ATOMISTIC CALCULATION OF PEIERLS­NABARRO STRESS IN A PLANAR

SQUARE LATTICE

W. R. Tyson

Trent University, Peterborough, Canada

ABSTRACT

Fully atomistic calculations for an edge dislocation have been per­formed using an empirical pairwise potential bonding atoms in a planar square array. The lattice friction stress for motion of a < 10> dislocation is much larger than the Peierls-Nabarro result and consistent with more recent theoretical approximations.

1 INTRODUCTION

A detailed knowledge of the structure and energy of the highly distorted region at the dislocation core is vital for a proper quantitative understanding of lattice friction stress, dislocation-solute interactions, cross-slip processes, and a host of other mechanisms thought to control

553 P. C. Gehlen et al. (eds.), Interatomic Potentials and Simulation of Lattice Defects© Plenum Press, New York 1972

554 W. R. TYSON

the flow stress of crystals. The first attempt to calculate atomic configura­tion in this region was due to Peierls l , and the resulting "Peierls-Nabarro model" of the dislocation core has received widespread attention. As well as providing a realistic picture of the core structure, the model leads2 to a lattice friction stress

_ k -41T 'p - I-v eXPT t (1)

where p. and v are the shear modulus and Poisson's ration, b is the Burgers vector, and 2~ = dl(l-v) is the dislocation width where d is the spacing between slip planes.

Eq. 1 predicts values of the friction stress, ~ 10-4 p.. Working within the framework of the same model, Foreman et if} have shown that wider cores and much lower values of 'pip. result if more realistic relations are used for the variation of the force acting between slip planes as they are sheared relative to each other. More recently, the method of calculating, p has been severely criticized, and Kuhlmann-Wilsdorf4 has argued that Eq. 1 underestimates , p by several orders of magnitude. Moreover, the Peierls­Nabarro model predicts that the two symmetric configurations for a <100> dislocation in a simple cubic lattice are unstable and of equal energy which means that the dislocation energy has a periodicity of one-half the lattice spacing. These features and the various modifications of the model have been reviewed by Nabarr05 and Hirth and Lothe6.

The unexpected results of the Peierls-Nabarro model may be due to the fact that it is semiatomistic only, in that the lattice on both sides of the slip plane is approximated by semi-infinite continua and that no account is taken of the position of other than first-neighbor rows of atoms across the slip plane. In this paper, the results of a truly atomistic calculation using a phenomenological interatomic potential to describe an edge dislocation in a two-dimensional lattice are compared with the Peierls-Nabarro model.

2 CHOICE OF POTENTIALS

As the elastic displacement field of a <100> edge dislocation in a simple cubic lattice is two-dimensional, all atoms lying in a plane perpen­dicular to the dislocation line remain in that plane. Hence, all essential features of the dislocation may be displayed in a two-dimensional lattice -the structure chosen for study in this work. Besides saving a considerable amount of computer time, this selection allowed atomic configurations to be easily visualized.

To further simplify the treatment, an interatomic potential was chosen so that the lattice would be elastically isotropic; this imposed

PEIERLS-NABARRO STRESS 555

conditions on the slope and curvature of the potential. The shape of the potential was chosen so that a square lattice would have the same cohesive energy as a close-packed (hexagonal) lattice, which has a considerably smaller atomic volume at equilibrium_ No volume-dependent term was considered. The potential is shown as Curve A in Fig. 1. Some calculations were also performed with potential B, which displays oscillations as would be expected for a "pseudopotential"; potential B produces a markedly anisotropic square lattice. Both potentials give very nearly sinusoidal force­displacement relations for shear of a perfect crystal (Fig. 2) .

Fig. 1. Interatomic potentials for the study of two-dimensional

square lattice; r l' r 2' and r 3 represent the positions of the first, second, and third nearest neighbors in this lattice.

Fig. 2. Force-displacement rela­tions for shear of a perfect crystal bonded by potentials A and 8 of

Fig. 1.

cp Uo

.20

.1 5

3 CORE STRUCTURE AND ENERGY

. 6

.4

.2

0 0

-.2

-.4

.. .... A

--- 8

-~SIN~

.2 .4 .6

>Vb

po If) 0 00

W .8 1.0

Dislocations with b = a[ 10] (a is the lattice parameter) were formed by displacing atoms to positions given by the elastic strain field of a

556 W. R. TYSON

continuum (anisotropic for potential B) from solutions presented by Hirth and Lothe6, and the lattice was then allowed to relax to equilibrium using a dynamic "quench" method (see, for example, Gehlen et a1.7). Using potential A, the relaxed configuration is not very different from the elastic solution, as shown in Fig. 3. However, a single function is not sufficient to describe displacements above and below the slip plane, and atoms above the slip plane were found to relax from the elastic positions considerably more than those below (Fig. 3). Thus, it is difficult to define a "width", although Fig. 3 indicates that the half-width is roughly ~ ~ l.3b which is somewhat larger than the Peierls estimate ~ = b/2(l-v) = O.67b. Atoms were relaxed within a radius of 10 lattice spacings of the dislocation ; as a check on convergence, results using a relaxation radius of 8 lattice spacings were found to be almost identical.

The total strain energy E within a radius r of the dislocation core for potential A is shown in Fig. 4 for the configuration labelled I (inset, Fig. 4). Points represent data for individual atoms, which contribute an amount to the strain energy bJi = 1/2 C1:.cf> - 1:.cf>0)' where the summation is taken over all neighbors within range of the potential; cf>o denotes the energy for the atomic positions in a perfect un strained lattice. The strain energy predicted from continuum elasticity may be expressed (Cotterill and Doyama8) as

·2

u b

·1

0 0 2

.'

. 3

E

,.. b'

• 1

0 . 2 .....- ,.

000 000 0·0 o 0

4 6

X/ b 8

00

Fig. 3. Displacements parallel with the Burgers vector for a dislocation in configuration I (see Fig. 4), potential A. Solid line shows results from linear elasticity; points show relaxed displacements for atoms

/0 above the slip plane (open) and below (filled).

, 1 ' 1 I 6 8 Ie

Fig. 4. Strain energy for configuration I (see inset), potential A, within radius r of the dislocation •

PEIERLS-NABARRO STRESS 557

2 E = p.b Qn_r_

47T( I-v) r eh

where the "equivalent hole radius" reh = blOl.p with 0I.l!. ~ 1 to 4 (Hirth and Lothe6); from Fig. 4, reh = .35 b and so OI.p = 2.8. Further, the "core radius" inside which nonlinear effects are important is r c ~ 3b, and the core energy within this radius is Uc ~ 0.6 Uo where the cohesive energy per atom U 0= l/2"1:,cJ>0 is 0.367 p.b 2 for potential A. With the dislocation in the other symmetrical configuration II midway between two rows in the upper half-plane, the corresponding results are r eh = 0.33 band OI.p = 3.1, with the core radius and energy indistinguishable from configuration I. Core relaxations were found to be much more significant for potential B, and OI.p was found to be 0.56 and 1.2 for configurations I and II, respectively.

4 PEIERLS BARRIER

Two properties are of interest regarding lattice resistance to disloca­tion motion: the uniform shear stress required to move the dislocation (Peierls stress T p)' and the energy that must be provided locally to move the dislocation to the next equilibrium site (Peierls energy Up). They may be separately determined by imposing different condItions during relaxation.

To find T p' a uniform shear strain 'Y was imposed on the whole lattice with the dislocation in configuration I. The atoms were then allowed to relax to their equilibrium positions. At a critical strain 'Yp ' the dislocation was found to move toward the boundary of the relaxed region, defining the Peierls stress T p = WYp; it was found that T pip. = 0.021 ± 0.001 for potential A.

To evaluate Up, the dislocation must be held in positions corre­sponding to core displacements between configurations I and II. This was achieved in the present case by displacing the dislocation to x = OI.b with ~<;0.5 and imposing elastic displacements corresponding to that posi­tion. Finally, the two atoms nearest the center of the dislocation (one above, one below the slip plane) were displaced to positions given by a linear interpolation between the corresponding fully relaxed positions at 01.

= 0 and 01. = 0.5 (configurations I and II), and held there during relaxation. The atomic configurations are shown in Fig. 5, and the increase in energy within the relaxed region above the value for 01. = 0 is plotted in Fig. 6. The . Peier~s energy is Up ~ 0.0038p.b 2, i.e., .Up'. ~ 0.017Uc. Also, conflguratlOn II appears to be metastable, a posslblhty for~seen by Hirth and Lothe6. The Peierls stress may be derived from the energy profile;

I dU Tp = b dx = 0.019p.

SS8 W. R. TYSON

ol=O

.2

.4

.16

.12

6U J-I-b2. .08

.04

\ \ \ \

.I

. 3

.5

\ .... \ \\

\ \ \ \

\ \ \ \

.006

.004

.002

o ~~ __ ~ ____ ~ ____ L-____ ~_'~~~ .2 .4 .6 .8 1.0

Fig. 5. Atomic arrangement in the core of a dislocation moving from configuration I (ex = 0) to configuration II (ex = 0.5) using potential A •

Fig. 6. Peierls barrier for

motion of dislocations using potentials A and B. Right­hand scale refers to potential A.

PEIERLS-NABARRO STRESS 559

for potential A (in reasonable agreement with the result found above). The derivative is taken at the inflection point. For comparison, T p calculated from Eq. 1 for this case is 0.00061J.l, a factor of 30 smaller than the fully atomistic result. Also shown in Fig. 6 is the Peierls barrier for potential B. In this case, Tp = 0.54J.l where J.l corresponds to (01)[ 10] shear; this enormous stress, a consequence of the marked relaxation in the core region, is discussed below.

The atomic configurations during glide of a [10] dislocation using potential B are shown in Fig. 7. At 0:: = 0, the atom directly above the dislocation line relaxes considerably in a direction normal to the Burgers vector b, and as 0:: increases to 0.5 this atom moves much further in this direction than it does parallel to b. The energy difference between 0:: = 0 and 0:: = 0.5 is about one-quarter of the core energy for this potential. Since the theoretical strength for (01) [10] shear is 0.14J.l which is

Fig. 7. Atomic arrangement in the core of a dislocation moving from configuration I (0:: = 0) to configuration /I (0:: = 0.5) using potential B.

ol=O

.2

.4

.1

.3

.5

560 W. R. TYSON

considerably less than T p.' the lattice will fail under an applied shear stress before the dislocation wIll move. The dislocation is effectively sessile.

To study the effect of different atomic arrangements across the slip plane, a dislocation with b = [11] was studied using potential A. It was found that each of the two symmetric positions of the dislocation is an unstable equilibrium state; the instability may correspond to the dissocia­tion [11] ~ [10] + [01].

5 SUMMARY

For potential A, the relaxed-core configuration is not greatly differ­ent from that given by linear elasticity, although for potential B consider­able relaxation was observed. For both potentials, the lattice friction stress is much larger than that predicted by the Peierls-N abarro result and more in line with more recent theoretical estimates (see, for example, Ref. 4).

Although quantitative predictions for any specific material require a knowledge of the appropriate interatomic potential and a sound theoret­ical justification for its form, the results presented here demonstrate that the friction stress for motion of an edge dislocation in a lattice held together by two-body central forces can be quite large.

REFERENCES

1. Peierls, R. E.: Proc. Phys. Soc. London 52: 34 (1940). 2. Nabarro, F.R.N.: Proc. Phys. Soc. 59: 256 (19471. 3. Foreman, A. J., et al.: Proc. Phys. Soc. 64A: 156 (19511. 4. Kuhlmann-Wilsdorf, D.: Phys. Rev. 120: 773 (1960). 5. Nabarro, F.R.N.: Theory of Crystal Dislocations, Oxford (1967). 6. Hirth, J. P. and J. Lothe: Theory of Dislocations, McGraw-Hili (1968), 7. Gehlen, P. C. et al.: J.A.P. 39: 5246 (19681. 8. Cotterill, R.M.J. and M. Doyama: Phys. Rev. 145: 465 (1966),