NONLINEAR THEORY OF THE POWER DISSIPATION DUE TO THE MOTION OF HEAVY INTERSITIALS IN OSCILLATING INHOMOGENEOUS FIELDS WITH STRONG STATIC BIAS

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NONLINEAR THEORY OF THE POWER

DISSIPATION DUE TO THE MOTION OF HEAVY INTERSITIALS IN OSCILLATING

INHOMOGENEOUS FIELDS WITH STRONG STATIC BIAS

T. Ö Oğurtani, A. Seeger

To cite this version:

T. Ö Oğurtani, A. Seeger. NONLINEAR THEORY OF THE POWER DISSIPATION DUE TO THE MOTION OF HEAVY INTERSITIALS IN OSCILLATING INHOMOGENEOUS FIELDS WITH STRONG STATIC BIAS. Journal de Physique Colloques, 1987, 48 (C8), pp.C8-167-C8-172.

�10.1051/jphyscol:1987822�. �jpa-00227126�

Colloque C8, supplbment n012, Tome dbcembre 1987

NONLINEAR THEORY OF THE POWER DISSIPATION DUE TO THE MOTION OF HEAVY INTERSTITIALS IN OSCILLATING INHOMOGENEOUS FIELDS WITH STRONG STATIC BIAS

T

.a.

O&JRTANI' * and A. SEEGER"

~ i d d l e East Technical University, Ankara, Turkey

. .

Max-Planck-Institut fiir Metallforschung, Institut fiir Physik, PO Box 800665, 0-7000 Stuttgart 80, F.R.G.

ABSTRACT

Les mouvements atomiques des interstitiels lourds (localisks dans des sites interstitiels octae'driques dans un rgseau cubique-centri) dans un champs arbitrairement debendent du temps et inhomogkne avec un fort biais statique sont dgcrits par un systsme d' 6quations diffkrentielles non-linkaires autonomes du premier ordre. La forme ~ineariske de ces kuations dans l e voisinage de ~'ktat kquilibre) thermodynamique est rbolue exactement en utilisant la transformation Fourier discrZte de l'espace k supplement& par une transformation Laplace par 6gard au temps. La dissipation de la force associde avec les sauts des atomes interstitiels pendant la przsence $06 une vibration simple harmonique o& des ondes acoustiques est determinle. Les implications de la thkorie non-lingaire pour I'enthalpie d'activation effective de la relaxation Snoek-Kbster sont 6laborges pour le cas limite des interactions ~oca~iskes ou d6localisks entre les interstitiels et les dislocations.

The atomic movements of heavy interstitials (located in octahedral interstices in a body-centred cubic lattice) in arbitrary time-dependent and inhomogeneous field with a strong static bias are described by a system of nonlinear autonomous first-order differential equations. The linearized form of these equations in the vicinity of the thermodynamic equilibrium state is solved exactly, using the discrete Fourier k-space transform supplemented by a Laplace transform with respect to time. The power dissipation associated with the hopping motion of the interstitial atoms in the presence of either a simple harmonic vibration or of propagating acoustic waves is determined.

The implications of the nonlinear theory for the effective activation enthalpy of the Snoek-Kbster relaxation are worked out for the limiting case of localized or delocalized interactions between the interstitials and dislocations.

I. INTRODUCTION

It has been known for a long time that the interaction between foreign atoms and dislocations plays an important rGle in the internal friction of metals. This is particularly so for body-centred cubic @cc) metals, in which the so-called heavy foreign interstitial atoms (carbon, nitrogen, and oxygen) occupy interstices surrounded by strain fields of tetragonal symmetry (preferentially octahedral interstices) and in which the interstitialdislocatioo interaction is therefore particularly strong. A mathematical formulation of the hopping motion of interstitial atoms in time-dependent inhomogeneous fields (due, e.g., to moving dislocations or kinks) under weak static bias with' or without2 chemical reactions has been developed by the present writers with the restriction that the magnitude of the interaction energies is small compared with the thermal energy k g T This restriction implies that the overwhelming majority of the sites available to the interstitials have to be vacant and that the original non-linear rate equations may be replaced by a set of linear coupled differential equations.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1987822

C8-168 JOURNAL DE PHYSIQUE

The present investigation lifts the restrictions on both the magnitude of the stuiic part of the field and on the concentration of the interstitials by maintaining the nonlinear character of the rate equations associated with the hopping motion. As shown elsewheres, after a suitable transformation the system of rate equations can nevertheless be linearized in the vicinity of thermodynamic equilibrium by assuming that the time-dependent part of the interaction energy is small compared with k g T In Sect. I1 the description of the hopping motion of octahedral interstitials in bcc lattices in terms of rate equations and their general solution in the discrete k space will be outlined. The power dissipation associated with the hopping motion of interstitials is calculated in Sect. 111. Sect. IV explores the effects of the site saturation and of the binding energy on the observed intensity and the activation enthalpy of the Snoek-Kaster relaxation through the results of the nonlinear theory and compares them with a more elementary approach.

11. NONLINEAR THEORY O F MICRODIFFCISION IN A BCC LATTICE

For simplicity. we confine ourselves to interstitials occupying octahedra1 interstices and capable of jumping between nearest-neighbour sites only. Such jumps are associated with a change of the crystallographic orientation of the tetragonal axis of the occupied interstices. The atomic concentrations in the three types of interstices with different orientations of the tetragonal axis are denoted by C j +1,2.3). The rate equations then read

aC1(RN+rl,t)/at = r21(RN+rl+rS~RN+'1) CZ(RN+rl+rS,t) [l-C1(RN+rl,l)l

+ ~ZI(RN+rl-rS.RN+rI) C2(RN+rI-rS.t) [I-Cl(RN+rl.t)I

-

~12(RN+rl,RN+rl+rs) CI(RN+rl,I) t 1 -C2(RN+rl+rS,t)l

-

rI1(RN+~l.RN+rI-rS) Cl(RN+rl,t) CI-C2(RN+rI-~S,t)I

plus (311 and (-131 terms. (1)

Equations for the time derivatives of C2 and C1 are obtained by cyclic permutation. The positions of the interstitials are characterized by that of the unit cell, RN. and their position vectors r j (j=1,2,3) within the unit cells. An analogous notation is used for the rate of the atomic jumps from i into j sites, Tij. This rate depends on the positions Rp and Rq of the adjacent inter- stices involved. By taking into account the principle of detailed balance, it may be written as

Tij(Rp,Rq) = V exp i CUi(Rp.t)-Uj(Rq,~)I/2kfl 7

where v is the jump rate in the absence of any external and internal fields and Ui(Rp,t) denotes the potential energy of an interstitial in orientation i at the site Rp due to such fields.

The time-dependent potentials may be separated into a static (bias) and a time-dependent (dynamical) part according to

Ui(Rp.t) = Ui(R&

+

ui(Rp,t)

.

⁽³⁾

In general, the static part (superscript S) consists of two different contributions, viz. the externally applied bias and the contributions of internal sources such as dislocations or precipitates. Similarly, the dynamical part (superscript d ) may be subdivided into a direct

The steady-state solution of the above set of nonlinear equations for a static field (giving rise to the potential energy

LI;)

may be written as

where p denotes the chemical potential of the interstitials (the "Fermi levelH), given by the equilibrium condition

p = kgT I~[c:(I-c;)~ ( 5 )

with

cp

cb/3, where

cb

denotes the bulk concentration of the interstitials.

In the neighbourhood of the state of thermodynamic equilibrium (1) may be linearized with respect to the dynamical part of the interaction field by retaining only first powers in the excess partial concentrations

ci(RP.t) = Ci(Rp,t)

-

Ci(RJ. _{( 6 )} The general solution of the linearized equations may be written asS

* ^d.. ci(Rp.f) fi(Rp.f) x ( R J . where

N

fj(t) = - P:(k) expE-e(k)il e:i(k)><e:i(k) icexp(P: (k)t91

(7

ud(t') 3dtSk

0

+ exp C-~o(k)il

1

epi(k)><epi(k)

/

?,(o). (8) In the above expression, which is given in terms of the discrete Fourier k space transforml~',

N

= [ k , ~ a ( - ~ f ) / a u f I ' I 2 . (9) is the Fermi-Dirac distribution energy derivative (FDDED)

[c;

^r Ci(Rp)]. This function is bell-shaped with a maximum at u:=jc. Therefore, oniy sites in the energy range near to the Fermi level can give appreciable contributions to the observed power dissipation. epi(k) are unit eigenvectors associated with the ith mode of the relaxation spectrum, P ~ ( L ) positive numbers2 related to the relaxation spectrum of the octahedral sites in a bcc lattice, lying in the closed interval C0.21. Their only zero occurs when i = 1 (acoustic branch), k = 0.

Ill. THE POWER DISSIPATION DUE TO THE HOPPING MOTION O F INTERSTITIALS In this section we shall outline the nonlinear theory of the power dissipation associated with the hopping motions of interstitials in the presence of harmonic excitations which are superimposed on an inhomogeneous static field. By introducing the discrete Fourier transform, the power input into a sampling supercell (periodic boundary conditions) may be written as

I(() = N.,.I~ [ Y U ~ I

-

katfkial.

where NT is the total number of unit cells in the sampling domain.

C8-170 JOURNAL DE PHYSIQUE

I n the present paper we consider two different time dependences of the interaction energy, namely

u a t ) = u p (expfiwt)

+

c.c.)/2 (11) and

u$t) = u p exp(-iwkt). (12)

Eq. (11) corresponds to simple harmonic oscillations, (12) may be used to describe kinks moving uniformly and rigidly along dislocation lines with the velocity V. In the first case we obtain for the dissipation part of the power input

and in the second case

Here it is assumed that

wk

= - w - ~ , which is the case for a kink moving with a constant velocity along the dislocation line'. The analysis presented in this section shows clearly that not only the dynamical part of the interaction fieldb@' but also the static part contributes to the energy dissipation. However, the latter contribution is an indirect effect. It takes place by the

.., ...

modification of the dynamic interaction field through

x.

In reality is a very complicated modifier depending on the temperature T , the composition CO(Rp). and the distribution of the static field in the bulk sample.

IV. DISCUSSION

Our previous treatment^',^ assumed that we were dealing with extremely dilute alloys and that not only the dynamical part of the interaction energy but also the static inhomogeneous interaction of the interstitial atoms was small compared with the thermal energy kBT. These two restrictions (especially the second one) resulted in a theory in which the power dissipation was independent of the static field arising from the imperfections with which the interstitials interact. As a consequence, the theory of kink dragging in the atmosphere of mobile interstitials based on these restrictions6-' leads to a n apparent activation enthalpy for the Snoek-KOster relaxationso9 which does not contain a contribution from the enthalpy of binding of the interstitials to the dislocations. However, such a contribution has been predicted by Seeger in his ad hoc treatments of the problem. The extensive experimental observations on Nb and Ta containing dilute oxygen interstitialsQ provide indeed strong evidence that the activation enthalpy of the Snoek-K6ster relaxation involves the binding enthalpy of the foreign interstitial atoms to < I l l > screw dislocations. In the case of heavy doping, such as Ta containing 740 atppm of oxygen, the measured activation enthalpy becomes independent of the enthalpy of binding due to the fact that virtually all available sites around the screw dislocations are occupied by interstitials, i.e. that saturation has been reached. In both Nb and Ta the binding enthalpies of oxygen to < I l l > screw

(0.03 eV) at the peak temperatures.

The viscous drag coefficient B for a uniformly moving kink due to the interaction with mobile interstitials can be calculated from the following expression valid in the linear Stokesian regime:

B = lim (I;/V' I

v-

This gives us

where w,

-

(k-?)v and

2

is a unit vector along the dislocation line and ^{T s} the relaxation time of the Snoek effect. The compact relationship (16) shows that any inhomogeneity in the static interaction field results in a positive contribution to the viscous drag coefficient B through the

'5

non-vanishing matrix elements of

z.

the Fourier k transform of the FDDED. A general discussion of the effects of the static field on the Snoek-KOster relaxation and on the expression (16) is rather complicated and requires detailed computer studies. However, (16) may be simplified by considering only the hydrostatic part of the static stress system associated with the kinked dislocation line.

In the temperature range well above the saturation temperature T,; for the dilute interstitial alloys the approximation

3

= ( ~ ~ ~ , ) ~ I b x p ( - u f / 2 k B ~ )

[IRsi

exp(-4/kBT)l -I1' (17) consistent with the classical Boltzmann distribution (non-degenerate case) is possible. For the effective activation enthalpy associated with the motion of kinks moving uniformly in the atmosphere of mobile interstitials,

HK," -d In Bk/d(l/kBT), (18)

we obtain

HK,., kBT

+

H;

+

M*, (T > T,J (19) or

HK.., = - k i T

+

H;

+

HiB, ( T > T,.). (20) Eq. (19) holds if the interaction between the interstitials and the kinked dislocation lines may be treated as delocalized, eq. (20) if we may treat it as localized. Below the saturation temperature we obtain for a localized static field

In (19-21) H M denotes the migration enthalpy and

~ f 3

the binding enthalpy of interstitials to the dislocations.

C8- 172 JOURNAL

DE

PHYSIQUE

According to Seeger's theory of the Snoek-Kbstpr extended later by Ogurtani and seeger6 by employing the proper expression for the viscous drag coefficient BK, the apparent activation enthalpy may be written as

HsK = HK,V + 2HK

-

kBT (22)

when NKL << 1, o r as

HsK HK, v + HK

-

kBT12 (23)

when NKL >> 1. Here L denotes the separation of unsurmountable obstacles for the kinks. 2HK the kink-pair formation enthalpy and NK the equilibrium density of thermal kinks. The result for apparent activation enthalpy of the Snoek-Kdster relaxation for a localized interaction potential reads

HSK = 2HK

+

^H~

+ pig -

^2kBT. NKL << I, (24) or

HSK = HK

+

H?

+ 4 -

^{3kgT/2, NKL >> I , (25)}

provided the peak temperature is well above the saturation temperature. These equalions are identical with the results given earlier9.

The present theory may be generalized to account for the occupancy of tetrahedral sites and -

for jumps to more distant sites. It covers thus a very wide range of experimental conditions.

We emphasize, however, that in order to study fine details and to recognize the full impact of the nonlinear theory on the interpretation of experiments, one has to perform thorough computer modelling studies using realistic interaction potentials.

REFERENCES

1. T. ~ x u r t a n i and A. Seeger, J. Chem. Phys.

z.

5041 (1983).

2. T. ozurtani and A. Seeger, J. Appl. Phys.

s,

3867 (1983).

3. T. Ogurtani and A. Seeger, J. Appl. Phys. 62, 852 (1987).

4. T. 01urtani and A. Seeger, J. Appl. Phys.

a,

2857 (1984).

5. T. Ozurtani and A. Seeger, J. Appl. Phys. j7, 193 (1985).

6. T. 01urtani and A. Seeger. J. Appl. Phys.

3,

4102 (1985).

7. T. Ogurtani and A. Seeger, Phys. Rev. B

x.

5044 (1985).

8. A. Seeger, phys. stat. sol. (a)

s,

369 (1979).

9. A. Seeger, M. Weller, J. Diehl, Zheng-Liang Pan, Jin-Xie Zhang. and Ting-Sui KG, Z.

Metallkde.

22,

1 (1982).