Static displacements and the electrical resistivity of interstitial alloys

STATIC

DISPLACEMENTS AND THE ELECTRICAL OF INTERSTITIAL ALLOYS* D.

W.

HOFFMAN:

MORRIS

and

RESISTJVITY

COHEN8

It appears that the predominant mechanism by which interstitial solute atoms influenoe the electrical resistivity of alloys, unlike substitutional solutes, is through the associated static displacements of the host atoms. A quantitative test of this hypothesis is possible in the case of tetr8gonal iron-carbon martensites, where the magnitude of the statio displacements is experimentally known. The resistivity component arising from the stetic displacements is estimated by 8nalogy to the mean-square thermal displaoements and aorresponding electrical resistivity of pure iron. The result agrees fairly well wit,h various meaeurements on the effect of dissolved carbon on the resistivity of martensitio alloys. The displacement hypothesis 8180 provides a logic81 explanation for the snomelous resistivity increase observed during the aging of iron-orubon and iron-nickel-carbon martensites. Spontaneous rearrangements of the interstitial atoms during aging tend to occur in such 8 manner as to increase the static dlsplacements, as illustrated by theoretical treatme@ of the formation of carbon-atom peirs. The normal aging behavior can be altered, however, by the prior introduction of a dense dislooation substructure through thermomechanical treatment, as in susformed martensites. DEPLACEMENTS

STATIQUES

ET RESISTIVITE INTERSTITIELS

ELECTRIQUE

DES

ALLIAGES

Le m&u&me ptidominant p&r lequel lea atomes en solution interstitielle, B la diff&enoe des atomes en substitution, ont une influence sur la r6sistiviti 6lectrique des alliages semble Btre constitu6 par les d&placements statiques associefl des 8tOmes du r6seau. 11 est possible de v&tier quantitativement c&e hypoth&e dens le c8s des martensites quedretiques for-carbone oti l’amplitude des dbplacements Une estimation de la oomposante de 18 r6sistivit6 rbultant stetiques est connue exp&mentalement. des d6plecements statiques eat effect&e par analogie 8vec les d6plaoements thermiques quedratiques moyens et avec 18 r6sistivit6 Blectrique oorrespondante du fer pup. Le r6sultat obtenu est en bon accord avec lea diff&entes mesures effect&es pour Studier l’influence du carbone dlssous sur la r6sistivit6 des alliages martensitiques. L’hypothBse des d6placements statiques fournit Bgalement une interpr&ation logique pour l’accroissement 8nonn81 de 18 tisistivit6 qui eat observe au oours du vieillissement des martensites fer-oarbone et fer-nickel-carbone. Des &rmngements spontan6s des atomes interstitiels au tours du vieillissement tendent B se prod&e de m8niBre B accroitre les d6plecements statiques, oomme l’illustre la thborie de la formation des paires d’etomes de aerbone. Le comportement normal du vieillissement peut &tre cependant alt&6 par l’introduction pr6fdeble d’un r&au dense de dislocations B l’aide d’un traitement thermomboanique, comme dans les marten&es (d6form6es). STATISCHE

VERSCHIEBUNGEN UND DER ELEKTRISCHE INTERSTITIELLEN LEGIERUNGEN

WIDERSTAND

VOX

Der dominierende Mech8nismus in Bezug auf eine Beeinflussung des elektrisohen Widerstandes von interstitiellen Legierungen ist-im Gegensatz zu substitutionellen Legierungen-die mit den interstltiellen Gastatomen verbundene statische Verschiebung. Ein quantitativer Test dieser Hypothese ist im Falle tetragonaler Eisen-Kohlenstoff-M8rtensite miiglioh, wo man den Betreg der stat&hen Verschiebung experlmentell ernutteln kann. Die durch die statischen Verschiebungen verursachte Widerstandskomponente wird in Analogie zu mittleren quadratischen therm&hen Verschiebungen und zum entsprechenden elektrischen Wider&and in reinem Eisen abgeschiitzt. Die Ergebnissa sind in befriedigender Vberemstimmung mit verschiedenen Messungen des Einflusses von gel&stem Kohlenstoff auf den elektrischen Wider&and mctrtensitischer Legierungen. Die Verschiebungshypothese erlaubt such eine logische Erkliirung filr die beim Auslagern beobachtete anomale Widerstandszunehme in Eisen-Kohlenstoffund E&n-Niokel-Kohlenstoff-M8rtensiten. Spontane Umlaperungen von Zwischengitteratomen b&m Auslagern erfolgen so, daD sie eine Zunuhme der statischen Versohiebungen bewirken; daa konnte durch eme theoretische Behandhmg der Bildung von Kohlenstoffatom-Paaren gezeigt werden. Das normale Auslagerungsverhalten kann jedoah durch eine vorher m einer thermomechanischen Behandlung erzeugte dichte Versetzungssubstruktur (wie in verformten Martensiten) geandert werden. 1. INTRODUCTION

The electrical resistirity of a dilute metallic alloy is generally a sensitive and increasing function of the solute concentration. The widespread use of electricalresistivity measurements to study second-phase precipitation in alloy crystals attests to that basic * Received November 30, 19;‘. t Part, of this paper IS based on a Ph.D. thesis submitted by D. W. Hoffman to the Department of Metallurgy and Materials Science at the 1lasnachusetts Institute of Technology. : Metallurgy Department. Sclcnt ~fic Research Staff, Ford Motor Company, Dearborn, Michigan, U.S.A. 5 Department of Met,allurgy and Materials Science, Massachusetts Institute of Technology, Cambridge, Massachusetts, U.S.A. ACTA METALLURGICA,

VOL. 21, SEPTEMBER

1973

fact. Although the depletion of solute from the primary phase will ultimately result in a resistivity decrease, the first change observed upon aging asupersaturated alloy is often a temporary increase, and may be regarded as anomalous .(l) The present investigation is concerned with the mechanism of these twoopposing resistivity changes as they occur in interstitial alloys. In particular, attention will be directed to the behavior of tetragonal iron-carbon marten&es in which the anomalous resistivity increase is clearly exhibited on aging.‘2) The effect of solutes on the electrical resistivity of a metal comes about through disturbances of the

1215

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METALLURGICA,

periodic potential field associated with the metallic ions among which the conduction electrons move.(3) This perturbation of the periodicity can be considered to arise from; (i) a local change in the depth of potential wells, owing to the charge-misfit of the foreign atoms, and (ii) a local change in the spa&y of the potential wells, owing to the size-misfit of the foreign atoms. Although both factors operate in substitutional alloys, the effect, of charge usually predominates.‘4’ Moreover, while the origin of the anomalous resistivity increase during the early aging of a supersaturated substitutional solid solution is still a subject of controversy, (5) the suggestion that local lattice distortions play a major role in the anomaly@) is no longer given much weight.(1*6) In interstitial solutions, however, the situation appears to be significantly different. Since an interstitial solute atom does not replace an atom of the host crystal, there should be no large effect on the depth of wells in the periodic potential array, although the Nordheim analysis (3) for the dependence of electrical resistivity on substitutional solutes has now been extended to interstitial s01utes.~~’ On the other hand, the extraordinary effect of interstitial atoms in distorting the host crystal is well documented.(s) The purpose of the present report is to examine the possibility that static displacements play an important role in determining the electrical resistivity of interstitial alloys. This approach yields an estimate of the scaling factor treated empirically in Ref.(‘) We shall deal mostly with tetragonal ferrous martensites, for which both the solute distortions and electrical-resistivity changes are rather well characterized. To make the point as simply as possible, we shall confine the argument to an elementary level of approximation. For instance, an analogy between thermal and static displacements will serve as a basis for assessing the contribution of the solutedistortion fields to the electrical resistivity of the virgin (random) solution. It will also be shown that solute redistribution during early aging, such as may occur to lower the elastic strain energy can increase thrmagnitudeofthedisplacementssufficiently to account for the initial rise in resistivity. Finally, we shall mtcrpret, the effect of mechanical treatment (ausformmg) on aging behavior in the light of this line of reasoning. 2. EXPERIMENTAL

OBSERVATIONS

The two aspects of the electrical resistivity of alloys alluded to above are evident in the behavior of tetragonal ferrous martensites, for Fig. 1 illustrates a typical result. The alloy

dilute aging which under

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AUSFORMED BY 5%REDUCTION

UNAUSFORMED

-200

-100 0 TEMPERING

300 100 200 TEMPERATUREK)

Fro. 1. Resistivity of 100 per cent martensite measured at - 190°C after 3 h of tempering. Fe-23% Ni-0.4 % C. contains 0.4 wt. % carbon and 23 wt. % nickel, the effect of the nickel being to depress the martensitic transformation temperature (M,) below the ambient to about -50°C so that inadvertent aging can be avoided. Curve 1 is for regular martensite formed from annealed au&en&e, while curve 2 indicates the effect of deforming the austenite (ausforming) prior to the subambient cooling. The results in Fig. 1 have been corrected for the presence of retained austenite.‘@ The overall decrease in electrical resistivity during aging reflects the removal of the interstitial carbon from solid solution by the precipitation of metal carbides,* and is therefore a measure of the basic effect of the solute on the transport of conduction electrons. The initial increase in resistivity brought about by low-temperature aging reflects the early stages of solute redistribution, and is apparently more affected by the ausforming treatment than is the overall decrease. We direct attention first to the overall decrease in electrical resistivity, which appears to be completed as far as carbon removal is concerned when the aging temperature exceeds 300°C. The magnitude of the observed overall decrease of 7.84 @ cm (8.55 pcla cm in the ausformed alloy) must be corrected for the presence of iron carbides in the fully aged structure in order to obtain a more exact picture of the effect of dissolved carbon on the electrical resistivity of the martensitic phase. Assuming the final carbide to be a coarse dispersion of cementite particles, the volume fraction of this phase will be about 7 per cent: if the carbide is taken to be

examination

l Part of the first decrease in resistwity may be due to the trapping of carbon atoms eb dlslocetions, but we are interested here in the overall decrease to the end-point of aging or tempering.

HOFFMAX

AND

COHEN:

RESISTIVITY

essentially nonconducting compared to the matrix, the overall resistivity decrease in the conducting phase then becomes 9.3 fi cm for the normal martensite and 10.0 ,I,& cm for the ausformed martensite. By comparison, the magnitudes of the maximum resistivit’y increases during early aging are 1.4 ,uQ cm and 0.33 ,uQ cm in the normal and ausformed martensites respectively, no correction for the presence of carbide particles being required at this stage of tempering. 3. ELECTRICAL RESISTIVITY STATIC DISPLACEMENTS MARTENSITE

OF

INTERSTITIAL

12

ALLOYS

I

I

1217

I /

PURE

&

IRON

aqh

@cm 4-

FROM IN 0

Inasmuch as the carbon content of iron in thermodynamic equilibrium with cementite is negligible for present purposes, the corrected reduction in the electrical resistivity of tetragonal martensite, caused by tempering at 3OO”C, measures the influence of the interstitial solute on the resistivity of the unaged material. We now seek to estimate the fractional contribution of the corresponding static displacements to the observed carbon-dependent resistivity. Here, we employ an analogy between static and thermal displacements, as adopted for the analysis of X-ray diffraction phenomena by Herbstein et aZ.“O) Precedent for the use of mean-square static displacements to interpret electrical-resistivity observations has been provided by Basinski et al.(“) in their study of the effect of dislocations in pure metals. These concepts suggest a simple proportionality:

(1) where the ratio of thermal resistivity (pa) to meansquare t’hermal displacement (2) calibrates the relation between mean-square static displacement (2) and the electrical resistivity (pat) caused thereby. That electrical resistivity should be proportional to the mean-square thermal displacement is the essence of Wien’s hypothesis. (3) While this relationship is admittedly an approximation, its acceptability is illustrated in Fig. 2 for pure a-iron. Values for the electrical resistivity were taken from the compilation of Ref. 3. The mean-square thermal displacements were computed from the Debye integra1:“2)

0

(2)

u. being the zero-point quantum vibrations, M the gram atomic weight, T the absolute temperature, v a dummy integration variable and 8,1 the “diffrac-

I

3

4

Fm. 2. Electrical resistivlty of pure iron plotted as a function of the mean-square thermal displacement, computed at the indicated temperatures by means of the Debye integral (equation 2).

tion” Debye temperature (409’K, Ref.(13)). It may be noted that an extensive linear region appears in the plot of equation (2). Deviation from linearity above 300°K signals the onset of appreciable magnetic disorder.‘3) For the calibrating factor in equation (1) we divide 9.8 ,uQ cm, the resistivity of pure iron at room temperature (295”K), by 0.00325 A2, the corresponding value of the mean-square thermal displacement, this ratio giving in essence the linear slope of the illustrated curve. The X-ray diffraction measurements of Moss”~) on martensite containing 1.33 wt. % carbon furnish the experimental determination of 7uSt needed for estimation of pst through equation (1). The extreme magnitudes of the principal static displacements around an interstitial carbon in iron prevent a true evaluation of the mean-square displacement by use of first-order attenuation theory.“s) At the same time, it becomes possible to deduce an effective value of the mean-square displacement more appropriate for the purpose at hand. From the attenuation of Bragg peak intensities in the X-ray diffraction spectrum, one determines experimentally a quantity L,,, related to the magnitudes and directions af the atomic displacements. When the displacements are small, as for thermal displacements, we have to a close approximation:(l4)

J&l = (units of A2)

I 2

I

2

%l

k&l

where z is the mean-square displacement in the [hkl] direction, and k,,:nkl is the corresponding diffraction vector. When the peak displacements are sufficiently large, as for some static displacements in an interstitial alloy, higher-order terms become significant

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on the right-hand side of equation (3) and the simple proportionality of the mean-square displacements to L,,, is lost. Nevertheless one can define an eflective value :

(G&&f= +

htl

(4)

of mean-square thermal displacement that would This effective cause an equivalent attentuation. value should, therefore, be a suitable quantity for comparing the influence of static and thermal displacements on the electrical resistivity. Using equation (4) we compute from the data of Mosso4) effective values for the mean-square displacements in directions of the crystal axes : 0.0371 As 0.049 = (zo),ft. = -(4_402)2 = 0.00235 A2

(z&r

(5a)* (5b)

for martensite with 1.33 wt. % carbon. Spherical averaging then gives the effective mean-square displacement along any direction in a polycr_ystalline sample : z(1.33

wt. % C) = +[0.0371 + 2(0.00253)] = 0.01405 A”.

(6a)

The effective displacements in dilute alloys of other compositions can be estimated by linear scaling to the composition of interest?, e.g. : C(O.4

wt. % C) =

Inserting this value for 2

(

0.4 9.01405 A2. 1.33 )

in equation (l), along with

the previously obtained values of pth and ~7, Pst =

(6b)

yields

(&J (+g) 0.01405

= 12.7 $L! cm

(7)

which is in reasonable agreement with the values of 9.3 p&l cm and 10.0 ,I& cm deduced in Section 2 from the overall decreases in resistivity on aging. That the estimate for pst exceeds the observed resistivity changes may in part be a consequence of the fact that Moss’ specimens experienced some room-temperature aging * The value of 0.640 for L,,,, derived from Moss’ Table 4’1di corresponds to his subsequent determination of 0.63 A for U”y’The scaling factor goes more properly as C,(l - C,)/ C*(1 - C,), where C, and C, are the stomio ratios of carbon in the alloys of interest. For dilute alloys, this is essentially CJC, and approximately (w/O),/(W/O)~.

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during observation. The rationale for this explanation will be developed presently. In the meantime, we can conclude that static displacements do account for a large portion of the electrical resistivity caused by the presence of interstitial carbon atoms in virgin martensite. Further corroboration of this conclusion is available from the work of King and Glover,f2’ who measured 21.9 ,uQ cm/wt.% C as the carbon dependence of resistivity in a series of quenched iron-carbon alloys, and from the observations of Shermanos) whose measurements indicate a value of 34.1 &l cm/wt. % C for a series of virgin martensites in Fe-23 Ni alloys. Cancelling the factor of 0.4 from equation (7) gives a corresponding estimate of 30.3 $J cmjwt. % C for the calculated carbon dependence of resistivity due to static displacements. Accordingly, the calculated result falls within the experimental range. 4. SPONTANEOUS INCREASE STATIC DISPLACEMENTS

OF

Having linked the electrical resistivity of ironcarbon martensites with the static displacements of matrix atoms caused by the presence of the intersitital solute atoms, it is appropriate to consider the first changes that occur when the temperature is elevated sufficiently for the interstitial atoms to migrate. With rising temperature of aging, the resistivity (measured at -196°C) increases. reaches a maximum, and then undergoes a much larger decrease.* Ideally one desires a theory capable of predicting the continuous course of such events, and it is possible that the static-displacement hypothesis could eventually serve as the basis for a development of this nature. Considerations would have to include a detailed treatment of the spontaneous course of solute redistribution (as driven to a large extent by the decrease in elastic free energy) and the diffraction of moving electrons by the prevailing static lattice displacements. Elements of both of these considerations are known, but such a detailed treatment is beyond the scope of the present work. Instead, we shall be content to show that during early aging the interstitial solute atoms will inevitably redistribute so as to increase the static displacements, and that the magnitude of this increase, as reflected by the change in mean-square displacement, is ample to account for the observed rise in electrical resistivity. The proposition that the spontaneous rearrangement * Some workers’iB’ report a small transient decrease in electrical rasistivity when vngin martensite is heated above about -100°C. While the cause of this behavior is not well established, it clearly precedes the onset of ordinary carbon diffusion. The experiments of Fig. 1 followed a low temperature pre-aging treatment designed to eliminate this effect.

HOFFJIAS

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rising lattice force meets the declining solute force at point %. The relaxation energy corresponds to the triangular area ABC (the difference between ABCD and ACD), and is equal to one-half the area of the enclosing rectangle ABED, as given by equation 8. The elastic free energy (F) of solution takes the form :(l*) where W (the “initial energy”) is the free energy of the alloy crystal in a hypothetical reference state where the host atoms are constrained from any displacement,s even in the presence of interstitial atoms. It is evident. then, that static displacements are the means by which an alloy crystal reduces its elastic free energy and. moreover, that further reduction of the elastic free energy requires increased displacements. This is illustrated by the primed case in Fig. 3 ; a state of lower elastic free energy (larger relaxation energy) is attained if the equilibrium displacement of a host (8) atom is increased from u to u’. This conclusion is the basis for our suggestion in Section 3 that Moss’ S-ray in the linear-force approximation (i.e. the variation of diffraction results lead to an estimate of pst somewhat solute and lattice forces are assumed linear with larger, because of inadvertent room-temperature displacement). While u designates an equilibrium aging, than the true value for the virgin martensite. displacement, &, is not an equilibrium force, but rather So far, we have not described how an increase in the a much higher value that the solute force would displacements could actually come about. Suffice it to approach if the host atoms were constrained to their say that the resolved lattice force which builds up in perfect lattice (unrelaxed) positions.(17) opposition to the solute force from a given interstitial Interpretation of equation (8) is clarified by reference will be affected by the presence of other interstitial to Fig. 3, where the displacement of a host atom solutes in its vicinity. In some configurations, the adjacent to an interstitial is depicted. While the gross lattice forces bearing against one interstitial &torn energy released by the solute force is represented by become shared partly by another, and the slope of the the trapezoidal area (ABCD) between the solute- resolved-force line is then lowered from AC to AC’, as force line and the horizontal axis out to some equilib- indicated in Fig. 3. Obviously the thermodynamic rium displacement (u), a portion of this energy is driving force favors those redistributions which will taken up in displacing the surrounding atoms, and this increase the equilibrium displacements and thereby portion is represented by the triangular area ACD lower the free energy. under the line for the resolved lattice force. Equilib5. CARBON PAIRING-A PROTOTYPE rium is established at the displacement (u) where the FOR AGING

of the interstitial atoms occurs so as to increase the static displacements of the solvent atoma may appear paradoxical at first sight. Clearly these static displacements require an input of energy which is stored by distortion of the bonds among the metal atoms on the host lattice, and which surely denotes an increase in the elastic energy of distortion. The question naturally arises as to how such an event can occur spontaneously, i.e. with a decrease in free energy. The answer resides in the fact that the static displacements of the host atoms take place because they are being pushed by solute forces from the interstitials. The work done by the solute forces during these displacements is greater than the elastic energy stored by distorting the host lattice. The difference is the so-called relaxation energy (R) given at equilibrium by :

F=W-R

(9)

Force

SOLUTE

FORCE

RESOLVED

FIG. 3. Force/displacement relations for solute and lattwe forces acting on a host atom in an interstitial alloy.

To illustrate the foregoing principles, we consider specifically the gathering of isolated interstitial carbon atoms into pairs. Other investigators(19-21) have already reported that elastic interactions favor the formation of interstitial pairs at neighboring sites in a given octahedral sublattice. Moreover, it is reasonable from the standpoint of kinetics to select carbon pairing as a prototype aging process. The average spacing of carbon atoms at a concentration of 0.4 wt. % in ferrous mertensite isabout lOA. Referring to Table 2 of Cohen’22) for carbon diffusion in ferrite and martensite, one sees that interstitials forming a pair could each migrate about half the average intervening distance in 3 hr at 0°C (the approximate time and temperature to achieve the peak resistivity in Fis. 1).

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The lattice Green’s-tensor method, reviewed recently by Magee et al., (1’) facilitates computation of the changes in energies and displacements caused by pairing. A linear approximation to the carbon-iron force law is also provided by those authors. Details of the computation are given in Appendix A. For the s&e of simplicity, consideration is arbitrarily restricted to first-neighbor solute forces, although the carboniron force law applies in principle to second neighbors as well. The Green’s-tensor components, however, incorporate iron-iron interactions to fifth neighbors, as determined experimentally by Brockhouse et aZ.f23) Values obtained for the first-shell displacements, the partial molar energy of relaxation, and the meansquare static displacements are listed in Table 1 for interstitials in both the isolated and paired configurations.* TABLE

I. Comparison of isoleted-carbon 8nd paired-carbon configuretions in iron Dipole distortion L‘v

Isolated Paired Difference

0.926 0.956 1-0.030

Meen-squ8re displacements (A’)

Relexation energy (eV/C 8tom) 5.15 5.32 +0.17

0.260 c(1 - c) 0.320 c( 1 - c) +0.060 c(1 - c)

e = atomic ratio of carbon in martensite.

The heading “Dipole Distortion” in the first column indicates that the displacements listed there refer to the change &n separation of the two first-shell iron atoms on opposite sides of an interstitial carbon atom. This nomenclature is chosen because the individual static displacements of the iron atoms become unsymmetric in the paired configuration, as the arrows of Fig. 4 are intended to show (drawn to scale). In the latter case, the carbon interstitial itself is displaced in order to remain centered between the two iron atoms, as dictated by the balance of forces. The relative displacement of the host and interstitial atoms, plotted in Fig. 3, is therefore one-half the dipole distortion. One sees that the dipole distortion and the relaxation energy do increase simultaneously, as expected, with the increase in relaxation energy being tantamount to a decrease in free energy. The carbonpairing process increases the mean-square displacement by some 23 per cent. At 0.4 wt. % carbon, 23 per cent of the carbon contribution (9.3 $2 cm) to the resistivity of virgin martensite is 2.1 ,uQ cm; the observed increase in resistivity during early aging * These values

are intended for comparative purposesonly; mterested m the more accurate velues obteineble for the isolated carbon mterstitial by molusion of second-neighbor SOlute forces are referred to Magee et aZ.(irJ

readers

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[o? .-;,

tJ

@

,.-.

‘._ )

b

Fro. 4. Displacement of atoms in the (110) plane caused by aarbon interstitials in 8 paired configuration.

(1.4 fi cm) is, therefore, within reach of the staticdisplacement mechanism. We do not suggest that the carbon atoms in martensite actually approach a totally paired state at any time in the course of aging. The aim of this discussion is merely to demonstrate plausibility, Le. to offer arguments by consideration of an example that changing static displacements during aging may, indeed, account for a resistivity increase of the observed magnitude, concomitant with a decrease in elastic free energy. It should be understood that the displacement hypothesis, as discussed here, provides no explanation for the resistivity maximum (in contrast to the increase itself). As aging proceeds, the mean-square displacement is expected to be an ever-increasing function of time, if coherency of the crystal lattice is maintained. In this light, it is tempting to associate the resistivity peak with a loss of coherence or with a competitive loss of carbon atoms to dislocations. The true explanation, however, may well be more subtle than that, considering the likelihood that the mean-square displacements alone become progressively inadequate for measuring the effect of static displacements upnn electrical resistivity when the solute distribution becomes increasingly non-random. Undoubtedly the scale of the developing nonuniformities or carbon clusters plays an important role, requiring a more sophisticated approach than the zero-th order method employed here. For this, one might hope eventually to draw on the theories of the thermal contribution to electrical resistivity as developed by Bloch and others.‘=)

HOFF?vfAhT 6. SUMMARIZING

AND

COHEN:

RESISTIVITY

DISCUSSION

Having dealt mostly with the electfical-resistivity behavior of regular martensite, it is now possible to draw some conclusions about the ausformed martensite, also represented in Fig. 1. We have interpreted the overall decrease in resistivity caused by tempering the subject alloy at 3OO’C to reflect the removal of essentially all the carbon from solution, and have found the observed resistivity change to be in fair agreement with the theoretical estimate for 0.4 wt. % carbon. Since the overall decrease observed for the ausformed martensite is slightly greater than for the normal martensite, we conclude that the prior deformation of the austenite (70 per cent reduction in area at 225’C) did not have the effect of precipitating the carbon at that stage. This is significant in light of previous evidence showing that fine carbides may form if the austenite is deformed at sufficiently high temperatures and contains strong carbide-forming solutes.(25) While ausforming does not affect the dissolved carbon concentration, the resultant substructure does appear to modify the initial course of aging. The resistivity peak for the ausformed martensite is only one-fifth that observed for the regular martensite. To all intents and purposes, ausforming virtually eliminates the initial resistivity increase in this 0.4 wt. % carbon martensite. From the vantage point of the earlier discussion, we interpret this to mean that interaction among the dissolved carbon atoms ceases to be the prime driving force for carbon redistribution such interactions would tend after ausforming; inevitably toward increased static displacements. Hence, the normal course of events during aging must be superseded by some overriding mechanism giving rise, if anything, to reduced static displacements. An obvious candidate is the trapping of the interstitial carbon atoms by the high density of dislocations in the ausformed substructure. The effect of carbon segregation to dislocations in lowering the electrical resistivity caused by interstitial carbon has been documented for low-carbon martensite by Speich.(26) On the other hand, the principle proposed above remains in operation; accommodation of the solute forces by the elastic strain field of a dislocation should, in effect, cause increased local displacements. The resolution of this apparent contradiction occurs, perhaps, when the interstitial enters the anomalous region of the dislocation core. Alternatively, the scale on which this segregation takes place may operate like carbon depletion, and thus result in only a minimal inorease in mean-square displacements during low-temperature aging. 3

OF INTERSTITIAL

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1221

In:summary, the following points merit emphasis : 1. Static displacements appear to be responsible for a major part of the electrical resistivity caused by the presence of interstitial carbon in ferrous martensites in particular, and probably by interstitial solutes in metallic alloys more generally. 2. Spontaneous rearrangement of interstitial solute atoms during the aging of a supersaturated crystal will, in reducing the elastic free energy, lead to increased static displacements. A natural explanation for the so-called anomalous resistivity increase during early aging thereby becomes available. 3. Quantitative evidence in support of the first point is gathered by using an analogy between static and thermal displacements to est,imate the component of electrical resistivity caused by static displacements in virgin iron-carbon martensites. 4. Quantitative evidence in support of the second point is obtained by examining, as an exercise, the energetics of carbon-pair formation and the associated changes in static displacements. 5. Thermomechanioal treatment, while not affecting the concentration of interstitial carbon, suppresses the aging process that leads t,o increased static displacements during the aging of ausformed iron-nickel carbon martensite. ACKNOWLEDGEMENTS

The authors are grateful to the following institutions for the sponsorship of ongoing research at M.I.T., of which the present investigation was a part: United States Air Force, Bethlehem Steel Corporation and Office of Naval Research. REFERENCES 1. A. KELLY and R. B. NICHOLSON, Prog. Mat. Sci. 18, 149 (1963). 2. H. W. RING and S. 0. GLOVEB,J.I.S.I. 198, 123 (1969); P. G. WINCHELL and M. COHEN, Trans. ASM 66, 347 (1962). 3. G. T. MEADEN, Electtil Reatitance of Metals. Plenum Press (1966). 4. P. DE FACET DE CASTELJNJ and J. FRIEDEL, J. Phye. Radium 17, 27 (1966). 5. J. C. GRIFFIN and M. C. JONES, Scsiptu Met. 8,415 (1972). 6. A. H. GEISLER,in Phaee TranaformationednSoiida, p. 387. (Edited by R. S~o~ncaows~~) John Wiley (1961). 7. J. A. PRYDE and I. S. T. TSON~, Acta Met. 19.1333 (1971). 8. H. A. WRIEDT and L. ZWELL, Tram. M&d. Sot. A.I.M.E. 224, 1242 (1962): E. J. FASISKA and H. WA~ENBLAST, Tram. MetaZSoc. A.I.M.E. a89, 1818 (1966). 9. D. W. HOFFMAN, Ph.D. Thesis, M.I.T. (1966). 10. F. H. HERBSTEIN, B. S. BORIE, JR. and B. L. AVERBACH, Acta Cryutdlog. 8,466 (1966). 11. Z. S. BASINSKI, J. S. DU~DALE and A. HOWIE, Phil. Mag. 8, 1989 (1963). 12. R. W. JAMES. The Owtical Princiwlea of the DiffracGon of X-rays, p. i20. Belf (1960). * w 13. F. H. HERBSTEIN and J. SMUTS, Phil. Nag. 8, 367 (1963). 14. S. C. Moss, A& Met. 16, 1816 (1967). 16. M. A. KRIVOOLAZ, Phys. &f&. Meldlog. 7, 11 ( 1960). 16. A. M. SHERMAN, Sc.D. Thesis, M.I.T. (1972). 17. C. L. MA~EE, D. W. HOFFMAN end R. G. DAVIES, Phil. Mag. 25, 1631 (1971).

ACTA

1222

METALLURGICA,

18. D. W. HOFFMAN, Acta Met. 18,819 (1970). 19. R. A. JOHNSON, G. J. DIENES and A. C. DABUSK, Acta Mel. la, 1215 (1964). A. G. KHACHATURYAN, Sov. Phys. SolidSt. 9.2249 (1968). J. W. FLOCKEN, Phys. Rev. B, 4, 1187 (1971). M. COHEN, Trane. Am. Inst. Min. Engrs. a,636 (1962). B. N. BROCKHOIJSE, H. E. ABOU-HELAL and E. D. IIALLMAN, Solid State &mm. 5,212 (1967). 24. J. M. ZIMAN, Electrona and Phonona. Clarendon Press (1960). 25. I-LIN CHEXG and G. THOMAS, Met. Trans. 8, 503 (1972). 26. G. R. SPEICH, Trans. Am. Inat. Min. Engw. e45, 2553 20. 21. 22. 23.

(1969).

APPENDIX. COMPUTATION OF RELAXATION ENERGY AND EQUILIBRIUM DISPLACEMENTS FOR ISOLATED-CARBON AND PAIRED-CARBON INTERSTITIALS IN IRON

(Al) Magee et al. use the Green’s_tensor method to determine a slope for the carbon-iron force law. Slight revision, according to the improved values of Table Al, yields & = 17.80 - 24.78~~~ (dynes x lOa) (A2) for the radial forces on the first-shell iron atoms surrounding a carbon interstitial, the radial displacement u,~ being in units of angstroms. This is, in fact, the force law depicted in Fig. 3. Forces on the secondshell iron atoms are determined by inserting (0.59 + u,J in place of u,,, 0.59 A being the difference in undisplaced radii of the first and second shells. The computation to be performed here is greatly simplified

(J-8’ (X/dsn)

!/ i

8-8’ GY;“’

973.55 0 o

/

Lattice

0

0

97i.75

1973

by omission of the second-neighbor solute forces, the error thereby incurred being insignificant for the present purpose. Consider the case of an isolated carbon interstitial in an octahedral site located at [OO+] in the bodycentered iron crystal (indices in terms of unit cell dimensions). Iron atoms at [000] and [OOl] will be displaced by equal amounts away from the carbon atom, forming a “distortion dipole.” Equation (Al) yields the magnitude of the displacement of either iron atom, say,

287.93 k69.41 169.41

0, 1, 1

+ Ggl &”

(A3)

where the equilibrium forces are equal and opposite, and given in turn by equation (A2) : $8”’ = 17.8 -

24.78up’

(dynes x 10”‘).

(A4)

Solving for the force : 17.8 x 10”’

g-J’ =

1 + 24.78 (GE0 -

@‘)

(A5)

x 10-b

gives, with the help of Table Al, a value of 6.33 x 10-d dynes and, by substituting back into equation (A3), a displacement of 0.463 A or a dipole distortion of 0.926 A. The relaxation energy is obtained by reference to equation (8). For first neighbors, the unrelaxed solute forces $,, have the magnitude 17.8 x lo-’ dynes given in equation (A2), and the two first neighbors of the isolated interstitial are displaced 0.463 A parallel to these forces as just determined. Thus ;, the relaxation energy is computed as : R = &2

x

17.8

x

lo4

x

0.463

x

lo-s)ergs

w

= 5.15 eV. with appropriate conversion of units. Consider now a pair of interstitial carbon atoms at [00&l and [4$1]. The nearest iron atoms are at [000], [OOl], [*$+I and [$!&I. Although these four iron atoms are first neighbors of the two carbon atoms,

Green’s tensor for b.c.c.

0 973.75

21,

uiol = qg” @”

The lattice Green’s_tensor method, reviewed recently by Magee et a1.,(17) is well suited for illustrating the effect of carbon pairing on the elastic energy and associated static displacements in iron. The Green’stensor component of GfjT8’simply gives the displacement of a host atom at site s in the i-th direction caused by application of a unit force to an atom at site s’ in the j-th direction. Green’s_tensor components of improved accuracy are listed for iron in Table Al. When forces 4;’ are exerted on more than one host atom, as in the vicinity of single or grouped interstitials, the total static displacement uia at s is ascertained by summing the individual contributions :

TABLE Al.

VOL.

iron extrapolated

for infimte cell

k69.41 287.93

i-69.41 69.41

220.87 0

0 220.87

0

69.41

287.93

0

0

24i.12

4, t, +

1. 1. 1

140.51

0

0

136.66

13.4

21.23

149.21

39.29

39.29

:

176.14 47.01

47.01 176.14

13.4 21.23

136.66 21.23

21.23 145.5

39.29 39.29

149.21 39.29

39.29 149.21

(A/dyn)

l Sign of (8 8’) elements elements are always positive.

influences off-diagonal

elements

in corresponding

row and column

as indicated.

Diagonal

HOFFMAX

AND

COHEN:

their displacements are not alike, owing asymmetry of t’he pair configurations : _g”

= $9

# ,go1 = _,gtt.

RESISTIVITY to

the

(A7)

Nevertheless, the solute forces exerted by the carbon interstitials on these first-neighbor atoms are all numerically equal,

because the carbon atoms will shift slightly to maintain a balance of forces. In other words, the carbon atoms will each be suspended midway between their respective, unequally-displaced first neighbors. Hence, in utilizing equation (A2) to determine the equilibrium solute force, one inserts one-half the difference of the displacements for either of the distortion dipoles: 4 = 17.8 -

24.78 x $(u!“’ -

@“‘)(dynes

x lOA). (A9)

The displacements are, in turn, obtained by explicit variants of equation (Al) :

@" = Go &"' + G&$00 + GE t$ffP + Gg* &"* WV

OF IFTERSTITIAL

ALLOYS

1223

algebraic difference, has the value of 0.956 A. As expected this is larger than for the isolated carbon interstitial, as is also the relaxation energy of 5.32 eV/carbon atom. Having obtained the equilibrium forces acting on the iron atoms, one can, in principle, obtain the meansquare displacement in a straight-forward manner by computing individually the displacements of iron atoms in the second, third and more-distant, shells, and summing their squares. This is, however, a tedious operation, and our approach for act’ual computation of the mean-square displacement is to use the Fourier method of Ref. (1*),wherein the summation over atoms in the real crystal is replaced by a summation over the f&t Brillouin zone in reciprocal space. Briefly, the operation is represented for t,he single interstitial by the equation : 7 %ingle =

h%ng,e

( i$

(Gt3E sin

++)2>k

(Al3)

where GIBEis the k-th Fourier coefficient of the Green’s tensor (also the inverse of Born’s dynamical matrix), (27&J is a component of the wave vector k, and ( )* calls for the mean value in reciprocal space. Similarly,

Simplifying equations (AlO) and (All) by means of equation (A8) and substituting into equation (A9) yields a solution for the solute force: +=

17.8 x lOA

1 + 4(24.78)(2@

-

(dynes)

2G$’ + G;j+ -

Gjj’) x lOA ’ (A12)

With the indicated components from Table Al, C$ is evaluated as 5.95 x 10-4 dynes, and by substitution back into equations (AlO) and (All), the displacements are computed as 0.436 and -0.520 A respectively. The iron distortion dipole, being the

The values obtained in this manner are listed, with the other results obtained above, in Table 1 of the text. Note added in proof Assessment of the scattering of conduction electrons by self-interstitiala in pure copper has been published by A. W. OVERHAUSER and R. L. GORMAN, Phys. Rev. 102,676 (1956). They conclude that static displacements are the principal cause of associated electrical resistivity.