Messerschmidt U. Dislocation Dynamics During Plastic Deformation

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90 4 Dislocation Motion

relatively low stress, the Seeger–Schiller term in (4.40) is not very important

so that ΔF

∗

fdk

=2ΔF

is a suﬃcient approximation, as suggested in [150],

τbh

exp



−

2ΔF



. (4.41)

For high stresses, the kinks of the pair are not clearly separated so that

the elastic formalism with the separation of kink formation and interaction

energies cannot be applied. Then the kink pair has to be treated as a small

bow-out considering the elastic energy and the potential energy of the Peierls

relief. Such theories [151, 152] lead to diﬀerent stress dependencies of the

kink-pair formation energy. In the high-stress limit, with τ being in the vicin-

ity of the Peierls stress τ

, the stress dependence of the activation energy

becomes independent of the particular form of the Peierls potential, that

is, ΔF

fdk

∝ (τ

− τ)

5/4

[153, 154]. This eﬀect was conﬁrmed experimentally,

for example, in [155, 156]. For a review see [157]. In the usual double-kink

model, the dislocation velocity is constant. However, at the presence of for-

eign atoms, the traveling distance Δx may not be proportional to the time

Δt.Instead,Δx ∝ Δt

with δ<1 [158]. This kind of kinetics is now being

discussed in diﬀerent ﬁelds of physics, chemistry, and biology, see for example

[159].

To summarize the above, double-kink formation is a process operating on a

level between the atomic one and that of elasticity theory. The elastic descrip-

tion applied is therefore an approximation. In particular, the free energies of

activation can be replaced by the energies as discussed above in the section of

elastic properties of kinks. For low stresses, the double-kink formation energy

can conveniently be replaced by twice the formation energy of isolated kinks

so that the formation rate is given by (4.41). For higher stresses, the stress

dependence of the formation energy according to the Seeger–Schiller approach

(4.40) should be used. In the high-stress range, other potentials after [151,152]

have to be applied. However, the experimental decision between these diﬀerent

cases is quite diﬃcult.

Dislocation Velocity in the Range of Double-Kink Nucleation

A straight dislocation in a Peierls valley is considered, which moves forward by

double-kink nucleation, by the spreading of the kinks of the pair in opposite

directions, followed by the mutual annihilation with kinks of the neighboring

kink pairs after sweeping a distance λ, or by the storage of kinks at obstacles,

which limit a dislocation segment of length λ

. The dislocation velocity is

controlled by the shorter distance of both. A suitable average between both

casesisgivenby

= h

λλ

λ + λ

. (4.42)

4.2 Lattice Friction 91

To determine the sweeping distance λ of the kink pairs, their lifetime t

∗

considered. It is given by

∗

. (4.43)

again is the kink velocity of (4.35). To ensure a steady state motion of the

dislocation, a new kink pair has to be created during the lifetime t

∗

along a

length λ/2 of the dislocation. Thus,

∗

λJ

. (4.44)

Solving for λ and using (4.35) and (4.38) yields

λ =2



=2a exp



ΔF



. (4.45)

After (4.42), the dislocation velocity is given by

=2h



2τbh

akT

exp



−

ΔF



(4.46)

for the case of long dislocations (λ

 λ). This equation is identical with

(4.37) for the drift of thermal kinks at low stresses, and therefore universal

over a wide range of stresses. For short dislocations, that is, λ

 λ,the

dislocation velocity becomes

= hλ

τbh

exp



−

2ΔF



. (4.47)

There are some remarkable diﬀerences between the two cases of (4.46) and

(4.47). At long dislocations, kinks store at the obstacles limiting the segments

so that these segments bow out while short dislocations maintain straight

segments. For short dislocations, the activation energy is higher by ΔF

,and

the dislocation velocity is proportional to the segment length. The latter eﬀect

was proved by in situ straining experiments on semiconductor single crystals

in the TEM, as mentioned in Chap. 6.

There are many reﬁnements of the double-kink model. As mentioned ear-

lier, some of them aim at more realistic Peierls potentials. In several metals

and intermetallic alloys, the dislocation cores are not planar (e.g. [160]) and

may have diﬀerent stable conﬁgurations. As an example, in b.c.c. transition

metals, the dislocations with 1/2111 Burgers vectors have cores of threefold

symmetry. This can be considered by a two-dimensional Peierls potential on

the {111} plane [161]. The plane on which the kink pairs will be formed then

depends on the orientation of the stress ﬁeld. The calculations reproduce the

plastic anisotropy of these materials. Mathematical methods to simulate the

dynamic behavior of dislocations are summarized in [162].

92 4 Dislocation Motion

4.2.3 Characteristics and Experimental Evidence

of the Double-Kink Model

The double-kink mechanism, often also designated as the Peierls mechanism,

describes the lattice friction of a dislocation in an otherwise undisturbed crys-

tal lattice at a ﬁnite temperature. The decisive quantity is the Peierls stress

, which is the ﬂow stress at temperatures approaching 0 K. In addition, it

determines the kink formation energy according to (4.26). The latter con-

trols the dislocation mobility in the range of the operation of the model.

Frequently, the contribution of the lattice friction to the total ﬂow stress at a

ﬁnite temperature is also called the Peierls stress.

Dislocation motion controlled by the lattice friction exhibits the following

features:

• The dislocations show straight segments oriented along the Peierls val-

leys, that is, low-index crystallographic directions, as demonstrated in

Fig. 4.6 for solar silicon. The corners between the straight segments may

be rounded.

• The dislocations move in a smooth viscous mode. Both features are shown

in Video 6.1.

• The macroscopic deformation behavior is characterized by the (at least

approximate) proportionality between the dislocation velocity and the

eﬀective stress after (4.46), which is valid for both low stresses with the

1 µm

Fig. 4.6. Dislocation structure formed during in situ deformation of a solar sil-

icon specimen in the HVEM. The dislocations are arranged parallel to the three

110 directions on the (1

1) slip plane. The viewing direction is [011], g =(1

11).

Micrograph from the work in [163]

4.3 Slip and Cross Slip 93

concentration of thermal kinks and high stresses with stress-assisted kink

nucleation. Considering the Orowan relation between the dislocation veloc-

ity and the strain rate (3.5) shows that also the strain rate is proportional

to the stress. It follows that the stress exponent m of (4.11) is equal or

close to unity. The consequence is a high strain rate sensitivity r approx-

imately equal to the eﬀective stress τ

∗

and a small activation volume V

of the order of magnitude of only a few b

. The small activation volume is

related to the size of the critical kink pair at kink nucleation.

The temperature range in which the double-kink mechanism operates is

determined by its activation energy, which, for long dislocations, is the energy

of kink formation ΔF

plus, for slowly moving kinks, the energy of kink

migration ΔF

. Usually, the double-kink mechanism controls the disloca-

tion mobility at the low-temperature end of the thermally activated range. In

closed-packed metals where atoms can easily be shifted with respect to each

other, the activation energy of the double-kink mechanism is small, which

restricts this mechanism to temperatures close to 0 K. On the other hand,

materials with directional bonds as semiconductors and ceramics have high

kink energies so that the lattice friction controls the deformation behavior also

at higher temperatures. As the double-kink mechanism shows also a strong

dependence on the temperature, the respective ﬂow stresses at low tempera-

tures may be higher than the fracture stress, resulting in a brittle behavior

with a well deﬁned brittle to ductile transition. An intermediate behavior is

observed in some oxide and alkali halide crystals and in b.c.c. metals.

4.3 Slip and Cross Slip

In crystalline materials, slip systems are selected having a low stress necessary

to overcome the lattice friction. As discussed earlier, these are the systems with

the shortest lattice vectors as Burgers vectors and low-index planes with large

spacings ((4.20) and (4.21)). For a certain orientation of the crystal or the

crystal grain, those possible slip systems are being activated, which have a

high orientation factor (2.3). If several slip systems of low friction stress have

high orientation factors, more than one system will operate leading to multiple

slip. In particular orientations, the orientation factor of one slip system is

remarkably higher than that of the others so that the crystal deforms in single

slip.

In materials with a low stacking fault energy, the dislocations may disso-

ciate into partial dislocations, including a stacking fault as described for f.c.c.

crystals by (3.50). The dissociation occurs by glide on the plane of the stacking

fault so that this plane contains the directions of the Burgers vectors of both

partial dislocations as well as that of the total Burgers vector. Thus, disloca-

tion glide will be restricted to this plane. Consequently, several criteria decide

on the activation of the particular slip systems. For an undissociated screw

94 4 Dislocation Motion

1µm

Fig. 4.7. Dislocation structure formed during in situ deformation of an MgO single

crystal in the HVEM showing extended cross slip of screw dislocation segments.

The viewing direction is 100. The trace of the {011} slip plane runs in horizontal

direction. b projection of the Burgers vector, T slip trails of moving dislocations.

The slip trails are imaged in weak contrast owing to elastic strains of the slip steps

in surface contamination layers. J jogs. From the work in [164]

dislocation with its Burgers vector parallel to its line vector, also other low-

index planes are possible slip planes. A change of a screw dislocation from the

preferred slip plane onto another one is called cross slip or cross glide. Cross

slip may occur as a single event on long screw dislocation segments, or with

frequent changes of the slip plane, or even in a non-crystallographic way. In the

latter cases, the slip traces on the surface are not straight but consist of short

straight segments, or they are even wavy (wavy slip). In most cases, cross slip

is a local phenomenon where only a relatively short screw segment changes to

another slip plane. An intermediate case is demonstrated in Fig. 4.7, showing

a slip band in an MgO single crystal during in situ deformation. Straight

screw segments leave the original slip plane with a horizontal surface trace.

The trails T of the dislocation motion run along non-crystallographic paths.

The cross slip is possibly initiated by the internal stresses of the slip band

and by image forces near the surface. As only the screw segments cross slip,

they are connected to the rest of the dislocations by jogs J, which pin the

dislocations and cause the neighboring segments to bow out.

Cross slip plays an important role in many phenomena of plastic defor-

mation. It may be responsible for both dislocation multiplication as well as

recovery by mutual annihilation of screw dislocations (Sect. 5.1). In inter-

metallic alloys, dissociated dislocations can get locked by cross slip of one

superpartial. On the other hand, cross slip can help to bypass glide obstacles

4.3 Slip and Cross Slip 95

like large precipitates. The ability to cross slip depends on the structure of

the dislocation core. Undissociated dislocations can easily cross slip onto other

planes of low lattice resistance. However, the partial dislocations of a disso-

ciated screw dislocation usually do not have screw character so that they

cannot leave the original slip plane. Either a dislocation reaction is necessary

resulting in a partial dislocation with a Burgers vector on the cross slip plane

[165], or the stacking fault has to constrict on the original plane before the

dislocation may extend on the cross slip plane [166].

Cross slip of dissociated dislocations in the f.c.c. structure is treated in

the model of Escaig [167] based on the suggestions of Friedel [168] and the

calculations by Stroh [169]. In this model described in detail in [11], the dis-

location has to form a constriction of the stacking fault in the original glide

plane of a certain length before dissociating again on the cross slip plane. The

ﬁnal conﬁguration is sketched in Fig. 4.8. Initial nuclei of constrictions may

pre-exist at jogs or dislocation nodes. Extending the constriction increases

the energy of the system. The energy diﬀerence is calculated considering the

stacking fault energies on both planes, the energies and line tensions of all

dislocation segments involved as well as their interaction energies. The total

energy diﬀerence depends on the widths of the dissociated dislocations and

the local stresses on both planes. Estimates for copper yield an energy, which

decreases with increasing stress and which amounts to about 1.1 eV at zero

stress for the case of pre-existing constrictions. Thus, the cross slip should be

a thermally activated process. The activation volume decreases from about

300b

at zero stress to some tens of b

at high stresses. Reﬁned calculations of

the activation parameters of cross slip are reviewed in [170]. Recently, cross

slip of dissociated dislocations has been modeled on an atomic scale by the

embedded atom method (EAM) [171] and by molecular dynamics simulations

[172, 173], essentially conﬁrming the results of the continuum models. Sch¨ock

[174] criticized several models of cross slip and derived the stress conditions

for the spreading of cross-slip.

The parameters of cross slip were studied experimentally in Cu applying a

particular technique [175,176]. The crystals were ﬁrst deformed in a symmetri-

cal multiple slip orientation into a stage of strong work-hardening. Afterwards,

smaller specimens were cut out of the predeformed samples and deformed in

SP SP

SF SFSF

CSP

Fig. 4.8. Cross slip of a segment of a dissociated dislocation. SP original slip plane,

CSP cross slip plane, SF stacking faults

96 4 Dislocation Motion

a single slip orientation, where the dislocations of the predeformation have a

high orientation factor on their cross slip plane but where the original slip sys-

tems are out of stress. It is assumed that the critical ﬂow stress of the second

deformation represents the stress necessary to activate cross slip of the dislo-

cations of the predeformation. In a narrow temperature range, the activation

energy agrees with the theoretical value quoted earlier. The described mea-

surements probably present the only experimental estimation of the activation

parameters of cross slip.

Similarly, also statistical data on the occurrence of cross slip are quite rare.

A method well suited to image cross slip events is the heavy metal decoration

technique of slip lines of individual dislocations on alkali halide cleavage sur-

faces described in Sect. 2.5.2, where the paths of individual dislocations with

a Burgers vector component out of the surface are imaged. A corresponding

example is presented in Fig. 4.9. During cleavage, some grown-in dislocations

are cut. At the emergence points S of these dislocations with a screw com-

ponent out of the surface, cleavage steps originate. The straight slip lines

mark the paths of individual dislocations, which had moved during plastic

deformation on {011} planes inclined with respect to the surface. When these

dislocations stop moving as indicated by E in the ﬁgure, they may shorten

1 µm

Fig. 4.9. Gold decoration surface replica of an NaCl single crystal deformed at

room temperature showing cleavage steps and slip steps of dislocations with 1/2011

Burgers vectors that had moved on {011} planes. 100 surface. Micrograph from

4.3 Slip and Cross Slip 97

their length by cross slipping on {010} planes under the action of image forces.

Using line tension arguments, the ﬂow stress on the cross slip plane can be

estimated from the cross slip distance on the surface. As a result, the ﬂow

stresses on the cross slip plane are approximately equal to those on the pri-

mary slip plane, indicating that both are controlled by the same mechanism

[178]. Segments of a certain length often cross slip during the dislocation

motion from the primary slip plane onto the cross slip plane and, after some

motion there, they move back to a plane parallel to the original slip plane.

Several of such events are marked by C. It should be noted that under the

applied stress acting in 001 direction there is no shear stress component on

the cross slip plane. For the respective geometry with {011} slip planes inter-

secting the {100} surface, the cross slip height h, that is, the distance between

the original and ﬁnal slip planes, is given by the displacement of the slip step

divided by

√

Figure 4.10 presents the frequency distribution H(h) of the cross slip

heights h in an NaCl single crystal with 32 ppm divalent foreign atoms

deformed at room temperature. In alkali halide crystals, the concentration

of divalent additions signiﬁcantly determines the yield stress. The histogram

shows frequencies strongly decreasing with increasing cross slip height. A very

simple theoretical model of the distribution of the cross slip heights is given

in [180], assuming that there is ﬁrst a certain ﬁxed probability per swept area

of the primary slip plane to leave this plane by cross slip, and second another

probability per swept area of the cross slip plane to return to a plane parallel

to the primary plane. Under these assumptions, the frequency distribution of

the cross slip heights becomes

Fig. 4.10. Frequency distribution of the cross slip heights of an NaCl single crystal

with 32 ppm divalent foreign atoms deformed along 001 at room temperature.

Total of 262 cross slip events. Data from [179]

98 4 Dislocation Motion

0.1

0.01

h (μm)

1 10 100

c (ppm)

(a)

1000

100

101

(b)

100

D (mm

–1

)

c (ppm)

Fig. 4.11. Dependence of cross slip height (a) and total density of cross slip events

along the slip line length (b) on the concentration of divalent cationic impuri-

ties in NaCl single crystals. Open squares: deformation at room temperature along

100. Ful l squares : liquid nitrogen temperature along 100. Open diamonds:room

temperature along 110. Data from [177]

H(h)=

exp



−



where

h is the average cross slip height. The theoretical distribution is included

in Fig. 4.10 as a solid line. It agrees roughly with the experimental distribution.

The main diﬀerence consists in the long tail for large heights in the experimen-

tal values. These properties are characteristic also of the crystals with other

concentrations of divalent additions. Figure 4.11a shows the dependence of

the average cross slip heights on the divalent impurity concentration c for two

orientations of the loading axis at room temperature, and for the 001 axis

at liquid nitrogen temperature. There is almost no dependence on the loading

axis and the deformation temperature. The values decrease slightly for small

impurity concentrations and more strongly for higher ones. In Fig. 4.11b, the

total density D of cross slip events per slip line length is plotted as a function

4.4 The Locking–Unlocking Mechanism 99

of c. There is a strong increase of D with increasing c but again no dependence

on the specimen orientation and temperature. As mentioned earlier, the con-

centration of divalent additions essentially determines the yield stress of the

alkali halide crystals. It also inﬂuences the dislocation density, which builds up

at the ﬁrst stage of deformation, and thus the intensity of the internal stresses.

As both the mean cross slip height and the total density depend on the impu-

rity concentration but not on the temperature, the cross slip processes seem

to be of athermal character and be inﬂuenced by the internal stresses rather

than by the total ﬂow stress.

A direct correlation between the local density of forest dislocations and the

density D of cross slip events was observed in the latent hardening study in

[181]. NaCl crystals were deformed consecutively along two diﬀerent compres-

sion axes so that the dislocations having formed during the ﬁrst deformation

were out of stress during the second one. As demonstrated in Fig. 4.12, a ﬁrst

surface decoration with the large gold nuclei revealed the emergence points F

of the dislocations generated during the ﬁrst deformation which, during the

second deformation, act as forest dislocations. The slip steps in horizontal

direction created during the second loading are marked by a second decora-

tion with smaller palladium nuclei. They show cross slip C and immobilization

events I. Individual micrographs reveal a local correlation between the forest

dislocation and cross slip densities, proving the role of internal stress ﬁelds as

a source of cross slip.

In summary, cross slip may show quite diﬀerent characteristics. On the

one hand, in materials of low stacking fault energy and with dissociated

dislocations, the initiation of cross slip seems to be a thermally activated pro-

cess. In contrast to that, in some materials with compact dislocation cores,

cross slip seems to be easily initiated and may be of athermal character.

The driving forces result from the internal stress ﬁeld of other dislocations

and from components of the applied stress. Further motion on the cross

slip plane is governed by these stresses and the usual mechanisms con-

trolling the dislocation mobility on the respective planes. Collective cross

slip of many dislocations is supposed to cause the decrease of the work-

hardening rate after heavy plastic deformation [182]. As will be discussed

in Sect. 5.1.1, individual cross slip events are a prerequisite to dislocation

multiplication.

4.4 The Locking–Unlocking Mechanism

As mentioned earlier, the cores of the dislocations need not be planar. In

particular, a dislocation with a certain total Burgers vector may have sev-

eral stable conﬁgurations extended on diﬀerent planes with diﬀerent energies.

These conﬁgurations may be glissile on a certain plane, or totally sessile, or

sessile with respect to the glide plane. To move the dislocation in a sessile