The chemical stress revisited

Scripta

METALLURGICA

Vol. 6, pp. 1 1 1 3 - 1 1 1 6 , 1972 P r i n t e d in the U n i t e d States

Pergamon

Press,

Inc

THE CHEMICAL STRESS REVISITED*

H. Wiedersich Argonne National Laboratory Argonne, Illinois 60439 (Received

October

2,

1972)

Confusion appears to persist regarding the "chemical stress" and the closely related "driving force" for mass transport in Nabarro-Herring creep.

Herring(l) has recently

clarified the latter by showing anew that the expressions he derived more than 20 years ago are indeed correct despite a number of criticisms that attempted to prove otherwise. Now Hesketh(2) in a comment on a note by Wiedersich and Herschbach on the chemical stress caused by vacancy and interstitial supersaturations(3) revives some of the arguments that led to the confusion.

The confusion arises mainly from the lack of a clear distinction

between bulk and local defect equilibria and, consequently, the proper use of both atomic and defect-formation volumes. The primary role of the local conditions at sinks and sources in Nabarro-Herring creep, or in a useful concept of the Bardeen-Herring chemical stress, must be emphasized.

If a

cube of edge length %1 = ~2 = ~3 is subjected to three (unequal) normal tractions o11, 022 , and 033, the work expended during transport of a volume dV = -dE I (= dE2

E2 " %3

~3 " hi) from surface i to surface 2 is obviously -dEI(E 2 ~3 °11) + d£2(£3 %1 022)

= dV (Oll - 022) or, expressed as work/atom, it is ~A (°11 - 022) with ~A the volume/atom. The latter work is precisely the amount of energy/atom (or "driving force") available for transport of atoms from surface i (source) to surface 2 (sink).

Note that no reference has

been made to the transport mechanism, e.g., vacancy diffusion, interstitial diffusion, or vapor transport.

Thus, the volume involved in the overall driving force is RA and not a

volume that characterizes a specific defect such as the vacancy (~V) or interstitial formation volume (~i) . (Hesketh's expression "activation volumes" for ~V and Ri are misnomers, in my opinion, The desire to link the overall driving force, i.e., ~A (°11 - 022)' to specific ~ransport mechanisms, e.g., atom transport by an opposite flow of vacancies, has led to the assumption that local equilibria of defects at the sources (surface i) and sinks (surface 2) are established.

The differences in values for the proper chemical potentials at sources

and sinks establish a diffusion field for mass flow.

Work supported by the U.S. Atomic Energy Commission.

1113

For chemical stress as well as for

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THE

CHEMICAL

STRESS

REVISITED

Vol.

6, No.

ll

Nabarro-Herrlng creep, the linking of the local forces on the sink interfaces with the local defect concentrations c is of primary importance.

Again using vacancies as an example, we

have, according to Herrlng(1) By " (ann - ~H ) nA = kT £n{Cv/Cov (OH)} ,

(i)

where Snn is the normal traction at the sink interface, and a H z ~1 (all + a22 + o33), which is the hydrostatic component of the stress tensor inside the body near the sink.

On the

rlght-hand side of Eq. (i), the approximation of unity activity coefficients for the defects has been made. The concentration c is the equilibrium concentration of defects in the ov bulk of the body under the imposed stress condition. Cov(OH) will, of course, have the form of Hesketh's Eq. (i) and, in particular, will contain the defect-formatlon volume fl and the v hydrostatic component of the stress oH. For interstltlals, an entirely equivalent expression applies

~i = - (°nn - oH) ~A = kT £n{ci/Coi(~H)}

,

(2)

where the sign of the work term has changed because the sink interface moves in the opposite direction when forming interstitials.

Thus, for a given (Onn - oH) , the local interstitial

concentration at the sink interface c i will deviate from its equilibrium bulk value Coi(SH) in the opposite direction as cv deviates from Cov(~H).

This holds regardless of the sign of ~i'

i.e., whether the volume of a crystal contracts or expands as a whole when atoms are removed from the surface and placed on interstices inside the crystal. The lo~al quantity (Onn - ~H ) , denoted symbolically as s in Ref.

(3), may be considered

as the negative of the chemical stress (or perhaps more properly designated as chemical force per unit area of sink interface) because it can prevent the net precipitation (or formation) of defects at the sink (or source) when a defect supersaturation (or undersaturation), S ~ C/Co(OH), exists relative to the bulk equilibrium concentration under the

pre~alZ~

8ire88 8~%e.

Note that this definition of a chemical stress includes both

magnitude and direction of a force exerted on the sink interface.

The direction of the

force is normal to the sink interface for two-dimensional sinks and is normal to both the Burgers vector and the line element for dislocations. A

concept of a chemical stress based on the hydrostatic stress required to make the

equilibrium concentration of defects throughout the bulk equal to the actual concentrations, as Hesketh(2) seems to suggest in the first two pages of his note, does not provide any information with regard to the "forces" exerted by excess defect concentrations on specific sinks.

For example, it is useless for calculating the equilibrium gas pressure of a bubble

(or its equilibrium size if the gas content is fixed)

at a prescribed vacancy supersat-

uration; similarly, information concerning the forces exerted on dislocations of various combinations of edge and screw character can not be obtained from this concept.

The reason

Vol.

6, No.

ii

THE

CHEMICAL

STRESS

REVISITED

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for the inadequacies is, oz course, that, under the hydrostatic pressure as defined by Eqs. (i) and (2) of Hesketh's note(2), the concentrations cv and c i are equilibrium concentrations throughout the body and, hence, the defects h a v e n o any sink regardless of its character and orientation.

tendency to precipitate on

The same type of shortcoming occurs

in a chemical stress concept based on hydrostatic pressure when applied to the dynamic case of simultaneous superSaturatlons of interstitials and vacancies. ceptual difficulty arises.

Here, an additional con-

When both formation volumes ~v and ~i had the same sign, e.g.,

positive, no hydrostatic stress exists that could counterbalance, i.e., prevent an excess interstitial influx to sinks.

If one uses local equilibria at sinks for the definition of

the chemical stress, the extension to the dynamic case does not encounter this difficulty, as shown in Reference 3.

References i.

C. Herring, Scrlpta Met. 5, 273 (1971).

2.

R. V. Hesketh, Scrlpta Met. 6, 995 (1972).

3.

H. Wiederslch and K. Herschbach, Scrlpta Met. 6, 453 (1972).