L) are also characteristic of the Leath version of a simple forest re problem, where the probability of occupied sites (burning trees) is speci ed a priori for the entire lattice. So it is not surprising that

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Fig. 9. Slope of the average spanning cluster size as a function of system size, L for s0 = 30. The smooth curve is a guide to the eye and is not model-based. At small L, the system is non-fractal (D=2); the eective exponent drops rapidly to a minimum for intermediate values of L, and then increases monotonically. The exponent is expected to level o at the standard percolation value of 1.895 but the necessary system sizes were larger than those we were able to explore.

no useful data could be obtained in the strain problem, either, for the limited system sizes accessible to us. The S(L) behavior discussed above is also highly dependent upon s0 . S(L) simulations were carried out over a matrix of s0 and a parameters. s0 was given values of 1.9, 2.2, and 5.0 while a was set at values 0.654, 0.653, 0.652, and 0.651. Not surprisingly, it was found that the small-L behavior shown in Fig. 8 depends strongly  upon s0 but only weakly on a. Finally, we explored the role of s0 on the behavior of the fractal dimension of the spanning clusters as a function of L. Once again, we nd that the behavior at small L depends strongly upon s0 . When s0 is much larger than the average strain, hsi ≈ 2, of a strained cell in large spanning cluster, the value of s clearly must decay to hsi. When s0 ≈ hsi, we expect the origin to approximate a typical site, and that is the reason for our choice of s0 = 2:2 in our standard plots: (However, even choosing such a value of strain for s0 does not ensure that the origin is a typical site, since a typical site will have a strain which uctuates about the mean value. So for any particular choice of s0 , the origin is always a special, not a typical site.). Fig. 6 already showed the behavior of D for the case where s0 = 2:2. The behavior was strictly power law down to very small L, and the exponent was consistent with standard percolation. Increasing s0 to 30 (a large value compared with the average value), we once again investigated the L dependence of the eective exponent by tting power laws to successive points in our simulations (see Fig. 9). This plot shows that at the lowest L, D is in the vicinity of 2, where the fractal character disappears. For intermediate values of L, the eective exponent drops below its expected value and then increases for large values of L towards its expected value. Even in Fig. 6,

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Fig. 10. Central region of a representative large spanning cluster, showing that the region near the origin is densely populated relative to the further reaches of the cluster. The initiation value was set to s0 = 100 to accentuate the eect at the origin.

s0 = 2:2, a careful exploration shows that the slope for low values of L is closer to D = 2 than to its expected value of D = 1:895, so the special character of the origin is observed even in this case. The decay of s around the origin for large s0 is more visibly demonstrated in Fig. 10, which shows the region near the origin of a large spanning cluster with s0 = 100. In this gure, the density of sites near the origin is fully dense, giving D = 2 for small L.

6. Conclusions We have developed a sum rule and mean eld approximation to the strain percolation problem, which predicts a percolation threshold at a critical value of a.  As in other critical phenomena, the mean eld prediction has only qualitative resemblance to the actual value observed in numerical simulations, but it does provide a useful physical picture for the percolation model. We have also explored the universality class for strain percolation in Case I, where the wall transmission includes a single term corresponding to simple source activation in the walls. For this case, we nd that the universality class in consistent with that of standard percolation. This conclusion appears to be robust because we have computed four independent exponents, when only two are truly independent. The percolation limit actually found (ac = 0:653) is close, but not exactly equal to the value predicted by the mean eld and sum rule versions, ac = 15 32 . The strain percolation problem is an example of correlated percolation, because the strain transmitted to a site depends on the value of strain in the neighboring site. Since

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the universality class for strain percolation is the same as standard percolation, we believe this is a case short-range correlation in the sense of Weinrib and Halperin [19] and Weinrib [20]. But the signi cant dierences between the standard theory considered by Weinrib and Halperin and the strain percolation problem makes it dicult to apply their work directly to our case. We have commented on the fact that strain percolation involves both a strain distribution and a geometrical distribution of strained sites. Thus, there exists a correlation function associated with the strain distribution, which was developed in the continuum limit for the mean eld version of the theory, and a separate correlation function associated with the purely geometrical aspects of the problem. The geometrical correlation is the correlation discussed in standard percolation theory [8]. The coexistence of these two variables is very reminiscent of the magnetic transition problem, where there are also two variables: the distribution of magnetic spin and the geometric clustering of the spin sites. In the magnetic case, the magnetic phase transition takes place at a dierent temperature from the percolation threshold. That is, the correlation functions for spin and for clustering are not the same. In the magnetic case, special constructions are invented to make the phase transition and the percolation threshold coincide. However, in the strain problem these variables have separate physical signi cance, and no such arti cial constructions would be appropriate. We leave the discussion of the physical consequences of this question for the deformation problem to a later paper.

Acknowledgements D. Stauer would like to thank S. Glotzer for making possible a visit to NIST during 1998, when the work on this paper was begun.

Appendix In this appendix, we consider a related percolation model suggested by our strain model, to which a variation of the Hoshen–Kopelman [24] algorithm can be applied. We present it here as a mathematical curiosity, because it does not correspond to the physical deformation problem. We still use the propagation law given by Eqs. (1) and (2). In the Hoshen–Kopelman algorithm [24], only one line of the square lattice needs to be stored at any one time. The algorithm goes through the system like a typewriter, rst from left to right, and then from top to bottom, to check if there is a cluster of nearest neighbors connecting top and bottom. Applied to our problem, we thus connect each site only with the previously investigated top and left neighbor through the random bond variable a explained in Section 2. Each site gets the maximum of the two s-values produced by the two neighbors. The whole top line gets s = s0 = 10. Via iterative dichotomy we vary the P1 parameter until we nd the threshold at which a spanning cluster rst appears.

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Averaging over up to 1000 dierent random number sequences, we determine the average threshold ac as well as the width
of the threshold distribution [8], which varies for large linear dimensions L of the lattice as L−1= . In this model we found that the average threshold, ac , has a maximum of nearly 0.68 at L ∼ 300 and can be extrapolated to 0:675 for L → ∞. (This value is to be compared to the “Stauer” threshold of 0.665 for the Leath algorithm of Section 5). The width varies roughly according to the random percolation exponent  = 43 . In these simulations, we set s = 0 if the s ¡ 1 as described in Section 2. If we do not introduce this cuto, the results barely change. With the cuto, but on the simple cubic instead of the square lattice, the threshold for P1 is about 1.30. Finally, we explore the eect in Case II, introduced in paper I, of varying the parameter P2 , where a = P1  + P2 exp(−=) and a ≈ P1 =2 + P2 . If we set P1 = 1 and  = 0:1, then we can check how the threshold for P2 depends on the initial strength s0 . We found it to be between 2.3 and 2.4 large s0 ¿ 1 and to increase drastically if s0 decreases further: P2 = 2:6, 4.2, 8.2 for s0 = 12 ; 14 , and 18 , from 100 lattice of size 1000 × 1000. This demonstrates the fact that the parameters P1 , etc., do not depend signi cantly on s0 . References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]

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