Some aspects of short-range order

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Some aspects of short-range order

J.B. Cohen, M. E. Fine

To cite this version:

J.B. Cohen, M. E. Fine. Some aspects of short-range order. J. Phys. Radium, 1962, 23 (10), pp.749-

762. �10.1051/jphysrad:019620023010074901�. �jpa-00236676�

REFERENCES [1] ^DRAIN (L. E.), ^Bulletin Ampère ^9e année, ^fasc. spé-

cial 425, 1960.

[2] ^CHENG (C. H.), ^WEI (C. T.) ^{and BECK} (P. A.), Phys.

Rev., 1960, 120, ^426.

[3] ^CHILDS (B. G.), ^GARDNER (W. E.) and PENFOLD (J.),

Phil. Mag., 1960, 5, 1267.

[4] ^TOWNES (C. H.), ^HERRING (C.) and KNIGHT (W. D.), Phys. Rev., 1950, 77, ^852.

[5] ^HANNA (S. S.), ^HEBERLE (J.), ^PERLOW (G. J.), ^PRES-

TON (J. S.) and VINCENT (D. H.), Phys. ^Rev. Letters, 1960, 4, ^513.

[6] ^MARSHALL (W.), Phys. Rev., 1958, 110, ^1280.

[7] ^GOODINGS (D. A.) ^{and HEINE} (V.), Phys. ^Rev. Letters, 1960, 5, 370.

[8] ^CHILDS (B. G.), ^GARDNER (W. E.) and PENFOLD (J.),

Phil. Mag., 1959, 4, 1126.

[9] ^DRAIN (L. E.), Faraday Society Discussions, 1955, 19,

200.

[10] ^BACON (G.), ^Acta Crystallographica, 1960, 14, ^823.

[11] ^BARNES (R. G.) and GRAHAM (T. P.), Phys. ^Rev.

Letters, 1962, 8, ^248.

[12] ^JONES (D. W.), ^PESSALL (N.) ^and MCQUILLAN (A. D.),

Phil. Mag., 1961, 6, ^455.

[13] ^VAN-VLECK (J. H.), Phys. Rev., 1948, 74, 1168.

[14] ^DRAIN (L. E.), ^{To be} published.

SOME ASPECTS OF SHORT-RANGE ORDER (*) By ^{J. B. COHEN and} M. E. FINE (**),

Résumé.

Les aspects de l’ordre à courte distance* qui sont discutés ici sont :

a) problèmes expérimentaux rencontrés dans la détermination de l’ordre à courte distance ; b) nature de l’ordre à courte distance ;

.

c) cinétique et mécanisme de l’augmentation de l’ordre a courte distance à basse température

dans les échantillons trempés ;

d) destruction de l’ordre à courte distance par déformation plastique, ^{obtention de} l’équation

fondamentale pour le durcissement dû à l’ordre à courte distance et ^sens de la comparaison ^avec

les résultats expérimentaux.

Abstract.

The aspects ^of short-range order discussed are : a) Experimental problems ^invol-

ved in determination of short-range order ; b) ^{Nature of} short-range order ; c) Kinetics and mechanism for increase in short-range ^{order at low} temperatures ⁱⁿ quenched specimens ; d) ^Des-

truction of short-range order by ^plastic deformation, derivation of the fundamental equation ^for short-range ^order strengthening, ^and significance ^{of the} comparison ^with experimental ^results.

LE JOURNAL DE PHYSIQUE ET LE RADIUM ^il TOME 23, ^OCTOBRE 1962,

Introduction.

Several excellent reviews [1-5]

have been written which deal with local atomic

arrangements in solid solutions. In this paper ^we shall consider primarily developments ^{since these}

and topics ^{that have not been} previously ^dealt

with in reviews. We shall discuss _progress ,in experimental methods, ^{some recent} data, and par-

ticularly ^the role of local order in changes in pro-

perties ^{at ambient} temperatures ^{and in} plastic

deformation.

I. The nature and détermination of short-range order.

In an alloy, ^{local order exists if the num-}

ber of short-range ^{unlike atom} pairs ^is greater ^than (*) This research was supported by the United States Ofiice of Naval Research and the Advanced Research Pro-

jects Agency ^{of the} Department ^of Defense, through ^the

Northwestern Materials Research Center.

(**) J. B. Cohen and M. E. Fine are Associate Professor and Professor, respectively, ^{in the} Department of Materials

Science, ^The Technological Institute, Northwestern Uni-

versity, Evanston, Illinois, ^{U. S. A.}

that in ^a random solution. In the absence of long-

range order, ^the probability ^{for an unlike atom} pair

tends to the random value for large interatomic

distances, ^{20 to 50 A or so.} Short-range ^order

results ⁱⁿ broad, ^diffuse x-ray scattering ^{in the} regions ^where super-structure peaks would appear with long-range order if it occurs. Warren [6] ^has expressed ^{this diffuse} intensity ^{in terms of a} single

Fourier series whose coefficients are the short- range order parameter

Pau ^{is the} probability ^{that a b atom is in the ith}

shell around an a atom, ^{mb and ma are the mole} fractions, ^i.e. the random probabilities, ^{and Ci is}

the coordination number of the ith shell. These

are probabilities averaged ^{over time and} position

in the sample.

^{(XI mb Ci} is the average ^excess

over the random number of b atoms in the ith shell around an a. As with all intensity ^measu-

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphysrad:019620023010074901

rements we can a priori ^learn something only ^about

the states of atomic _occupancy as functions of interatomic distances, ^but nothing ^{about the actual}

location of atoms.

There are several other diffuse scattering ^effects

which ai e present ^{in the} diffraction pattern ^{from an} alloy : (a) Compton ^modified scattering, (b) ^ther-

mal eff ects, ^and (c) scattering ^{due to} différence in atomic size of the atoms. Thermal effects result in (1) ^a reduction in peak intensity, ^and (2) ^a modulation ^{of each ai} by ^{an e7-31 term} [7, 8]. ^The

redistribution of intensity ^from (1) ^{occurs throu-} ghout ^the pattern ^but peaks ^{near the} fundamental reflections. This modulation distorts the short- range order peak. Furthermore, since details of the nature of the vibrations of different atom types

in a solid solution are not known, ^{it is not} possible

to calculate this effect with certainty. It is better to make measurements at low temperatures ^or

near to the origin ^of reciprocal space ^to minimize this error. The most accurate available data to date employed powder methods and measurements at room tempei ^{atur e or} lower. The detailed single crystal ^{data of} Cowley [6] ^on Cu3Au, ^for example,

involved measurements near the 300 position ^at

400°C and higher and, therefore, ^the ais ^are pro-

bably ^{in error} by appreciable ^amounts.

In making measurements at low temperatures attention must be paid ^{to heat treatment of the} specimen. ^In slowly ^cooled specimens ^{an uncer-} tainty ^{exists as to what} equilibrium temperature ^is associated with the measm ed ocis. ^In quenched specimens cognizance ^{must be} given ^{to the} possi- bility ^of changes during ^the quench.

Stiain _{eneigy is} an important factor in deter-

mining local atomic arrangements. Différences in the atomic sizes of the species produce ^diffuse scattering eifects similar to (1) ^and (2) ^{and also lead}

to a slowly varying modulation of the diffuse inten-

Fie. 1.

Schematic representation ^of the contributions from a face-centered cubic alloy crystal ^{to diffuse} x-ray

scattering ^of short-range ^order and atomic size. The shaded area represents ^the depression ^{of the} short-range

order peaks due to thermal vibrations and différence in atomic size.

sity [7, 9]. ^The coefficients of this modulation ^are related to the distances between a atom pairs ^{and b}

atom pairs spaced ^at various interatomic distances.

All of these effects are illustrated roughly ^{to scale}

in figure ^1.

Workers studying ^{solid solutions are indebted to}

Warren and Averbach and their students for their

quantitative treatments and investigations ^{of these}

diffuse effects ; ^{it is a difficult} experimental ^and

theoretical problem ^{to sort them out.} They are, in fact, easy ^to miss even qualitatively except ^with

the most careful techniques.

’

Borie [7] using isotropic ^elastic theory, ^{has shown}

that all the size effects can be expressed ^{in terms}

of a single ^elastic distortion coefficient, ^{so that it}

is possible ^to obtain information on the actual indi- vidual atomic sizes from either the peak depression

or the modulated diffuse intensity. ^{If this} assump-

tion is not valid, the modulation must be known ^as

a function of alloy composition ^{to sort out the dis-}

tortions or separations ^{of the} individual aa and bb

pairs and then with lattice parameters it is pos- sible to calculate the separation ^{in ab} pairs. ^In systems where there is little différence in atomic

scattering ^{factor so the diffuse} scattering ^{is weak it}

is still possible ^to obtain information on individual atomic sizes from the depression ^{of the} fundamental

peaks although ^{this last} concept ^{has not} yet ^been applied ^{in a} systematic study [2]. ^{It should be}

mentioned though ^that only from the diffuse modu- lations can sizes be obtained as a function of inter- atomic distance.

It is clear from the above discussion, particularly concerning ^the temperature ^{and size} effects, ^that portions of the available data are subject ^{to error.}

However, ^the present ^{data have far out distanced}

our understanding of solid solutions. Size effect data have indicated that the différent atom types

in a solid solution may diff er in size from that in the pure ^state and from each other. Recent calcu- lations using elasticity theory ^more general ^than isotropic ^are improving ^our ability ^{to calculate the}

strain contribution to free energy [10,11].

ln addition to important ^strain energy eff ects,

there are electronic effects. Some of these may be of the nearest neighbour type. ^For example, ^the

atomic distances .between nearest Mo atoms in Ti-Mo alloys [12] ^are smaller than in pure Mo

suggesting ^an electronic transfer. In Li-Mg alloys [13] ^{there is a} decrease in the Li-Mg ^dis-

tance suggesting additional bonding. It is inte-

resting ^{to note that these are the} only ^{two cases}

where this effect has been experimentally ^observed

and both alloys ^are body-centered ^{cubic. The} large

deviations from Vegard’s ^law in the Fe-Ni [14] ^and

Fe-Co [15] systeins ^{indicate that a} sudy ^of peak depressions might ^{show a} large change. ^{It is also}

well known that there are long-range ^electronic

factors as well. Perhaps the classic example ^of

tbis is Cu-Pt [16]. ^{In the ordered state the} alloy

consists of alternate 111 planes ^{of Cu and Pt in a} slightly ^distorted face-centered cubic structure, ^so

that there are 6Pt and 6Cu around each atom, just

as would be expected ^{for a random} alloy. ^Further-

more, above the critical temperature, a1 is still zero, i.e., ^{there are still 6Cu} and 6Pt around any atom. Nearest neighbour interactions cannot be

contiolling ^{here. We can} probably expect ^then

that arrangements involving groups of atoms, ^not just pairs, ^are important ⁱⁿ determining ^{the overall} properties ^and thermodynamics of many, if ^not all, systems. Such groups ^must alsô be important ⁱⁿ

the local distortions. Being ^{able to} specify ^the probability ^of finding ^{an a atom at a certain dis-}

tance from a b is not enough ; it is also necessary ^to

specify ^{what atoms are} around the ^a atom, i.e.,

what influence the presence of ^{an a} has on its

surroundings [7]. ^This will, ^of course, influence the strain fields acting ^on b and therefore the actual sizes. Unf ortunately, ^the pair probabilities

or ai,s are the only quantities ^{we can so far}

measure.

As a result we do not really ^know exactly ^what short-range ^{order means} in any system. ^There

seem to be three possibilities : (1) regions ^of high

order and low order, perhaps differing in compo-

sition ; (2) tiny, highly ^ordered antiphase domains,

and (3) ^a liquid-like arrangement ^{of unlike} neigh-

bors around each atom.

The experimental problem ^{is the direct} analogue

of the major ^{barbier in structure} determinations where one can from- the measurements of peak

intensities in general ^determine only ^the proba- bility ^{of atomic} occupancy ^{at various} interatomic distances and not the actual atomic arrangement, simply ^{because it is not} possible ^{to measure the} phase ^{of the} amplitude from any ^{one atom.}

Although ^{a number of methods have been deve-} loped ^to solve this phase proble,n ^{in studies of} structure, ther e has been no cori esponding ^success-

ful effort for diffuse scattering ^{such as is} present

with short-range order. There is, however, ^a

small body of evidence which when considered all

together suggests ^{the nature of} short-range ^order.

a) ^Raethei [17], using ^electron diffraction obser- ved faint _supei lattice ⁱ eflections from thin fil_ns of

Cu3Au ^at temperatures up ^to 100°C above the cri- tical te.nperature. ^These spots ^{appear too} sharp

and intense ^to attribute ^{them to local} order ; they

appear instead ^to be due ^to small highly ^ordered regions. ^There is, however, evidence that the orde-

ring pattern ^{is différent in thin filmes} than in the bulk. Scott [18] ^{observed a} différent diffuse scat-

tering ^at the surface of a single crystal ^of Cu3Au

than in the interior. (This difference, however,

could also be due ^to differences in composition ^at

the surface ^{as a} result ^of etching prior ^{to the orde-} ring treatment.)

b) ^Warren [1] pointed ^{out that the diffuse} peak

in CU3Au ^is displaced ^{to a} higher angle ^{than that} expected ^{from the} short-range ^order theory, ^{as if}

the peak ^was coming from small highly ordered regions ^or antiphase domains which would have a

smaller lattice parameter than the average value.

None of the récent improvements ^{in the} theory ^of scattering discussed above can account for this since they ^{do not} predict any such shift.

c) ^{In a recent} study ^{of Cu-Pt} alloys [19], ^{even at}

37 at. % ^{Pt the} short-range ^{order at} high tempe-

ratures is characteristic of the layer-like ^arran- gement ^{at the} equiatomic composition. oc, and _a3

are small but _OC2 is large ^and negative ^and a4 is

positive. (For ^ordered Cu.Pt (l1 is large and nega- tive and _OC2 positive, ^{while for CuPt} a1 is 0, cx’2 is

negative, a3 is 0, ^and a4 is positive.) ^At high ^tem- peratures a large ^{number of small} antiphase ^do-

mains of the structure of 2CuPt face-centered cubic cells ^{on an} edge ^{with a} long-range order para- meter of about 0.5 can qualitatively explain ^the

results in these alloys. ^At températures ^{closer to}

the critical, a1 increases, suggesting ^{a mixture of} regions ^of Cu3Pt ^{and CuPt.}

If there are tiny ^ordered regions ^{then the breadth}

of the diffuse short-range ^order peak might ^{be due}

to the size of the small regions. ^For example, ⁱⁿ

Cu-14 at. % Al, this size would be the order of 12 À. Let us assume that the. e are highly ^ordered regions ^{of CuAl with a CsCI} type structure. Appro- ximately 1/4 of all the atoms in a unit volume must then be in these particles ^{to account for the obser-}

ved _oc, of - 0.137 [20]. Assuming spherical ^re- gions ^{of 12} À diameter (containing ^{about 70} atoms)

there would have ^to be about 1019 to 102° of these per cubic centimeter (other structures which might

be assumed would yield ^a gieater density). ^Tb.e

matrix computes ^to pure copper. Further calcu- lations indicate that a strong slnall-angle scattering

would result of the order of 5 to 30 _cps under normal conditions. In an experiment ^{to detect}

. such scattering, ^{none was observed.} Furthermore,

from changes ⁱⁿ strength ^with stiain-rate at 4.2 °K

on this alloy [21], ^the activation volume through

which the stress must act was found ^to be 2 X 10-21 cm3. Assuming that this is due to the

cutting ^of spherical ^ordered particles, and that the activation distance at 4.2 DK is b, ^the spacing ^of particles computes ^{to 250} A, corr esponding ^{to a} density ^of only 1017/em3. ^{This number is so small}

that there would have to be about the same order in the matrix as the measured value.

In _sum, there are a few indications that short- range order is ^not simply ^a liquid-like distribution of the atoms. In Cu-14 at. % ^{Al a} large ^number (r-oJ 1019) ^of highly ^ordered regions enriched in Al surrounded by ^an impoverished ^and nearly ^ran-

dom matrix is not a good ^model. However, ¹⁰¹⁷

per cm3 particles ^of higher order than the average

alloy ^{may be} present. Perhaps ^{the best} way of

regarding ^an alloy ^with short-range ^{order is to}

think of it as being made up of contiguous imper- fectly ordered domains of differing degrees ^{of order} statistically distributed.

A closer examination of a(0), usually ^{assumed to}

be unity, may give ^more information about short- range order. Being essentially ^{the area under the} short-range ^order peak, ^{it is difficult to} evaluate,

but it is a rneasure of the number of atoms involved in local order.

II. Local ordering ^{at low} températures.

^It

has long ^been known that ordering ^{can occur at} relatively low temperatures. Sykes ^{and Jones} [22]

from calorimetric measurements found that in ^a

specimen ^of Cu3Au water-quenched ^{from 550°C} ordering began ^at 60 °C. Brinkman et al. [23]

measured the change ⁱⁿ electrical resistance at

150 °C in a Cu3Au specimen quenched ^{from various} temperatures. ^The ordering ^{rate increased when}

thé prior annealing temperature ^{was raised from}

542 to 648 °C. They ^{also observed} changes ^{as low}

at 110 OC, ^and very recently resistivity changes ^at

room temperature have been detected in quenched specimens [24].

Similar effects occur in a number of other

alloys [25]. ^In quenched ^{Cu-14 at.} % ^Al [26] ^the resistivity ^{is maximum for a} particular quenching temperatm ^e indicating ^minimum short-range ^order

for this temperature. Similarly ^{the modulus is}

minimum for a particular quench [27] ; the modulus increases and the resistivity [26] ^{decreases on} aging.

The former was studied at temperatures ^{as low}

as 21 °C. Wechsler and Kernohan [26] ^observed

that the effective migration energy, Em, increased

as aging progressed. Their initial value was

0.65 eV and ^their final value 1.2 eV. They ^sug- gested that several different kinds of defects are

involved in the diffusion process giving ^a spectrum

of sws. The defect with the largest mobility ^is presumed ^{to anneal out} faster ; ^{the effective s.}

would then increase with time. cm determined from the modulus study ^{is in} substantial agreement

with that determined from the resistivity change.

FIG. 2.

Young’s modulus of CU3Au ^{as function of} quenching temperature (Ref. [28]).

Fie. 3.

Change ⁱⁿ Young’s ^modulus

of CusAu ^versus temperature.

Flc. 4.

Resonant frequency ^{of a} Cu3Au specimen ^as function of aging time at various temperatures.

Young’s ^{modulus of} Cu3Au ^{has also} been studied after quenching ^{from différent} temperatures [28].

The data are shown in figure ^{2. The} resistivity

at 77 OK also shows a minimum [29]. The increase

beyond ^{500°C may} be attributed to ordering during

the quench, because, ^{as shown in} figure 3, ^{there is}

no increase in modulus beyond ^{500 °C if the measu-}

rements are made at temperature. ^{This con-}

clusion has recently been confirmed by ^{Feder and}

Nowick [30] through measurement of electrical

resistivity.

The modulus of quenched Cu3Au ^{has also been}

found to increase with aging ^{at low} temperatures (fins ^{4 and} 5). ^The range of aging temperatures

FIG. 5.

Resonant frequency ^{of a} Cu,Au specimen ^as

function of aging ^{time at various} temperatures.

studied is 21 to 125°C. Effective activation ener-

gies ^were determined by making ^sudden changes ⁱⁿ

the aging temperature ; s ^{increased from about}

0. 6 eV for the beginning of the process ^to 1. 2 eV for the end of the experiment. It should be noted that the use of temperature transitions to measure

the activation _energy probably ^{comes close to} giving c., the energy of motion ^of the diffusion con-

trolling defects, ^{if the} extrapolation ^{to the tran-}

sition time is carried out carefully. Note that the process ^{was not} carried to completion ^at any tempe-

rature. The rates of aging ^at 21 and 50 eC (figs ⁴

and 5), appear ^to be maximum for the quench

from 440°C. _{This may} correspond ^{to the least}

amount of quenched-in short-range ^order.

It is of interest to attempt ^{to ascertain the}

amount of short-range ^order associated with the increase in modulus which ^{occurs on} aging quen- ched Cu3Au specimens. ^The fractional change ⁱⁿ

modulus (AMIM -- 2A f / f ) ^{for the} specimen quen- ched from 440 OC ( fig. 5) ^{is 17 X 10-3. The varia-}

tion of S11, ^with temperature, ^from Siegel [31], ^is

shown in figure ^6. Cowley [6] ^measured a1 at 405 and 460°C; 1(X11 ^{is 0.004. From} figure 6,

FiG. 6.

Elastic compliance, S11,

of Cu3Au ^versus temperature.

ilSlllSl1 corresponding ^{to the} change ⁱⁿ ordering

between 405 and 460 °C ^was estimated to be 30 X 10-3 by subtracting ^{the dashed} extrapolated

curve from the experimental ^{curve. Thus a} AMIM ^of 17 X 10-3 corresponds roughly ^to laocli

of 0.002. Diffuse _x-ray scattering ^{was measured}

in a specimen quenched from 440 °C before and after a 12 hour anneal ^at 115 °C ; any change ⁱⁿ a1

was less than the experimental ^{error of 10} %.

A change ⁱⁿ al of 0.002 corresponds roughly ^for Cu3Au ^{to a} change ^{in the fraction of Au-Cu bonds} by ^{0.007. With} ocl = - 0.15 there must be 1. 7 vacancy jumps per 103 bonds ^or 2 per ^{102 atoms}

assuming ^the jumps ^occur randomly. Assuming

the equilibrium concentration of vacancies, the pro-

bability ^{that a} given ^atom interchanges ^{with a}

vacancy in time t, pt ^{= t v} e-Em/kT e-Ef/kT, neglec- ting ^the entropy ^{factor. Ef is the formation} energy of the vacancy. We note that roughly 1/5 ^{of the}

total change ⁱⁿ figure 5b occurred in 23 hours at 50 °C. Assuming ^Em + ^ef

^{2 eV} [32, 33], pt ^com-

putes ^{to 4 in} 1013 ; therefore, ^{there must} be many

more vacancies than the equilibrium concentration.

If sm is 0. 7 eV, roughly ^{the measured} value, ^{then .}

the required ^defect concentration Gv at 50 °C is 4 X 10-11. The changes ^{in cm as the} aging process progresses appears ^{to show} here as well as with the Cu-Al alloys ^{that more than one kind} of vacancy is

present.

Finally, ^from figure ^{2 it is} possible ^to roughly

estimate Ef of a vacancy. The ^amount of ordering during ^the quench ^will depend ^{on the initial Cv as}

well as the history ^{of the} quench. ^{The latter will}

be neglected. ^{The extra} modulus for annealing

above Tc is given by ^the différence between the lower dashed and ^the solid curves in figure ^{2. In} figure ^{7 In AM is} plotted ^versus 11T ; ^{an effective}

activation _energy of 1.3 eV is indicated which is

FIG. 7.

ln AM versus 1 IT ^for Cu3Au. ^{AM is the}

différence between the lower dashed line and the solid curve in figure ^2.

about the accepted ^{value of -’si in} [Au [or [Cu,

1.1 eV. For comparison, taking ^{Em of a} single

vacancy as ^1.1 eV (from figs ^{4 and} 5), ^{an et of}

0.9 eV is computed ^{from the} activation _energy for

diffusion, ² eV, determined from ereep [33] ^of

Au-27 at. % ^{Cu as well as diffusion of Au in Cu} [32].

Quenched-in vacancies thus do not all anneal out

rapidly ⁱⁿ Cu3Au ^{or Cu-14 at.} % ^Al. They ^must

remain for long ^{times and over a} range of tempe-

rature well above room temperature. ^{Sources for}

atomic motion above room temperature ^revealed by these studies must play ^an important ^{role in}

other diffusion controlled processes ^{such as in the} formation of G-P zones inAl-Zn after reversion [34].

Here the kinetics of zone formation is aff ected by

the thermal history prior ^to the direct aging ^even

after reversion.

III. Eflects of short-range ^{order on mechanical} properties. - Over 40 years ago Griffith [35] poin-

ted out that slip through ^{an ordered} alloy ^would change ^{the structure in the} neighborhood ^{of the} glide plane. ^{Since the new state is one of} higher potential, ^Griffith suggested ^{that an} alloy ^with

order would have a higher yield ^{stress than its}

most ductile component. ^Cottrell [36] computed

the stress required ^{to disorder an} alloy ^with long-

range order ; ^{the value is} extremely high. ^{But it}

is well known that an alloy ^{with near} perfect ^order

is relatively weak, being essentially ^not stronger than the component ^metals. Superlattice ^dislo-

cations whose motions did not destroy ^{order were}

invented [37, 36] ^to explain ^this apparent ^discre-

pancy. Such associated dislocations have now

been directly observed with the electron microscope by Marcinkowski et al. [38].

But in most solid solutions in equilibrium ^there

are only short-range departures ^{from randomness}

in the atomic arrangements. ^The fact that disor-

dering ^occurs during deformation has been esta- blished [39, 40]. ^The important ^{role of local order}

in the deformation _process can be ^seen from

figure ^{8. Here the} stored energy is plotted ^vs.

true strain for wire drawing ^of Ag, Au3Ag ^and Cu3Au quenched ^{from above Tc. The} data, ^from

Bever and coworkers [41, 42, 43], ^{were obtained in a}

very similar _way, allowing ^direct comparison.

The _oc, values and the nearest neighbour ^ener-

FIG. 8. - Stored energy ^versus true strain in wires of Ag, Au3Ag, ^and Cu,,Au (quenched from above Te) ^{drawn at}

300 oK. Data from Bever, Schottky, Titchener, and Cohen, Refs. [41], [42], [43]. ai and V, ^{are from}

Refs. [6] ^and [44], Cowley, Norman and Warrenj ^values

for Au3Ag ^{are estimated.}

FIG. 9.

Ratio of stored to expended energy ^versus true strain in wires of Cu3Au (quenched from above Tc) ^and Au3Ag ^drawn at 300 OK. Data from Bever, Titchener, and Cohen, ^Refs. [42], [43]. ai and V, ^are from Refs. [6]

and [44], Cowley, Norman and Warren ; values for Au.,Ag

are estimated.

gies (Vi) [6, 44] ^{are also} given. As order is des-

troyed ^{the amount} of stored energy, theory predicts, ^is proportional ^to V1 (a1. ^{For a} given

strain the ratio Qf stored energy ⁱⁿ Cu.Au ^{to that}

in Au3Ag ^is equal-to the ratio of V1 a1 for each.

In figure ^{9 the} ratio.of stored ^to expended energy is given. ^More energy is stored in Cu3Au ^{than in} Au3Ag ^for comparable deformation because more

energy is required-to destroy ^{local order in} Cu3Au.

The arrangement ^{of atoms in the two} planes

below a splipping ^atom ,in the face-centered cubic structure is shown in figure ^{10. The numbers}

FIG. 10.

Arrangement ^of neighbours in the two (111) planes ^{below an atom} undergoing slip ^{in a} face-centered cubic crystal. The initial position of the atom under-

going slip ^is represented by the shaded circle. The numbers give the shells of its neighbours ^{in the two} planes.

The new neighbours resulting ^from slip ^{are indicated} by

the marked path.

FIG. 11.

Arrangement ^of neighbors in the two (110) planes ^{below an atom} undergoing slip ^{in a} body-centered

cubic crystal. The initial position of the atom under-

going slip ^is represented by the shaded circle. The num-

bers give the shells of its neighbours in the two planes.

The new neighbours resulting ^from slip ^{are indicated by}

the marked path.

refer to the neighbouring ^shells around this atom.

During slip, ^indicated by the heavy ^{arrow, thé distri-}

bution ^of nearest neighbours changes. ^{The àrran-} gement ^{for the} body-centered ^{cubic structure is}

shown in figure ^11. Changes ⁱⁿ oc, with the slip- ped ^{atom as} origin may be calculated ; ^formulae

for the first three shells (i

1, 2, 3) ^{fer both}

structures are given ^{in Table I.} In the f ace- centered cubic structure the first full unit of slip, a/2 ¹¹⁰ >, changes only ^two of the twelve nearest neighbours ^{and from} figure ^{10 it is} readily

seen that oc’ 1

(10xi + OC2 + oc3) /12. ^{Table 1}

also includes formulae for charges ⁱⁿ aisp, the order

parameter computed by counting only ^the neigh-

bours across the slip plane. ^{With the} face-centered cubic structure the formulae for the new a3 after

slip is différent if the slip ^is a /2 110 > or

a/6 ²¹¹ >, sincethenumberof thirdneighbours

in the second plane ^{below the} slip plane changes

after motion of a partial dislocation.

With the formulae in Table I and published

values of oci in several systems [6, 20, 44, 45]

changes ^{in m of a} slipped ^{atom were} computed

TABLE I

FORMULAE FOR CALCULATING WARREN SHORT-RANGE ORDER PARAMETERS (aj) ^{AFTER SLIP}

(*) ^ce; corresponds ^to a; after a given number, n, of units of slip ; ^letters refer to the new neighbours ^{in the} slipped position : ^a, b, ^{c are new} first nearest neighbors,

new second neighbours ^are d, e, f, ^{etc. See} figures ^{10 and 11.}

(**) _(Xis ^{is ce,} calculated considering only ^{bonds across}

the slip plane.

versus the number of units of slip ^{n, and} plotted

in figures ^{12 and 13. The} changes ^{in ais for} a/2 ¹¹⁰ > slip ^{alone can be} ascertained by connecting points ^of integral ^{values of n. The ai,s}

do not fall smoothly ^to limiting ^{values but} large

oscillations occur (except ⁱⁿ Ag3Au ^where ai, i > 1,

is very small). ^Thus during the déformation _pro-

cess at certain stages energy is released. After many units of slip across one plane il and _a3 are

changed by ^{about a} third ; OC2 is changed. by ⁵⁰ %.

Experimental determinations have ^so far reported changes only ^{in a1} [39, 40]. ^The oscillatory change

in ou during deformation has important ^conse-

quences ^and will be discussed later.

Fisher [46] considered dislocation motion in a

solid solution with,-hort-range ^order. A disordered interface of free _{energy, y per} unit _{area, is} pro- duced by passage of ^a dislocation. Fisher’s treat- ment has been extended by ^Rudman [47],

Flinn [48], ^Suzuki [49], ^Sumino [50], Seeger ^and Ranziger [51], ^and Hendrickson [52].

The following ^{treatment contains a number of} improvements ^and corrections over the previous

treatments of hardening ^{due to local} order. First

Fie. 12.

Variation of short-range ^order parameters ^with slip ⁱⁿ Cu3Au ^and Ag3Au. C’ti’sl> ^{was obtained} by conting only

the bonds across the slip plane. The formulae for computing ^{ai, and} C’ti’sl> ^are given in Table I. Values of xi ^are from Refs. [6] ^and [44], Cowley, Norman and Warren ; n represents ^units of 2 110 > slip ; n /2 represents

units of 6 211 > slip.

FIG. 13.

Variation of short-range ^order parameters ^with slip in Cu-14.5 at. % ^{Al and} (3-Ag ^{Zn. For} explanation ^see

caption ^for figure ^{12. Values of} oci ^are from Refs. [20] ^and [45], Houska and Averbach, and Suoninen and Warren.

of all the equation ^for short-range ^order streng- thening, To, in terms of _y, the free energy increase per unit ^area of slip ^{is :}

where fi ^{is a} geometrical factor. In a face-centered cubic crystal ^{.for b}

(1 j2) a ¹¹⁰ >, To b3

(yb2 V3-)12 ^{and g} = 0/2. y may be

expressed ^{in terms} of Af, ^the change in free energy per ^atom slipped ; y

A//.A ^{where A is the area}

on the slip plane equivalent ^{to one atom of} slip,

V3 b2/2. For motion of a partial ^the Burgers

vector is b’

(1 j6) a 211 >

b/V3. ^For

this case g ^is B/-3 12 ^{and A is} b2 j2 B/3. ^{The inter-}

nal energy part of Af,

In (2) ^{ma and mb are the} atomic fractions of a and b, Vi ^{is the} interaction _{energy of} an atom in the ith shell with the central atom

C; is the coordination number of the ith shell and àai ^{is the} change ^{in the Warren} short-range ^order parameter [6] ^{of an atom} in the ith shell for one

Burgers ^{vector of} slip. ^This expression ^{is based}

on Cowley’s theory [6] ^{which is a} quasi-chemical type formulation containing ^{a V; for each ith shell.}

The actual Vrs ^{in a} given alloy ^are obtained from the measured ows and thus these V;-s really repre- sent a set of empirical ^{constants which are con-}

sistent with the observed otrs. The entropy, às,

may also be expressed in terms of the ocis using

the expression developed by Cowley [6].

In (3) ^{k is} Boltzman’s constant. The summation is over every bond which is changed during slip ^of

one atom over a distance b. i represents ^{the ini-}

tial (X’s, j the final oc’s. There will be an entropy

term in (3) for every ^mi in figure ¹⁰ changed ^{to oci}

such as a1 ^to (X2, al ^{to a3,} a2 ^to a1 and _{a3 to a7.}

In comparing ^the theory ^{with the} observed cri- tical resolved shear stress, ^the Au-Ag system,

where r, was measured by Sachs and Weerts [53]

at room temperature, ^{has attracted} particular

interest. The short-range ^{order in two} alloys ^was

measured by ^{Norman and Warren} [44] ; V1 ^com- putes ^to + 0. 007 ^{eV and} a1, for Ag ^{Au is} - 0. OS,

a2 is + 0.01. Flinn [48] computed ^the strength

due to both Suzuki locking ^and short-range ^order

’ hardening versus composition, neglecting ai, i > 1,

and As. The computed ^curves when added agree

reasonably well with the measured critical resolved shear stress, ^{Te. In} computing ^{al versus} compo- sition he used the approximate ^formula

assuming T for the measured ’4’S corresponded ^to

600 oK. Rudman [47] computed To for the lead

partial (otherwise ^his calculation is similar to

Flinn’s) ^and predicted ^a sufficiently large ^{value for}

ma = 0. 5 ^to completely ^account for the observed To

but another mechanism would have to be evoked for dilute solutions.

Cottrell [36] suggested ^that short-range ^order hardening ^would result, ⁱⁿ principally, ^a yield drop

eff ect since successive dislocations moving along

a slip plane ^are required ^to produce less ^{and less}

disorder. Ardley [54] ^{observed a} yield drop, Clrt, ^and

strain aging effect above Ta in CU3Au. However,

since the time necessary ^to give ^{the maximum}

Aï at 565 °C was about 12 hours at tempera- ture, ^{it seems} likely that the eff ect is not ^{due to} change ^{in local} order ; long range diffusion _appears

to be involved, ^and Cottrell’s suggestion ^needed

further confirmation.

The Cu base Al system ^is particularly ^attractive

for attempting ^to get experimental guidance ^for theory. ^{Houska and Averbach} [20] ^{found exten-}

sive ordering ^{in a} slowly cooled Cu-14.5 at. % ^Al alloy ^and reported ai ^{to i}

^7. Further, ^{as shown} by ^Cahn and Davies [55], ^{the amount of short-}

range order may be varied by ^heat treatment. For these reasons an investigation ^{of Cu-Al} alloys ^was

undertaken in this laboratory by ^{T. J.} Koppe-

naal [27]. ^The critical resolved shear stress was

measured ^{as a} function of temperature ^{down to}

4.2 OK. A yield drop ^{effect was} observed. Some of the results are shown in figure 14 ; ^Te (4.2 °K)

- To (297 °K), ^Te (297 °K), ^{and OT} (77 OK), ^the yield drop, areplotted ^as functions of Al content to 14 at. % (tc ^{refers to the lower} resolved shear

stress).

Theoretical curves for 297 DK are also shown in

figure ^{14 for both} (1/2) a ¹¹⁰ > and

(1 j6) a 211 > slip. V1, V2, ^and V3 were ^com- puted ^{to be 0.83} kT, ^-0.31. kT, ^{and -0.14} kT,

where T refers to the temperature ^{at which} the

observed ocîs are the equilibrium values. As ^t he

sample ^{used to} determine the oc’s was cooled slowly

this was taken to be 500 OK. Using ^the simplified expression :

0:1 ^was computed ^{versus ma. The} computed a1’s

are nearly ^a linear function of m. The observed

values of ou for Cu-14.5 at. % AI ^{were then} plotted

and curves of the same form as the computed ^curve

were drawn. The ai,s at other values of ma for com-

puting ^{-r’s were} taken from such curves. The

change ^{in internal} energy and entropy per ^atom for

one Burgèrs ^{vector of} slip ^{was then} computed

from (2) ^and (3) considering changes ^{in the first}

three shells about an atom just ^{above the} slip plane.

The computed ^{stress from} (1) ^for (116) a ^{211 >}

slip ^{in a} face-centered cubic crystal ^{will be some-}

what of an overestimate since the two partials ^do

not move independently ; energy is required ^to change ^the equilibrium separation. ^The trailing partial ^which requires by ^itself less energy ^{to move} would then push ^{the lead} partial. Taking ^the

second and third shells into consideration in com-

puting ^the strength for Cu-14 at. % ^{Al adds}

about 30 % ; taking ^T AS ^into consideration at 297 OK ( fcg. 14), decreases the computed strength by ^{about 20} %.

FIG. 14.

Thoretical and measured strengths ^{of Cu-,base}

A1 alloys ^{to 14 at.} % ^Al. To, dashed curves, are the

computed theoretical contributions of short-range ^order

to the critical resolved shear stress detailed in the text.

The upper ^curve considers the motion of the lead partial alone, the lower curve considers motion of a perfect

undissociated dislocation. Experimental ^results given

are (Koppenaal ^and Fine, ^Refs. [21] ^and [67]) ^the yield drop, At’, ^{at 77} °K, the critical resolved shear stress,

To, at 297 DK, and the difference between Te at 4.2 °K and iro-at 297,oK.

Comparison ofthe ^{curves in} figure ^{14 shows that} only A-r, ^the yield drop, ^{has the same form for the}

variation ^with composition ^{as the} predicted ^short-

range order strengthening ^{but its} magnitude is very much smaller. Other experimental ^{reasons exist}

for believing ^{that àr} is related to the short-range

order in the sample [27]. ^{àr increases with heat}

treatment designed to ^increase 1 ail ; ^{’r297 -]Ç does not}

change obviously although ^as will be mentioned

subsequently ^a small effect is expected. ^{A strain} aging phenomenon ^occurs which appears ^to be associated with reordering. ^{This occurs at room} temperature which is 200 DK lower than where

Ardley [54] ^{observed strain} aging ⁱⁿ Cu3Au.

Above 373 OK, ^the lower resolved yield ^stress actually ^{increases on} heating. ^{This was} attributed

to reordering ^{near the} slip plane ^{between successive}

dislocation _passes. Also, the lower yield ^stress

increases as the strain rate is lowered. An asso-

ciated phenomenon, ^{d’t" varies} rapidly ^with tempe-

rature between 200 and 300 DK [27] ( fig. 15).

FIG. 15.

The yield drop, ^{Ar, versus} temperature ^for

Cu-14 at. % ^{Al, Cu-10 at.} % Al, and Cu-5 at. % ^Al.

Refs. [21] ^and [67], Koppenaal ^{and Fine.}

Below 200,DK, ^{àr does not} vary much with tempe-

rature. One does ^not expect short-range ^order hardening ^to vary much with temperature except through ^{the T As} term ; for 14.5 at. % ^Al only

a 15 % ^{increase in} To is predicted ^on cooling ^from

300 to 77 °K.

Evidence that changes in local order after defor- mation can occur at ^room temperature ⁱⁿ Cu-,3Au ^is presented ⁱⁿ figure ¹⁶ [43, 45]. ^After deforming quenched CU3Au ^{at 78 OK where the} resistivity increases ; ^the resistivity ^{falls on} warming ^{to room} temperature. ^{The amount of} decrease becomes

larger ^as the deformation is increased. The resul-

ting ^{curve of} resistance versus deformation is

roughly equivalent ^{to the curve obtained by car-}

rying ^{.out the} deformation ^{at room} température

(fig. 16b). However, ^an anomaly exists here. Incre-

asing ^the short-range ^order by changing the quen-

c.hing temperature ^{causes an} increase in resistivity [29]. ^{Yet here} presumably, increasing ^{the short-}

range order by annealing after cold work gives ^a drop ⁱⁿ resistivity.

FIG. 16.

Resistivity ^of Cu3Au, quenched from above Té,

measured at 77 °K, ^versus true strain on drawing wires

at a) ⁷⁸ OK, ^Ref. [56] (Roessler ^and Bever) ; b) ³⁰⁰ OK,

Ref. [43] (Cohen ^and Bever).

In Cu-Al, 1l’r, ^{but not the} general alloy streng- thening at ^room température, appears due ^to short- range ordering. ^Two complications ^{exist :} (1) ^The

measured àr is about 1/7 ^{of the} computed ^short-

range order strengthening. (2) The deformation propagates during ^easy glide by ^{a Lüders band} type deformation and thus fresh material might ^be supposed ^{to be} undergoing slip ^and there should not be a yield drop. Considering (2) first, Koppe-

naal and Fine [27] using ^an interférence microscope

found that only ^{about 50} % ^{of the} slip during ^the yield drop is associated with the Lüders bands.

The rest of the slip gives ^{lines not visible with a} light microscope. They suggested ^{that the} slip during easy glide propagated ^on previously ^active slip planes ; therefore, ^{it is} plausible ^{that the} yield drop arises because of partial destruction of order in the regions ^which undergo slip ^{ahead of the}

Lüders band. This may be one reason why ^the

measured àr is smaller than the predicted ^ro.

The heat treatment selected for the data in

figure 14, ^an air cool from 523 °K, ^{did not} give

maximum order or maximum ^àr [27]. ^{A 48 hour}

anneal at 433 OK doubled Ar. Houska and Aver- bach [20] slowly ^{cooled their} alloys ^{and their ou-s,}

from which _To ^was computed, ^may correspond ^to

higher short-range order than given by ^{an air cool}

from 523 °K.

The difference between the measured àr and the

computed To may arise from still another source.

The changes ^{in oc,, a2, and} a3 with slip ^{for the}

Cu-14.5 at. % ^{Al are shown in} figure ^{13. In}

all aîs a large ^{Aou occurs} during the first pass ^cau-

sing ^ai to " overshoot " the final value after a

large ^number of passes. Thus subsequent passes

require ^{less stress and after} roughly ^two dislocation

passes through ^a region, ^{the next} dislocation would

seem to move through spontaneously ^{because it}

releases energy. The ^net effect is that the dislo- cations should move through ⁱⁿ groups ^{at a stress} smaller than the maximum value required ^{to move}

a single dislocation. This might ^{cause the calou-}

lated _{To to} be too high by ^{as much as 40} %.

The mechanism for formation of the dislocation group may be as follows. A stress sufficiently large

to move the lead dislocation will move the follo-

wing dislocations. The second dislocation through

a given region ^{in Cu-14. 5 at.} % Al, ^for example,

will require only ^{about 10} % ^{of the stress needed}

to move the first so that its velocity ^{will be consi-} derably higher (by ^{a factor of} 10). ^{The third will}

release energy from the recreation of order ; ^{it will}

move still faster. The dislocations will move closer

together ^{until a} steady ^state separation ^{is set} up and then they ^{will move as a} group with the trai-

ling dislocations " pushing " ^{the lead} dislocation.

Swann and Nutting [57] observed with ^an elec- tron microscope ^{that in Cu-15 at.} % ^{Al the dislo-}

cations were paired. They believed these to be

superlattice dislocations. An explanation ^{of the}

sort proposed ^{here seems much more reasonable}

since it is highly unlikely ^{that in this} alloy ^there

are large régions ^of long-range ^order.

Consider next the increase in r,. on cooling ^to

4.2 °K [27] ( fcg. 14). In many solid solutions

t4. 2 ^{- ’t’297 0K} increases rapidly ^on alloying. ^This

has been attributed to clusters of solute atoms in Al-Zn [58] ^{and Al-Cu} [59] alloys ^{and the} widening

of the forest dislocations due to decrease in stacking

fault energy in Ag-Al [60] ^{and Cu-Al} [27] alloys.

In Ag3Au ^Suzuki [61 ] also ^observed a large increase in

Te on cooling. ^From information ^on filings [62, 63]

it does not appeat that the stacking fault energy in this system changes ^{much on} alloying. ^{Thus it is}

attractive to propose that the increase in ^{Te on}

cooling ⁱⁿ Ag3Au ^{is due to localized} regions ^{of rela-} tively high short-range ^order.

Returning ^to Cu-base Al alloys ^{at lower Al con-}

centrations the large increase in _r4.2 -,r2o7 -r. ^we

believe is due only ^{to the} stacking fault energy eff ect. At higher ^{.A1 the} stacking ^fault enérgy [64]

does not change ^{much with} alloying ^{and we} pro- pose ^that localized regions ^of higher short-range

order also contribute to the low temperature alloy

strengthening. ^That is, ^{in 14 at.} % ^{AI the increase}