ELASTO-ATOMISTIC COUPLING PROCEDURE TO DETERMINE THE CORE CONFIGURATION OF EDGE DISLOCATIONS IN IONIC CRYSTALS

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ELASTO-ATOMISTIC COUPLING PROCEDURE TO

DETERMINE THE CORE CONFIGURATION OF

EDGE DISLOCATIONS IN IONIC CRYSTALS

P. Petrasch, V. Belzner

To cite this version:

JOURNAL DE PHYSIQUE Colloque C7, suppliment au no 12, Tome 37, D6cembre 1976, page C7-553

ELASTO-ATOMISTIC COUPLING PROCEDURE TO DETERMINE

THE CORE CONFIGURATION OF EDGE DISLOCATIONS

IN IONIC CRYSTALS

P. PETRASCH and V, BELZNER

Institut fiir Angewandte Physik der Universitat FrankfurtlMain, Robert-Mayer-Str. 2-4, FRG

R6sum6. - Nous prksentons une mkthode qui nous donne la possibilitk de calculer, d'une maniere semi-discrete, Ia configuration ionique autour du centre d'une dislocation coin. Nous avons applique les formules de la thkorie non lineaire de l'klasticite pour la region extkrieure et des poten- tiels rkalistes pour la region intkrieure, les deux regions ayant une partie commune. Les rksultats d'un calcul pour NaCl (AgCI) sont discutb.

Abstract.

-

A method is described to compute the ionic configuration around the centre of a straight edge dislocation by using semi-discrete techniques applying formulae of non-linear theory of elasticity for the outer and realistic interaction potentials for the inner region, both regions overlapping each other. Results of computations for NaCl (AgCI) are presented.

1. Introduction.

-

In order to describe the ionic configuration of the surroundings of edge dislocation centres it is useful to apply semi-discrete methods. The next neighbourhood of the dislocation line is treated as a discrete lattice whereas the remainder of the crystal is considered as an elastic continuum.

In early applications of this model, e. g. [l to 31, the boundary between these two regions was kept rigid, the atoms in the outer region being fixed in their positions as given by linear thcory of elasticity thus neglecting the core boundary conditions. Apply- ing this method, however, brings about some artificial effects like the zero mean value of volume dilatation and stress discontinuities across the boundary. On the other hand, extending the inner region is time- wasting because there is a rather slow convergence for the atomic positions even near the very centre of the dislocation core.

To avoid these inconveniences, several variants of the semi-discrete method using a flexible boundary have been proposed [4 to 61.

In the present paper, we report on results obtained by realizing ideas of A. Seeger, Stuttgart, and C. Teo- dosiu, Bucharest, applying formulae of nonlinear theory of elasticity as well as a special semi-discrete method, the so-called method of the overlapping

regions [5].

2. The method of overlapping regions. - As indicat- ed by its name, this method implies a ring-like zone between .the two regions mentioned in section 1 which, alternately, is treated atomistically and elas- tically.

Figure 1 may serve to facilitate the understanding Here once more, what we need to start the calcula- tions :

a) The formula for the nonlinear elastic field to obtain the atomic positions in the regions I11 (and 11),

b) realistic interaction potentials for region I

(and 11), and finally,

c) a mechanism to synchronize those two different ways of computation and their results, viz., the method of overlapping regions, in brief : The overlap-method.

FIG. I. - Model to explain the method of the overlapping

regions.

2.1 ELASTIC COMPUTATIONS.

-

The historic (linear)

elastic field [l to 31 giving a rough core configuration has t o be supplemented by terms supplied by non- linear theory of elasticity as, e. g., prepared in 171.

C7-554 P. PETRASCH AND V. BELZNER

The nonlinear formula, which is too voluminous to where the ions i and k are next neighbours of the be repeated here, is represented by expression like ion j. There are two constant

E(j)

and

60')

depending

l/r, (log. r)/r where r is the position vector. The on whether the ions are situated on a line or form an complete formula for the elastic field then, in abbre- angle (see Table I). Adding such three-body potentials, viated terms, reads as follows : as for the first time proposed by Sarkar and Sengupta

U = U, 4- U*,

+

U*

,

(1) 181, yields the desired violation of the Cauchy relation.

where U, is the well-known (linear) Volterra field,

U,, the nonlinear field, and U, that part of U whose coefficients, in accordance with the unknown stresses on the boundary, are not given by theory and therefore must be determined in an other way (as will be des- cribed in section 2.3).

2.2 ATOMISTIC COMPUTATIONS.

-

When describing the structure of ionic crystals, primarily electrostatic and short range repulsion (Born-Mayer) interactions are taken into account. When improving the model, it often (e. g., in AgCl) is useful to add potentials for Van der Waals interaction. These two-body (central) interaction potentials, however, bring out, when fitting their parameters to elastic constants, such constants for whose the Cauchy relations (i. e.,

c,, = c,,) are fulfilled. But experimental results show considerable deviations from the validity of these relations for many crystals.

In order to overcome these difficulties the simple Born-Mayer model has to be supplemented by addi- tional terms, say three-body potentials. Such many- body potentials are introduced in a way that three ions i, j, k give a contribution to the total energy of

names

-

2 . 3 THE OVERLAP-METHOD. - AS mentioned in section 2.1, the nonlinear elastic field was not quite complete yet. The still unknown term U , is given by the sum

where the B, are fourfold sets of complex constants. If we confine ourselves to the physically most important part of U, (i. e., the terms with n = l), U, is simplified to

The constants dl, d2, p, and p, and the variables zi

(i = 1, 2, 3, 4) are well-known complex constants or

variables, respectively, which appear also in U, and U,,. It is the complex constants Bi (i = 1,2, 3, 4) that are unknown and must still be determined. We, therefore, may take them as parameters to be eva- luated in such a way that the atomistically and elastic- ally calculated displacements (or stresses, what is equivalent) correspond to each other in region I1 as well as possible. This problem is solved by the overlap method. It is explained, with the help of figure 1, in

Collection of equations of potentials and potential parameters

Born-Mayer repulsion between next (Bij) and next nearest (Bjj) neigh- bours

pij

: Paulingfactor, ri, r j : Ionic radii, p : hardness parameter

Van der Waals attraction

aligned triples 0-S-0 i j K angular triples 0 4 ~ i

l

0

k Potential formulae

-

ri

+

r j

-

rij

U?' =

pij

exp

(

CORE CONFIGURATION OF EDGE DISLOCATIONS

Final values of the constants B, (i = 1 to 8) obtained when reaching the last cycle. The relations B, = etc. are due to the high-degree symmetry of cubic crystals

Numbers of cycles

-

B1 -

-

B3

B5

B7

-

NaCl 13 (1.814, 1.480) (0.667,

-

0.853) (10.432,2.995) (7.194,

-

2.999) AgCl 9 (0.596,

-

-

0.500)

-

(0.278, 0.233) (- 6.575, 1.802) (- 3.707,

-

0.348) B, = B1,B4 = B3,B6 =

-

&,Bs =

-

The subscripts 5 to 8 belong to the terms with n = 2 in Eq. (3) which were used here in addition to (4).

The following scheme : number of iteration cycles. A good convergence can

1) Elastic computation of the ionic positions in clearly be recognized. Table 11 gives an impression of regions I, I1 and 111. the magnitudes of these complex constants at the and

2) Atomistic variation starting with the results ofthecom~utations- of (1) in regions I and I1 (R = 6 b ; b = Burgers

10

-

d

vector). 0 0

0 . O 0

3) Determination of the unknown constants, B , 9 - o O

from the difference between atomistically and elastic- 0

8 . 0

ally computed displacements, AU = U,, along the 0 0

curve S by means of the Fourier transform of Eq. (4) : 7 .

ak eiek B,

+

bk eiek Bz

+

ck eiek B3 -l-

+

dk eiek B, = 6, eaek

.

(6)

Setting, subsequently, the subscripts k equal to

-

2,

-

1, 1, and 2, we have, thus comparing the first four Fourier coefficients left and right of the equal sign, a system of four equations with the unknowns B, ;

in matrix notation :

A B = C ,

- (7)

where B is the solution vector readily to be determined by :

B =

_-

A - ' . C . _- (8)

FIG. 2.

-

A selection of some constants Bi of NaCl determined

by the overlap-method converging with the number of compu- tation cycles : a) real part of Bg ; 6 ) modulus of imaginary part

of B3 ; c) real part of B1 ; d) real part of Bs.

0.9

.

0.7 -

0.5

-

Table 111 presents some values of coreenergies determined by means of different methods. As easily to be seen, three-body interactions play an important

b O O o o O o o O 0 0 O O O Q O O O 0 0 O O a O

part in the case of silver chloride as was to be expected 4) Elastic recomputation of the ionic positions in

because of the violation of the Cauchy relation, regions I, 11, and 111, this time including the term U,

contrary to sodium chloride, where c,, % c,,. A

of the elastic field as determined by the last step.

2 4 6 8 10 12

So we have returned to the starting point of this scheme. This procedure is repeated till reaching self- consistency.

3. Results and discussion.

-

In order to test the overlap-method, core configurations have been com- puted for edge dislocations in sodium- and silver chlorid. Results will be presented mainly for the former.

In figure 2, real and imaginary pax ts of some of the constants of the function U, are plotted versus the

Energies (shares and total) of 3 b-cores quoted in eV per plane

Mad Rep VdW 3-body Total

-

-

-

NaCl 0.485 9 0.583 6

-

0.098 2

-

0.000 25 0.971 1 AgCl (*) 0.730 7 1.575 1

-

0.821 8

-

0.509 3 0.974 6

AgCl(**) 0.710 3 0.494 0

-

0.668 0

-

0.566 3 0.969 9

C7-556 P. PETRASCH AND V. BELZNER

general trend of reducing core energies is observed when applying the method of overlapping regions. One important criterion how to check the accor- dance between the atomistically computed ionic positions and the ones in the surrounding elastic continuum is, for example, to examine the (modulus of the) force acting on an ion when crossing the boundary. This is illustrated in figure 3, where the

regions brings about some progress, i. e., a more realistic description of the surroundings of the dislocat- ion centre : The dashed line describes the force on an ion after the variation procedure using the improved potentials but the boundary kept rigid. The dotted curve, however, was obtained using, besides applying nonlinear anisotropic theory, flexible boundaries according to the overlap-method.

At first sight, the striking things are, of course, the discontinuities of both curves when crossing the 6-b-border. We certainly need not, in this place, argue about the jump appearing in the dashed line. Its origin is well understood. Let us notice instead

the improvement reached after using the overlap- method. There is, of course, still a discontinuity but, comparing with the dashed line, it has diminished considerably : It is only about one tenth of what it had been before. Moreover, calling our attention to the core ions themselves, it can be noticed that the forces on the ions within the core have become very small, i. e., there are almost no forces any more what seems quite satisfactory.

20

FIG. 3.

-

Forces on ions in arbitrary units versus distance from

the dislocation centre in NaCI. (The angle between the path and 4. The authors are grateful the x-axis was chosen 450.) The dashed curve was obtained after to Prof. A. Seeger, Stuttgart, Prof. F. Granzer,

...., 1

& - - - - - - _ - _ A _ _ _ _ _ _ - _ _ _ _ _{... . ...}

.

=,---, . ' ,. .

- - _ L

variation with rigid boundary, the dotted one after 12 cycles of _{Frankfurt/Main, and Prof. C . Teodosiu, Bucharest,} variation applying the overlap-method. _{fort their constant interest and many helpful dis-}

cussions. This work was supported by the Deutsche gradient of the potential,

I

V U

I,

is plotted versus the Forschungsgemeinschaft, Bonn-Bad Godesberg, and distance from the dislocation line. was carried out with the help of the Hochschulreche-

This plot reveals that the method of overlapping nzentrum, university of Frankfurt.

10 20 6b j o r [a]

References

[ l ] HUNTINGTON, H . B., DICKEY, J. E. and THOMPSON, R., [5] TEODOSIU, C. and NICOLAE, V., Rev. Roum. Sci. Tech.,

Phys. Rev. 100 (1955) 1117. sCr. m6c. appl. 17 (1972) 919.

[2] GRANZER, F., WAGNER, G . and E I ~ E N ~ L X ~ R , J., l61 GEHLEN, P. C., HIRTH, J. P., HOAGLAND, R. G. and KAN-

Status Solidi 30 (1968) 587. NINEN, M. F., J. Appl. Phys. 34 (1972) 3921.

[7] SEEGER, A., TEODOSIU, C. and PETRASCH, P., Phys. Status

[3] EISENBLXTTER, J., Phys. SSfatus Solidi 31 (1969) 71 ; 31 (1969) _{Solidi (b) 67 (1975) 207.}

87. _{181 SARKAR, A.'K. and SENGUPTA, S., Solid State Cornrnun. 7}