Collective behavior and superdislocation motion in L12 alloys

82

Materials Science and Engineering, A164 (1993) 82-92

Collective behavior and superdislocation motion in L 12 alloys D. C. Chrzan and M. J. Mills Sandia National Laboratories, Livermore, CA 94551-0969 (USA) (Received August 30, 1992; in revised form October 22, 1992)

Abstract A simulation of superdislocation motion applicable to L12 intermetallic alloys displaying the yield stress anomaly has been developed. The simulations incorporate a cross-slip-pinning mechanism and allow direct study of the spatial and temporal correlations between pinning events. The predicted dislocation structures agree well with available experimental observations. The relationship between the pinning-point correlations and the mechanical properties of the alloys is explicitly investigated. The superdislocations are shown to undergo a stress-driven, non-equilibrium phase transition. The critical exponents of the transition are shown to be related to strain hardening. Theoretical predictions for the exponents are in excellent agreement with the algebraic form of primary creep transients for NiaA1 within the temperature range of the flow strength anomaly. The importance of treating dislocation-level phenomena on an equal footing with detailed microscopic mechanisms is stressed.

I. Introduction

It has been claimed that the observed anomalous increase in the yield strength with temperature observed for many L12 intermetallic alloys is directly related to the thermally activated cross slip of screw superpartial dislocations from the easy glide {111} planes to the {001} planes [1]. This claim has spawned numerous efforts to understand the precise nature of the cross-slip process as well as to determine how the details of the cross-slip process influence the mechanical properties of the alloys (for a recent review see ref. 2). Most of these efforts have stressed the microscopic understanding of a cross-slip event, the formation of a pinning "point" [3] or segment [4], and spend comparably less effort incorporating these events into dislocation-level models. The dislocation-level models typically assume steady state conditions and a very regular spatial array of pinning points. It is shown below that this assumed array differs significantly from the arrangement of pinning points that arises naturally. Furthermore, it is demonstrated that the correlations in the formation of the pinning points, their so-called collective behavior, are the origin of the experimentally observed algebraic primary creep transients. The purpose of this work, then, is to eliminate the assumptions made in prior dislocation-level modeling efforts in order that the true collective behavior of the pinning points can be explored. Phase transitions represent the most striking example of the importance of collective behavior. As an example of this, one may 0921-5093/93/$6.00

consider a ferromagnet. The macroscopic physical properties (most notably the spontaneous magnetization) above and below the ferromagnetic-paramagnetic transition temperature are substantially different, even though the physical properties of the individual spins remain unchanged through the transition. The change in macroscopic properties arises from the collective behavior of the spins. Superconductivity is another example of a macroscopic property (e.g. the Meissner effect, zero resistivity) being determined by the collective behavior of microscopic elements, in this case the electrons. In a similar fashion, it is possible that some of the macroscopic physical properties of L12 alloys derive from the collective behavior of the cross-slip-pinning points. This notion is confirmed below, where it is demonstrated that the superdislocations in the L12 intermetallics undergo a non-equilibrium phase transition, and that the simple algebraic form of the primary creep transient stems from the scale invariance of the transition point. The remainder of this paper is organized as follows. Section 2 presents a model for dislocation motion that imposes no artificial correlations between pinning events. The proposed model is contrasted with the work of Paidar et al. [3] (henceforth PPV). Section 3 contains a review of select results obtained from the proposed model, with particular emphasis on the dynamical features. In Section 4 it is argued that the superdislocations undergo a non-equilibrium phase transition and the critical exponents of the transition © 1993 - Elsevier Sequoia. All rights reserved

D. C. Chrzan, M. J. Mills

are directly related to the primary creep transient. In Section 5, a very simple model for the collective behavior of the pinning points is described and used to calculate the critical exponents of the transition, thereby predicting the algebraic form of the primary creep transient. A discussion and the conclusions form Section 6.

2. The model As stated above, it has been proposed that the yieldstrength anomaly is related to the thermally activated cross slip of screw superdislocations from {111 }planes, where the superdislocations are highly mobile, to {001} planes. A careful, detailed analysis of this process has been carried out by PPV [3]. They argue that the crossslip-pinning events have the following properties. (1) The activation enthalpy for cross slip is estimated to be 0.3 eV. (2) In order for a segment of cross-slipped superdislocation to be stable, it must be at least 5 A long. In a continuum description of superdislocation motion, this minimum length translates into a minimum degree of screw character in order for cross slip to occur and an increasing probability of cross slip for increasing screw character (3). The presence of a crossslip-pinning point causes the adjacent segments remaining on the ( 111 ) plane to bow about the pinning point. As the segments bow, they exert a force on the pinning point. Eventually, this force reaches a critical value, represented by a critical bowing angle (Fig. 1 ), at which point the cross-slipped segment dissolves. The description of a pinning point was incorporated by PPV into a dislocation-level model developed by Takeuchi and Kuramoto [5]. In this model, the pinning points form in a regular array in predetermined positions and at predetermined times. This configuration of pinning points allows direct calculation of the yield strength based entirely on the microscopic nature of an individual pinning event and knowledge of how a free segment of superdislocation with a specified length moves under an applied stress. The yield strength, so calculated, agrees well with experiment, and further refinements in the description of the pinning points have led to even better agreement with experiment [6]. The model described below differs in two very important respects from that proposed by PPV. First, the model imposes no artificial correlations on the formation of the pinning points. The points are allowed to form subject only to physical constraints arising from a microscopic understanding of the cross-slip event. The formation of the pinning points also allows for the randomness inherent in the pinning process. The second difference is that an equation of motion for the free segments is explicitly included in the model.

/

Superdislocation in L I: alloys

83

This inclusion is necessitated by the fact that the pinning points can now create free segments of varying length which evolve differently under the applied stress. The model is constructed as follows. The superdislocation is represented as an array of points. At each time step of the simulation, three distinct procedures are performed. First, each point is assigned a pinning probability, based on the local screw character of the superdislocation and the temperature, as dictated by PPV's description of a pinning point. The pinning probability increases with both temperature and superdislocation screw character. A random number distributed uniformly between 0 and 1 is generated for each of the points. If the random number is less than the pinning probability, that point is said to be pinned and its velocity is fixed to be zero until dissolution of the pinning point. Second, each pinned point is checked for dissolution by calculating the bowing angle (Fig. 1) of the adjacent free segments. If the bowing angle of one of the adjacent segments exceeds the critical value (chosen to be 30°), the pinning point dissolves, and that point of the superdislocation is free to advance according to the equation of motion. The third procedure is the integration of the equation of motion for all of the free points. The equation of motion is that studed by Frost and Ashby [7]:

v=( zab-ATc)/B

(1)

where v is the superdislocation velocity normal to the superdislocation segment, ra is the applied Shear stress, b is the Burgers vector of the superdislocation, A is the superdislocation line-tension (assumed isotropic), r is the local circular curvature of the superdislocation segment, and B is the drag. The values for the param-

pinning point Fig. l. A schematic representation of a pinning "point". The thick line-segment represents a cross-slip-pinning "point" as envisioned by Paidar et al. [3]. The presence of the pinned segment forces the adjacent superdislocation segments to bow under the applied stress. As the segments bow, they exert a force on the pinned segment. The magnitude of the force exerted by a single segment increases with the bowing angle--the angle the adjacent segment makes relative to the screw orientation. Eventually, this force reaches a critical value at which point the segment unpins. This critical force corresponds to a critical bowing angle, indicated by 0 c in the figure.

84

D.C. Chrzan, M. J. Mills

Superdislocation in L12 alloys

TABLE 1. Simulation parameters

b

1 ktm

Parameter

Symbol

Value

Magnitude of the Burgers vector Shear modulus Superdislocation line tension Drag

b

5.0 × 1O- "~m

/a A B

5.0 x 10 j° Pa pb 2= 1.25 x 10- u) N 2.0 x 10 -3 kg m - l s- l

It

c

eters entering the theory are displayed in Table 1. The value of B is difficult to determine from experiment. It has been chosen to give a superdislocation a free flight velocity of 100 m s-] under an applied stress of 400 MPa, independent of temperature. Complete details of this model can be found in ref. 8. The initial superdislocation configuration was chosen to be a circular loop 1 p m in diameter. This loop was allowed to expand under the applied stress until it reached a predetermined size. At that time, the edge-character ends of the loop were eliminated and the remaining mostly screw-character superdislocation segments were allowed to advance further. The physical motivation for this procedure is the following. As the loops expand, the edge-character portions advance more rapidly than the screw-character portions. Eventually, the edge portions of the loops move completely out of the crystal leaving the slower moving, mostly screw-character portions of the loop to accommodate further strain.

) ) ) })

lll

(a)

(b)

/

)

(c) Fig. 2. Simulation of the expansion of dislocation loops for a temperature of 300 °C and stresses of (a) 175 MPa, (b) 240 MPa, and (c) 350 MPa. Configurations after uniform time steps of 3 x 10 -7 s, 2 x 10 -7 s and 1.5 x 10 -7 s respectively, are shown. The pinning points are represented by blackened points. The structural differences between the superdisiocations advancing under an applied stress of 240 MPa and 350 MPa are indicative of a change in the collective behavior of the pinning points.

S

3. Results

Figure 2 contains typical results for loops expanding under an applied stress of (a) 175 MPa, (b) 240 MPa and (c) 350 MPa at a temperature of 300 °C. The nested loops are "snapshots" taken at constant time intervals. At 175 MPa, the pinning along the mostly screw-character segments is so rapid and dense that the edge segments of the loop actually become shorter as the loop expands. Eventually, the edge-segments are no longer able to overcome the bowing stress, and the loop ceases to expand. At 240 MPa (Fig. 2(b)), the expanding loops appear substantially different from their lower stress counterparts. The loops still contain long segments of highly pinned, near-screw-character superdislocation, but in addition, there is a substantial number of long unpinned segments of mixed character, connecting the highly pinned segments. The nearscrew-character portions of the superdislocation are now able to advance under the applied stress, through the lateral motion of these mixed segments, which will be referred to as "superkinks". Lateral superkink motion occurs rapidly, although not as rapidly as motion of the

/ KW

S /

KW

S KW

KW

1 ixm

Fig. 3. An illustration of the structure of post-deformation superdislocations for crystals deformed in the anomalous yield strength regime. The segments labeled KW represent the Kear-Wilsdorf locks and the segments labeled S are the superkinks.

pure edge ends of the loops. At 350 MPa (Fig. 2(c)), the screw portions of the loop advance at a rate comparable with that of the edge portions, resulting in the oval shape. In addition, while there is still a significant amount of pinning along the screw portions, it is less dense and does not lead to the formation of superkinks. Figure 3 illustrates a typical superdislocation configuration observed (using transmission electron microscopy (TEM)) after deformation at lower temperatures in the anomalous flow strength regime. Note that extended regions of superdislocation have cross

D. C. Chrzan, M. J. Mills

/

slipped to form the so-called "Kear-Wilsdorf locks" [1]. Mixed segments lying on the primary {111} slip plane link these locks. These mixed segments, which have been designated as "superkinks" in the literature [4, 9-11], are remarkably similar to the mixed segments present in Fig. 2(b). Another striking feature of Fig. 2(b) is the presence of long highly pinned regions aligned nearly along the screw direction. Although similar in appearance, these structures are not precisely the same as the Kear-Wilsdorf locks observed in experiment. The similarity suggests, however, that this is the stress regime of relevance to experiments. (Figure 4 compares the structure of the PPV pinning point with the structure of the Kear-Wilsdorf locks. The PPV pinning point is only partially dissociated on the cube plane whereas the Kear-Wilsdorf lock is completely dissociated.) In addition, the highly pinned regions influence the motion of the simulated superkinks in a fashion very similar to how the Kear-Wilsdorf locks must affect the motion of the real superkinks. The pinned regions are thus the dynamical equivalent of Kear-Wilsdorf locks. The correlations that develop between pinning events are intrinsic to the motion of the superdislocations and are discussed in detail below. Couret et al. [12] measured an exponential distribution of superkink heights in Ni3Ga. A related quantity easily accessible from the simulations is the distribution of superkink lengths. Let p ( 1 ) d l be the probability that a superkink's length lies between l and l+dl. Figure 5 shows a plot of p(l) as a function of 1 calculated from simulations conducted at an applied stress of 220 MPa and a temperature of 300 °C. The small l portion of this distribution is accurately described by

Superdislocation in LI : alloys

85

an exponential, which is consistent with the experiments of Couret et al. The distribution, however, shows a second regime at longer lengths. The transition between the two regimes occurs very near to the critical length at which a segment oriented along a perfect screw direction is able to overcome its pinning points and be mobile. In essence, the number density of the mobile kinks as a function of length differs in functional form from the number density of immobile kinks. The majority of superkinks belong to the immobile branch. This suggests that the observations of Couret et al. apply to the distribution of immobile superkinks, particularly because these superkinks are shorter and more likely to be completely contained within a T E M foil. Perhaps the most important feature of the distribution of superkinks is that the average superkink is immobile. In order to understand the motion of the superdislocations in these materials it is therefore necessary to study distinctly non-typical superkinks. The simulations also allow the complete study of the temperature and stress dependence of the distribution of superkink lengths. The dynamics of superdislocation motion, however, are more relevant to the current work and the remainder of the paper is devoted to the study of these dynamics. The most prominent feature of the superdislocation motion is that the mostly screw oriented superdislocations advance normal to the Burgers vector through the lateral motion of superkinks. This mode of motion is in marked contrast to that expected from a dislocationlevel model in which the pinning points are uniformly spaced. 10 °

SP 2

SP 1

A_A_

A-A-

10 -j i

10 .2 l

i0.3 1o]

10 .4

i

O00000DOQO0000

i

1 0 .5 (b)

Sp 1

Fig. 4. Panel (a) is a schematic view of the PPV pinning "point". Panel (b) depicts the experimentally observed Kear-Wilsdorf lock. Note that the superdislocation in the Kear-Wilsdorf lock configuration is completely dissociated on the (010) plane whereas for the PPV pinning "point" the superdislocation is only partially dissociated on the (010) plane. It is still an outstanding question as to how the PPV pinning point becomes a Kear-Wilsdorf lock. The simulations, however, suggest that the collective behavior of the pinning points allows more time for the conversion process than that expected from the PPV model of superdislocation motion. See ref. 8 for details.

OOtO0000

| i|

O0 OQ

1 0 .6

50O

lOSO0 I nm

15~00

20~00

Fig. 5. A plot of p(l) as a function of / calculated under the conditions of an applied stress of 220 MPa and at a temperature of 300 °C. The dashed line represents an exponential distribution, p(l)-exp(-l/lo), with l0 equal to 13 nm. The shorter superkinks, distributed according to this exponential, are immobile. The simulation results show a significant deviation from the simple exponential form at large lengths. These larger length, mobile superkinks are responsible for superdislocation motion.

D. C. Chrzan, M. J. Mills

86

/

A second prominent feature of the simulated superdislocations is the highly pinned regions resembling Kear-Wilsdorf locks. The dynamic origin of these structures is described in Fig. 6. As mentioned above, the distribution of superkinks contains both mobile and immobile kinks. Immobile superkinks are created when two pinning points form very near each other, and the stress required to bow the short segment connecting them beyond the critical angle exceeds the applied stress. It follows that there is only a small region of space near an existing pinning point in which a new pinning point can form and create an immobile segment. This region of space is indicated (schematically) by the shaded region in Fig. 6. The size of the shaded region is larger for lower stresses and, in addition, at lower stresses the free superdislocation segment spends more time in the shaded region. Hence, there is a substantial probability of forming an immobile superdislocation segment. This results in the highly pinned regions of superdislocation resembling Kear-Wilsdorf locks. As the stress is increased, two effects become important: (1) the size of the shaded region shrinks and (2) the increased velocity of the free segment implies that it spends disproportionately less time in the smaller shaded region. Thus at high applied stresses, the probability of an adjacent segment becoming pinned decreases, and the highly pinned regions of superdislocation no longer form--the correlations between the pinning events are stress dependent. The correlations between pinning events also lead directly to the formation of superkinks. Figure 7 illustrates the dynamic process by which the superkinks are created. At intermediate stresses, the highly pinned regions assume an overall slope, as indicated in the figure. As the superkink moves laterally across the

Superdislocation in L12 alloys

crystal, it creates highly pinned regions connected by immobile superkinks. Occasionally, a moving superkink will spawn a mobile superkink traveling in the opposite direction. This newly created superkink is likely to travel a significant distance because of the average slope of the highly pinned structure created by the passage of the initial superkink. As this newly spawned superkink advances, it creates a new highly pinned structure and the process continues. The above description highlights two important features. First, it establishes that the existence of superkinks is direct evidence of the collective behavior of the pinning points. Second, it demonstrates the randomness inherent in the creation of superkinks. Had the initial superkink in Fig. 7 made it to the edge of the crystal before generating a superkink moving in the opposite direction, the entire superdislocation would have become immobile. Thus as a result of a fluctuation, the superdislocation motion can become completely exhausted. The simulations reveal that in the intermediate stress regime (the regime of Fig. 2(b)), exhaustion of superdislocation motion is the rule rather than the exception. This result, coupled with observation of post-deformation substructures, strongly suggests that exhaustion of superdislocation motion is the major source of strain hardening in these alloys [13, 14]. The primary creep transient at lower temperatures provides clear evidence of this strain hardening [15] and is the experimental condition most amenable to modeling, as is described in the next section. Based on the simulations described above and previously [8], the following qualitative features of the exhaustion process are noted: (1) simulated superdis2

bowing superdislocation

II ~

.

.

.

.

.

~ _

~

spawned mobile superkink

_

pinning points

~ b

immobilizing region Fig. 6. The dynamic origins of the highly pinned regions of Fig. 2(b). The region of space in which a pinning point can form and result in an immobile superdislocation segment is indicated by the shaded region. As low stresses, the superdislocation spends a significant time in the pinning region and the highly pinned structure arises naturally. As the stress is increased, the size of the gray region decreases and, concomitantly, the superdislocation moves more rapidly. Both effects reduce the probability of forming an immobile segment leading to the less correlated pinning of Fig. 2(c).

Fig. 7. The dynamic process by which superkinks are created. As the superkink labeled 1 in the lower panel advances to the left, it produces the highly pinned region indicated in the figure. Occasionally, a pinning point capable of creating an immobile segment fails to form and a mobile superkink is spawned. This newly spawned mobile superkink will then move to the right in the figure and, because of the overall slope in the highly pinned region, it will increase in length. The structure of the highly pinned regions thus ensures that the newly spawned superkinks will travel for a significant distance before their motion is exhausted.

D. C. Chrzan, M. J. Mills

/

locations advancing under a high applied stress (e.g. 350 MPa at 300 °C, Fig. 2(c)) have not been observed to exhaust; (2) the randomness inherent in the formation of pinning points implies that the exhaustion times and areas swept out vary from superdislocation to superdislocation; (3) the transition between the regime in which exhaustion is observed and that in which it is not occurs over a fairly narrow range (approximately 30 MPa) of stresses. These observations are indicative of a significant change in superdislocation motion as the stress is increased. It is argued below that the change is a direct result of a non-equilibrium phase transition in the motion of the superdislocations. The proposed transition is from a "phase" in which exhaustion is prevhlent and superkinks are found to a "phase" in which the motion of the superdislocations does not appear to exhaust and superkinks are not created. The transition is driven by the applied stress. An alternative view of the two non-equilibrium phases is that the low-stress phase, the highly correlated phase, is one in which the collective behavior of the pinning points determines the nature of superdislocation motion, and the highstress phase is one in which the pinning points do not display strong collective behavior. The implications of this transition are discussed more thoroughly below.

4. Non-equilibrium phase transition and primary creep In this section of the paper, the arguments for the existence of a non-equilibrium phase transition in the superdislocations in the L12 intermetallics are made more formal. As a starting point, a simplified description of the proposed transition is presented. The simplified description is followed by a more detailed analysis. The analysis of creep is expected to lend insight into other mechanical tests, including stressrelaxation experiments and constant strain-rate tests. 4.1. S i m p l i f i e d description o f the transition

The simulations described above demonstrate that in the regime of interest (intermediate stress), the segments responsible for dislocation motion are the mobile superkinks. It is possible, then, to formulate a rough theory based on their properties. Let the number of mobile superkinks in the alloy be given by N(t). The simulations show that there are three processes which can change the number of mobile superkinks (under constant stress conditions). (1) A mobile superkink can spawn another mobile superkink. (2) A mobile superkink can become immobile because it is moving "uphill" relative to the highly pinned structure. (A

Superdislocation in L1 z alloys

87

superkink spawned by the superkink labeled 2 in Fig. 7 would necessarily move from right to left in the figure. When this newly spawned superkink reaches the point in the highly pinned structure where the average slope of the pinned structure changes sign, the length of the advancing superkink begins to decrease. If this "uphill" motion continues, the superkink will reach the length below the length required to assure further motion.) (3) Superkink-superkink collisions result in the annihilation of a mobile superkink. These three processes can be incorporated into a phenomenological rate equation describing the number of mobile superkinks as a function of time. The rates of processes (1) and (2) are independent of the number of superkinks. The rate of process (3) is proportional to the number of mobile superkinks. (The rate of superkink-superkink collisions scales inversely with the average superkinksuperkink separation. The average separation is inversely proportional to the density of superkinks along a single superdislocation.) The rate equation then becomes dN dt

--=

[i~ - ~oN( t)]N( t)

(2)

where the term in brackets on the right represents the net rate at which mobile superkinks are created or annihilated. The combined rate of processes (1) and (2) is designated/a, and can be either positive or negative. The rate of process (3) is proportional to ~o which must be positive, as a superkink-superkink collision cannot increase the number of mobile superkinks. This equation is similar to that proposed and studied by Li [16] for the description of pure metals. In that situation, however,/~ is always assumed positive and represents dislocation multiplication. Consider a creep test performed on a prestrained sample initially containing N o superkinks (prestraining creates the initial population of superkinks). If/a < 0, the number of mobile superkinks ultimately goes to zero, and the sample stops creeping. If ,u >0, the number of mobile superkinks approaches the fixed value/t/~o. The creep rate, assumed to be proportional to the number of mobile superkinks, becomes independent of time and yields steady-state creep. These two different regimes are the manifestation of the nonequilibrium phase transition (for a discussion of a similar transition in a system of chemical reactions, see ref. 17). For low stresses/~ <0, and at high stresses/~ > 0. The stress for which/a = 0 is the critical stress of the transition. The number of mobile superkinks as a function of time becomes N(t) -

1

1

1/N, + ~ot

~ot

(3)

D.C. Chrzan, M. J. Mills / Superdislocation in L12 alloys

88

where the final form on the right assumes N o -, oo. At the transition, the average creep rate, again assumed proportional to the number of mobile superkinks, reduces to a simple power law and results in logarithmic creep. Although the above description is oversimplified (e.g. the description neglects the fact that mobile superkinks have varying lengths, etc.), it does establish several salient features of the proposed transition. Below the critical stress, all superdislocations eventually exhaust. Above the critical stress, some finite number of superdislocations remains mobile. At the critical stress, the creep-rate becomes proportional to a simple power-law of the time, in this case 1/t. These features are also apparent in the more complete analysis presented below.

Based on the arguments above, it is assumed that creep deformation is controlled by the motion of finite lengths of predominantly screw superdislocations (the highly mobile edge portions of the loops have rapidly exited the crystal). The crystal continues to creep as long as the predominantly screw superdislocations continue to move, enabling the dislocation sources to remain active. However, at intermediate stresses, exhaustion of the motion of all the superdislocations eventually impedes additional superdislocation generation from the sources, and the crystal ceases to creep. In this view, the form of the creep curve is determined by the exhaustion spectrum for the screw dislocations. Let n(a, t) da dt be the number of superdislocations which sweep out an area between a and a + da before exhausting at a time between t and t+dt. Since the strain at time t is proportional to the area swept out by the superdislocations between time 0 and t, n( a, t) can be used to determine the strain as a function of time ~(t)--the primary creep transient. The total strain at time t arises from two contributions: (1) the superdislocations which exhaust before and at time t, and (2) the superkinks destined to exhaust at a time greater than t. By considering both contributions, one arrives at: dy(t) dt

co

f dt'/t' f dan(a,t') t

).n(a,t) = n().~a, 2~t)

(4)

0

An interesting feature of systems displaying secondorder thermodynamic phase transitions is that at the transition point, the so-called critical point, they display spatial scale invariance [18]. Scale invariance is typically associated with the appearance of simple power-laws. In the simplified description above, an important feature of N(t) at the critical point is that its time dependence reduces to a simple power-law. It is

(5)

where a and 6 are the scaling exponents. It can be easily shown that eqns. (4) and (5) imply that the creeprate in these materials should have a simple power-law time dependence: dy(t) dt

4.2. Detailed analysis of the transition

oo

therefore conjectured that at the transition point, n(a, t) becomes scale invariant, but because this is a nonequilibrium phase transition, the distribution is both spatially and temporally scale invariant. Practically, this means that there is no intrinsic time- or area-scale in the distribution of exhaustion events; there is no "typical" exhaustion time nor a "typical" area swept out. Mathematically, the scale invariance is written

t_ a

A=-(l+2a)/6

(6)

where A is defined to be the creep-rate exponent. The experiments of Thornton et al. [15] performed in 1970, directly measure a power-law dependence on time for the creep rate within the anomalous flow strength regime, and hence determine the critical exponent A. The relevance of these experiments to the simulations is evident from the work of Hemker et al. [19] which showed that primary creep in Ni3A1 at intermediate (and presumably lower) temperatures is due to the motion of superdislocations on {111 }planes. Furthermore, these superdislocations are typically observed in the "stepped" structure similar to that arising in the simulations. To summarize, the superdislocations are described as undergoing a non-equilibrium phase transition from a phase in which the motion of all superdislocations is mediated by the lateral motion of superkinks and is eventually exhausted, to a phase in which superkinks are not created and a finite number of superdislocations continue to propagate for extended times. The transition is suggested to be akin to a second-order thermodynamic phase transition in that the system passes through a critical point displaying scale invariance. The scale invariance of the critical point is reflected in the distribution of exhaustion events, n(a, t), and hence is directly related to the observation of power-law time dependence for the creep rate. It is further argued that the creep-rate exponent A is a critical exponent of the transition. The simulations provide ample evidence that the above description is directly applicable to the L12 intermetallics. The differing superdislocation structures at high and low stesses are indicative of the transition. In addition, there are several features common to second-order thermodynamic phase transitions which are also present in these simulation results.

D. C Chrzan, M. J. Mills

/

Consider the quantity fi(t) defined as fi(t) = ~ da n(a, t)

(7)

0

Physically, fi(t) is the total number of superdislocations which are mobile for a time t before they are immobilized. Figure 8 is a plot of h(t) as a function of time for an applied stress of 410 MPa at a temperature of 500 °C. A pronounced feature of this plot is the exponential dependence of fi(t): ~(t) - e x p ( - t/to)

t ~ oo

(8)

Careful examination of Fig. 8 also reveals that at the shortest times, there is a deviation from the exponential form. Although a single point here, calculations with finer time increments indicate that this is indeed a significant deviation. A feature of second-order thermodynamic phase transitions is that the correlation functions, near the transition point, have the OrnsteinZernike form [18]. The equivalent statement for the non-equilibrium transition proposed here is that

h( t) - t-~ e x p ( - t/to)

(9)

In this form, the quantity to is the equivalent of the thermodynamic correlation length, and so it is hypothesized that at the transition, to diverges according to a simple power-law dependence of the stress

to(r ) - I r c - rl -w

(10)

where r is the applied stress, rc is the critical stress, and ~p is one of the critical exponents of the transition. Figure 9 is a plot of the stress dependence of to, and the corresponding fit to eqn. (10). The critical stress r c obtained from the fit is 450 MPa, and the value of ~p so

Superdislocation in L I 2 alloys

89

obtained is 1.8. The quoted values of the parameters r c and ~p are not to be taken too seriously, as the statistics required to place strict error bars on their values are not available using these simulations. (Reliable values of ~ are even more difficult to obtain, so ~ was not calculated.) The divergence of to, however, is readily apparent. The major point of this discussion is that the simulations provide substantial evidence for the proposed non-equilibrium phase transition. The similarity of the proposed transition to a thermodynamic phase transition is exploited below, where the algebraic form of the primary creep transient is predicted from a simplified version of the simulations.

5. Simplified superdislocation model A remarkable feature of critical exponents is that they are universal--they do not depend on the precise microscopic details of a process, but rather arise from symmetries inherent in the system [20]. If the creeprate exponent A is truly a critical exponent, it can be determined by a simple model possessing the correct symmetry. This determination, then, constitutes prediction of the algebraic form of the primary creep transient. Ideally, a model with the proper symmetries should be derived from a microscopic description of the important processes. In this case, one needs a thorough understanding of the properties of the superkinks. This understanding has been obtained through the study of simulated superdislocations. The following simplified model has been constructed to reflect the features

0.8

-

0.6

-

0.4

-

7 o

6-

5~ 4 0.22

oo

o

1o

o

0,0O

o

i

....

0.0

i ....

J

0.1

. ,

i

0.2

380

w i

I

I

I

I

390

400

410

420

430

440

r "

0,3

time [106sec] Fig. 8. The distribution of superdislocation exhaustion times as defined by eqn. (7) of the text calculated for a superdislocation advancing under an applied stress of 410 MPa at a temperature of 500 °C. The most prominent feature of this plot is the exponential dependence of ri(t) for long times (eqn. (8) of the text).

stress [MPa] Fig. 9. A plot of the characteristic exhaustion time t0 as a function of the applied stress for simulations of superdislocations at 500 °C. The solid line is a fit to the form of eqn. (10) with • ¢ = 4 5 0 MPa and V = 1.8. The quoted values of z¢ and V do not reflect sufficient statistical accuracy and are quoted for illustration only.

D. C Chrzan, M. J. Mills

90

/ Superdislocation in L12 alloys

believed to be important to superkink motion, but also to enable the accumulation of adequate statistics for determining the critical exponents. The superdislocation is represented by a sequence of L segments, labeled by the index i. The position of segment i is designated by an integer, y~. At each step of the simulation, each segment is assigned an advancement probability p~. L random numbers are generated and compared with the p~. If the random number is less than Pi, segment i is advanced by one unit, Yi-"Y~+I. The physics of superkink motion is contained in the specification of the advancement probabilities {Pi}. In the model, the advancement probability of site i is restricted to depend only on the positions of the segments at site i - 1 , i, and i + 1. The local three-point configurations which enter the model are depicted in Fig. 10. The black dots represent superdislocation segments, not pinning points. The simulations described above indicate that a superkink does not advance if it does not have sufficient edge character. The edge character of a segment is determined by its position relative to its neighboring segments. A segment is therefore defined to be of insufficient edge character, and consequently immobile, if both lYi-z-Yil and lYi+~-Yi[ are less than two. For this reason, configurations lO(a)-(f) are immobile and have pi=O. The configurations labeled (g) are mobile because they have sufficient edge character, both lYi-1-Yil and lYi+l-Yil are equal to two. Dissolution of a pinning point occurs when the force exerted on it

:

:

:

(a)

(k)~ (c) (d)

(e)

7/ slope = 2

s,o:X Fig. 10. The three-point configurations which enter the simple model. The black points represent superdislocation segments, not pinning points. The rules governing the motion of these segments under the applied stress are outlined in the text.

exceeds a critical value. The force exerted on a pinning point by a superkink is proportional to the bowing angle, which depends on the position of the adjacent segments. The forces acting on segment i are given by (Yi-1 -Yi) and (Ys+l -Yi). If either of these quantities is equal to two, the pinning point dissolves and segment i advances. The advancement probabilities of configurations 10(h)-(k) are therefore set equal to unity. The configurations 10(1)-(o) represent the upper edge of the superkinks. Their advancement probabilities are chosen to depend on two parameters, O and r. O is a parameter which reflects the temperature dependence of the pinning frequency. An increase in O is equivalent to an increase in temperature, r is the parameter representing the applied stress in the simple model. (Both r and ® are assumed to have dimensions of [length] - l, so that Pi defined by eqn. (11 ) is dimensionless.) The advancement probabilities are given by p, = max[0,1 - O / ( r - r,)]

(11)

where ~csis the circular curvature at site i, calculated by replacing appropriate derivatives in its definition with their discrete counterparts based on the positions of the neighboring segments. The precise form for the advancement probability stems from the fact that the amount of time spent in the pinning region, and hence the total pinning probability, scales inversely with the segment velocity. The segment velocity is, in turn, proportional to the difference between the applied stress and the bowing stress. The distribution of exhaustion events, n(a,t), is calculated as follows. An initial configuration is generated based on the configurations 10(a)-(f). A segment is chosen at random and advanced by one unit. The procedure outlined above is executed until the superdislocation ceases to advance. The area swept out and the total exhaustion time are then recorded, and the process is repeated starting from the exhausted configuration. Given the distribution n(a, t), it is possible to determine A. Figure 11 is a plot of A as a function of r as determined from the simple model for L = 100 and O = 50. The inset of Fig. 11 shows the (reanalyzed) data of Thornton et al. [15]. Note that both the theory and the experiment have A ~ 1 for a range of lower stresses, implying logarithmic creep. At the critical stress, which is estimated to be r c = 100, A = 0.56, which is consistent with experimental observations. Thornton et aL attribute the variation of A to a temperature dependence. The theory presented here suggests that A is determined through a balance of the stress-induced segment advancement and the temperature-induced pinning. The implication is that variations in A with stress should be observed for creep experiments run at constant temperature.

D. C. Chrzan, M. J. Mills

/

A

1.0

A

A

AA A

A A

A A

0.9

A A

0.8

1.0 -4111

Ill

0.9 -4

0.8 -4

0.7

A

0.7 -I

OO

0.6 -4

A

0.5 q

0.6

• 21)0

A

A A

400

A A~

stress [MPal

0.5

61)

7t)

80

91)

100

Fig. 11. A plot of the creep-rate exponent A as a function of the applied stress calculated within the simple model with L = 100 and O = 50. (Note that the units of r are chosen to make the probabilities in eqn. (11) dimensionless.) The inset shows the data of Thornton et al. [15] reanalyzed within the context of the present theory. The filled squares are data taken at 25 °C, the filled circles are data taken at 399 °C, and the filled triangles are data taken at 626 °C, whereas the theoretical results are taken from simulations performed at a constant "temperature". Thornton et al. attributed the variation of A to a dependence on temperature whereas the present work suggests that the value A is determined by the balance of the applied-stress-induced motion (represented by r) of the superdislocations and the temperature-induced formation of pinning points (represented byO).

6. Discussion and conclusions The simulations described above have provided significant insight into the nature of superdislocation motion in L 12 intermetallic alloys displaying the yield stress anomaly. It has been argued, based on the results from the simulations, that proper treatment of the collective behavior of cross-slip-pinning "points" provides a natural explanation for many of the observed experimental properties. The simulations produce dislocation structures that are similar to those observed using TEM. The distribution of superkink lengths is in excellent agreement with experiments performed on Ni3Ga [12]. It is also established that the typical superkink is immobile, and that a proper understanding of superdislocation motion requires an understanding of distinctly non-typical superkinks. More importantly, the simulations allow the direct study of the dynamics of superdislocation motion. The

Superdislocation in L I : alloys

91

experimentally observed superkinks arise naturally from correlations in the formation of the pinning points. Without the proper treatment of the correlations, the origins of the superkinks and the distribution of their lengths could not be understood. Thus the microstructure of the superdislocations is determined by the collective behavior of the pinning points. An even more profound observation is that in the regime in which superkinks are present, the major source of strain hardening is the exhaustion of superdislocation motion. A particular superdislocation is found to exhaust when it no longer contains any mobile superkinks. In the stress regime of interest, exhaustion is the rule rather than the exception. Characterization of the exhaustion process is thus warranted. Careful analysis of the properties of the simulated superdislocations at high and low stresses indicates substantial differences. Most notably, at high stresses, the superkink structure does not develop. In addition, superdislocations advancing at high stresses are never observed to exhaust. In the stress regime in which exhaustion is prevalent, the randomness inherent in the formation of the pinning points ensures that different superdislocations advance for different times before exhausting. The distribution of exhaustion times contains a characteristic time t0 which diverges as the stress increases. These observations suggest that the superdislocations undergo a non-equilibrium phase transition. The ramifications of the proposed non-equilibrium phase transition have been explored. The critical exponents of the transition are related to the algebraic form of the primary creep transient. Since the critical exponents are supposed to be universal, their values have been deduced through study of a simplified model. The simple model, constructed to mimic as many properties of the simulated superdislocations as possible, but still remain computationally tractable, displays the proposed transition. The critical exponents of the exhaustion spectra obtained from the model provide excellent agreement with those deduced from measurement of the primary creep transient. This agreement with experiment provides strong evidence for the existence of the proposed non-equilibrium phase transition. Finally, it is noted that dislocation exhaustion and strain hardening are also very important characteristics of deformation under constant strain rate conditions, and presumably, during stress relaxation tests. Modeling these conditions is inherently more difficult than the modeling of creep experiments. However, the success achieved in describing the primary creep transients suggests that other flow behaviors in these alloys might be treated in a similar manner.

92

D. C. Chrzan, M. J. Mills

/

In conclusion, the proper treatment of the collective behavior of the cross-slip-pinning events leads to many predictions concerning both the microstructure and the mechanical properties of the L12 intermetallic alloys displaying the yield strength anomaly which have been experimentally verified. Perhaps the most striking success is that the proper understanding of the collective behavior of the cross-slip-pinning events leads directly to the prediction of the algebraic form of the primary creep transient. This success highlights the importance of treating collective behavior of the pinning events on equal footing with the microscopic description of the pinning events.

Acknowledgment This research was supported by the US Department of Energy, Office of Basic Energy Sciences, Division of Materials Sciences under Contract No. DE-AC0476DP00789.

References 1 B. H. Kear and H. G. E Wilsdorf, Trans. TMS-AIME, 224 (1962)382.

Superdislocation in L12 alloys

2 D.M. Dimiduk, J. Phys. Ili, 1 (1991) 1025. 3 V. Paidar, D. P. Pope and V. Vitek, Acta Metall., 32 (1984) 435. 4 P.B. Hirsch, Philos. Mag. A, 65(1992) 569. 5 S. Takeuchi and E. Kuramoto, Acta Metall., 21 (1973) 415. 6 V. Vitek and Y. Sodani, Scripta Metall. Mater., 25 (1991) 939. 7 H.J. Frost and M. F. Ashby, J. Appl. Phys., 42 ( 1971 ) 5273. 8 M.J. Mills and D. C. Chrzan, Acta Metall. Mater., 40 (1992) 3051. 9 Y. Q. Sun and P. M. Hazzledine, Philos. Mag. A, 58 (1988) 603. 10 P. Veyssibre, Mater. Res. Soc. Symp. Proc., 133 (1989) 175. 11 M. J. Mills, N. Baluc and H. P. Karnthaler, Mater. Res. Soc. Symp. Proc., 133 (1989) 203. 12 A. Couret, Y. Sun and P. M. Hazzledine, Mater Res. Soc. Symp. Proc., 213(1991) 317. 13 A.E. Steton-Bevan, Philos. Mag. A, 47(1983) 939. 14 P. M. Hazzledine and Y. Q. Sun, in T. C. Lowe, A. D. Rollett, P. S. Follansbee and G. S. Daehn (eds.), Modeling the Deformation of Crystalline Solids, The Minerals, Metals and Materials Society, Warrendale, PA, 1991, p. 395. 15 P. H. Thornton, R. G. Davies and T. L. Johnston, Metall. Trans., 1 (1970) 207. 16 J.C.M. Li, Acta Metall., 11 (1963) 1269. 17 H. Haken, Synergetics, Springer, New York, 1983, p. 274, 3rd edn. 18 H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, New York, 1971. 19 K.J. Hemker, M. J. Mills and W. D. Nix, Acta Metall. Mater., 39(1991) 1901. 20 S.-K. Ma, Modern Theory of Critical Phenomena, Benjamin/ Cummings, Reading, MA, 1978, p. 33.