the role of strain energy during precipitation of copper and gold from alpha iron

THE ROLE OF STRAIN ENERGY DURING PRECIPITATION OF COPPER

AND GOLD FROM ALPHA IRON*

E. HORNBOGENt

A comparative study was made of the precipitation of copper and gold from alpha iron. The solutes differ principally in their atom sizes (TC&Q = 1.003; rbu/me = 1.13). These ratios influence strongly the strain energy generated during nucleation and growth of precipitates. Accordingly, copper precipitates by general nucleation and spherical particles are formed. Under the same circumstances, gold nucleates almost entirely on dislocations and the particles are thin plates with {loo}, habit.

LE ROLE DE L’ENERGIE DE DEFORMATION PENDANT LA PRECIPITATION DU CUIVRE ET DE L’OR DANS LE FER ALPHA

L’auteur a pro&d6 B une Etude comparative de la pr6cipitation du cuivre et de l’or dans le fer alpha. Les atomes dissous diffbrent principalement par leurs dimensions (r&-pe = 1.003; T ,,/r~~ = 1.13). Ces rapports influencent fortement 1’6nergie de deformation mise en jeu pendant la germination et la croissance des pr&ipit&. Le cuivre p&ipite par germination g&n&ale, et donne des prBcipit& de forme sphbrique. Dans les m&es circonstances, l’or forme des germes presque entierement SUP les dislocations, et les particules qui se ferment sont de minces plaques avec {lOO}G comme plan d’accolement.

DIE BEDEUTUNG DER VERZERRUNGSENERGIE WAHREND DER AUSSCHEIDUNG VON KUPFER UND GOLD AUS or_EISEN

iiber die Ausscheidung von Kupfer und Gold aus a-Eisen wurde eine vergleichende Untersuchung durchgefiihrt. Die gel&ten Stoffe unterscheiden sich hauptsilchlich in der AtomgrijDe (rcu/rpe .= 1,003; T_&F~ = 1,13), wodurch die von Keimbildung und Wachstum der Ausscheidungen hervorgerufene Verzerrungsenergie stark beeinfluBt wird. DemgemiiW erfolgt die Ausscheidung von Kupfer nach normaler Keimbildung in kugelfiirmigen Teilchen, wiihrend unter den gleichen Bedingungen die Keime van Gold fast nur an Versetzungen entstehen und zu diinnen, pliittchenfiirmigen Ausscheidungen in { 100).Ebenen des a-Eisens fiihren.

INTRODUCTION

It is well established that the size ratio between

solute and solvent atoms in a solid solution is important

in determining solubility and solution strengthening.

If the chemical interaction between the atoms is

neglected it is possible to explain these effects by the

elastic energy created by the solute atoms regarded as

point defects.(l) The atomic size is not a unique value

for it depends on type of bonding and co-ordination in

the particular crystal structure.(2) To investigate the

effect of atomic size on nucleation and growth of

precipitated particles the behavior of a solute atom of

the same size as the solvent atom should be compared

with that of an atom with a large difference in size.

The simplest case would be if the precipitate has the

same crystal structure as the matrix and a composition

of 100 per cent of the solute element. In most practical

cases it has to be considered that the precipitate phase

has not the crystal structure of the matrix and does

not consist of one kind of atom only, which has a slight

effect on the atomic size values. Assuming a solid

solution which is not supersaturated, a redistribution

of the atoms takes place around dislocations. This

distribution of solute atoms is a function of the

temperature but also of the difference in size between

solute and solvent atom. The strain energy depends

on F = (rn - rA)/rA, the relative size difference of

atomic radii between solute and solvent. This energy

* Received Mav 26, 1961; revised August 7, 1061. i- Edgar C. B&n Laboratory for Fundamental Research,

U.S. Steel Corporation, Research Center, Monroeville, Pennsylvania.

ACTA METALLURGICA, VOL. 10, MAY 1962 62.7

can be reduced by interaction with the stress field of a

dislocation. If an edge dislocation with Burgers

vector b is present in the lattice, an energy FI can be

gained which is the sum of the interaction energies of

the solute atoms with the dislocationc3) which is

approximately

% where p = shear modulus

R, = distance from the dislocation line

Larger solute atoms will segregate on the tension side,

smaller ones on the compression side of the dislocation,

and F, is a measure for the tendency of a particular

solute atom to segregate.

This segregated state can be taken as the starting

condition for nucleation if the solution becomes super-

saturated. The energy for nucleation is

AF = aoi’i’ + bi(AF, - FE) (2)

if strain energy F, is needed for formation of the

nucleus.(4) The other terms are the usual ones from

nucleation theory; G = surface energy, AF, = the

chemical energy gained by formation of the nucleus,

i is the number of atoms in the nucleus and a and b are

constants that depend on its shape. Equation (2)

shows that no nucleation will take place unless the

condition AF, + F, < 0 is fulfilled.

The strain energy in the matrix was calculated by

Nabarro’5J6) assuming a spherical shape of the nucleus

F, = G,LLU’F~ (3)

526 ACTA ~ETALLURGICA, VOL. IO, 1962

where V = the volume of the sphere. Other assumptions made are that the material is an isotropic elastic medium and that elastic equations of a continuous medium can be applied. Nabarro also calculated on the same basis that FE can be reduced to about 20 per cent of the value given in equation (3) by change of the shape of the nucleus to a thin disk if it is coherent with the matrix. The assumptions involved in this equation allow only a qualitative evaluation of F,, but the dependence on E is clearly shown. A solute atom with a high tendency to segregate to dislocations (equation 1) will also impede nucleation.

It has been observed in many instances that nucleation occurs much more easily on dislocations than in the matrix. If the core of the dislocation plays no role in the formation of the nucleus, the interaction energy FI is the only force that leads to larger fluctuation around the dislocation than in the matrix. Rut it must be assumed, that in most instances of nucleation on dislocations, interaction with the core can not be neglected. (‘) A simple case is the formation of a h.c.p. precipitate from a f.c.c. matrix in the AI-Ag system.(s) Here the formation of a partial dislocation can change the stacking in the (111) planes of the matrix so that one layer of the h.o.p. precipitate is created. In most other systems precipitate phases can not be formed by such a simple atomic movement. Nevertheless, the dislocation will always be effective if it provides a way to decrease surface and strain energy. It is evident that they will be most effective if the crystal structure of matrix and precipitate are closely related. Crystal structures with large elemen- tary cells will form on dislocations only due to the larger extension of the fluctuation. No general rules can be given indicating how a dislocation will affect nucleation. It depends on the lattice structure and dimensions of the phases that can be formed from a matrix, which are different in any individual system.

To investigate the influence of strain energy on nucleation and growth of precipitates, a comparative study of precipitation of copper and gold from K-iron was chosen for the following reasons:

(1) Copper and gold have about the same solubihty in cr-iron.

(2) Copper and gold both precipita~ as f.c.c. phases, the chemical similarity between the two elements may allow one to ignore chemical interaction effects in a comparative study.

(3) The atomic size ratios of copper and gold in a-iron are(“)

r% ZZ 1 003 . 2 2 = 1.126 rl?e YF.5

The copper atom has about the same size as the iron atom, while the gold atom is much larger. Both copper and gold precipitate as f.c.c. solid solutions from the b.c.c. iron matrix so that the difference in specific volume between matrix and precipitate can be rather accurately obtained from the atomic size data, From Equation (1) it follows that gold has a much higher tendency to segregate to a dislocation than copper, but the strain energy required for its nude- ation in the matrix is higher also (Equation 3). For Fe-Au, we estimate a value of FE N 6000 Cal/g-atom;

for Fe-Cu it is < 10 oaljg-atom. There is necessarily a large uncertainty in the absolute values, but the difference in order of magnitude of the strain energy in the two cases is probably correct.

Nucleation and growth in an iron-copper and an iron-gold solid solution were observed mainly by transmission electron microscopy. Supersaturation and heat treatments were chosen so that strain energy was left as the only pa.rameter that was significantly different in the two systems.

EXPERIMENTAL PROCEDURES AND RESULTS

I?“o-on&gold

1. Alloy and heat treatment

The similarity of copper and gold is reflected in the similarity of their binary phase diagrams with iron.(lOP)

The iron-gold phase diagram shows a miscibility gap between the iron-rich and the gold-rich solid solutions. The gap becomes rather small between gamma iron and gold. No intermetallic compound of iron and gold exists.(ls,12) The maximum solubility of gold in alpha iron is 2.3 at.% at 903°C. The maximum solubility in gamma iron is 8.0 at.% at 1168°C. The f.c.c. iron-gold alloys transform to alpha iron without decomposition so that a highly supersaturated alpha iron can be obtained by quenching from the gamma field. The change in hardness and coercive force during aging of such an alloy has already been investigated.(13)

In order to avoid the complex imperfection structure which is introduced by the gamma + alpha transformation, an ahoy was chosen for this work which could be obtained as a homogeneous solid solution by quenching from the alpha field. It was intended to be of the same solute~solvent ratio as the iron-copper alloy of a previous investigation.u4) The alloy con- tained 1.14 at.% Au (3.90 wt.%) and was vacuum melted and cast in a copper mold. The ingot was cold- rolled to sheets of 1. mm thickness. All heat treatments were done in lead pot furnaces. Thin films were prepared after the alloy had been aged as samples of

HORNBOGEN: STRAIN ENERGY AND PRECIPITATION 527

RELATIONSHIP

RELATIONSHIP

0.001 per cent, the carbon content (0.002 per cent.

A study that has been made of this alloy with the

addition of nitrogen up to 0.018 per cent showed that

there was no influence of this impurity on the pre-

cipitation of the f.c.c. gold precipitates.

The cold rolled alloy was solution treated 60 hr at

840°C and quenched into ice water. The solid solution

had the following characteristics:

grain size = -1O-2 cm

subgrain size = -lo4 cm

lattice parameter = 2.875 A

2.66 I I hardness (DPH) = 148

0 I 2 3 at. % SOLUTE

average dislocation density = ~5 x 108/cm2

FIG. 1. Lattice parameters of Fe-& and Fe-Au solid The microstructure of the homogeneous solid solution

solutions. is shown in Fig. 2(a) by light microscopy and in

Fig. 3(a) by transmission electron microscopy.

1 mm thickness. The purity of the electrolytic iron The lattice parameters of the iron-gold and the

was about 99.94 per cent and of the gold about 99.98 iron-copper solid solutions, including some values

per cent. The nitrogen content of the as-cast alloy was taken from the literature(10y12’15t16) are shown in Fig. 1.

FIG. 2. :h. x 350.

2. Nucleation

The supersaturated solid solution was aged at 500,

600 and 700°C. The precipitation process was followed by microscopy.

The light micrographs (Fig. 2) indicate that sub-

boundaries play an important role in nucleating the

precipitate, but the details of the process can be

followed better by transmission electron microscopy.

Fig. 3(a) shows the quenched solid solution. The

microstructure contains single dislocations and others

forming networks and sub-boundaries. Most of these

dislocations were introduced by the gamma-alpha

transformation during the previous thermal history of

the alloy. The solution temperature of 840°C was not

suficient to anneal out the substructure. Fig. 3(b) and (c) show the changes in dislocation lines after aging at

500°C. The images of the dislocation lines thicken

probably due to the segregation of gold atoms, then

knots appear about 300 A apart. In Fig. 3(c) the knots appear as rings that grow in connection with the dis-

location lines. A few rings also have been formed without apparent connection with dislocations,

mainly in zones of low dislocation density. The

tendency for matrix nucleation increased with decreas-

ing aging temperature. At 700 and 600°C it was very

rare.

Single dislocations usually showed a higher density

of the rings than of dislocations in a network, probably

because of rapid depletion of gold from the matrix

within a network. On a given dislocation the rings

formed on the three possible (100) planes (Fig. 3~).

Figure 3(d) shows the formation of very thin gold

plates (~30 A thickness) at the sites of the former

rings. This is the earliest stage at which f.c.c. particles

can be extracted. In Fig. 4 the two stages, formation

of the rings and growth of the particles, have been

indicated on the 500°C growth curve. It is difficult to decide whether the rings are discrete particles. Some

rather uncertain evidence that they are not particles is

their higher rate of growth compared to known

528 ACTA METALLURGICA, VOL. 10, 1962

3 24HR 5 00°C

c 90HR 500°C 0 6HR 600%

FIG. 3. Nucleation of gold particles on dislocations in m-iron. Transmission electron micrographs x 21,000.

HORNBOGEN: STRAIN ENERGY AND PRECIPITATION 529

600”~ ______ - -WC- ___--- -f

Fk. 4. Two-dimensional growth of precipitated pIates in Fe-Au alloys.

particles, also the failure to extract particles and to obtain electron diffraction patterns, both of which are easily accomplished even with very thin goId particles.

There is no evidence from transmission electron microscopy for homogeneous clustering of gold in alpha iron preceding nucleation on dislocations.

The particles grow as very thin plates. Fig. 5 shows the particles at an aging stage where precipitation is almost completed. The ratio of plate thickness to the two lateral dimensions was about 1: 50 but the thickness is difficult to determine. The habit of the plates scatters around (loo), (Figs. 5a and b). The plates are not disks, but rectangular. All the interfaces are {lOO},,, planes. The diagonals, the directions of maximum growth, are 4(110),, directions. The orientation relationship (110), 11 (ill),, & 4’ was found. This indicated that the planes with the highest density of atoms and the best matching are not the habit planes, nor are the close packed directions the directions of maximum growth of the particle. An f.c.c. crystal structure was found for all the particles that could be extracted and investigated by electron diffra.~tion. If an interm~ia~ step exists between the b.c.c. and the f.c.c. lattice, it must exist only in the range of a few atomic layers during formation of the rings on the dislocation lines.

sufficiently accurate for lattice parameter measure- ments. These plates were extremely thin (20-30 A). The iron content of the particles varies greatly for different aging temperatures, but was constant for different periods of aging at the same temperature at a value somewhat higher in iron content than indicated by the phase diagram.(lO) The growth of the gold-rich plates was determined by extraction and surfacereplica techniques during aging at 500, 600 and 700°C (Fig. 4). The extraction replica technique is ideal in this case for determination of the particle dimensions, except thickness. Fig. 4 shows that during the early stages of ~owth, while the particles are growing into a supersaturated solid solution, the size is proportional to the square root of the aging time.

After the start of continuous precipitation on dislocations a second discontinuous process begins at the grain boundaries. Some of the high angle boundaries move, leaving behind an aggregate of ferrite and very coarse particles. This second process was observed in the quenched alloy after aging 2 hr at 600°C. The microstructure after 10 hr at 600% is shown in Fig. 2(c). The nature of the driving force for this reaction is not quite clear yet, and will be the subject of a# special iIlvestigation.

B , Iron-copper

1. Alloy and solution treatment

The lattice parameters of extracted particles formed at 600 and 700°C were measured, with the following results:

aBOO = 4.01 $- 0.01 A; ~25 at.% Fe

u,00 = 3.95 + 0.01 8; ~38 at.% Fe

The data for nlates nrecinitated at 5OO’C were not

The precipitation of copper has been described in an earlier publication.(14) Therefore, only a few additional experiments were done by thin film electron microscopy to gain information on the distribution of the nuclei. As previously reported, nucleation of spherical f.c.c. particles of a size less than 100 A occurs at a very

1 I I high rate. The growth of these particles in the early


FIG. 5. Complete precipitation of gold in cc-iron, 92 hr at 6OO”C, surface plane near {loo}. (a) Transmission electron micrograph x 21,000 (b) Electron diffraction

from the lower right. grain.

stages of precipitation has been interpreted as diffusion limited (see equation 6 of the earlier publication). For the new experiments, an alloy of iron with I.09 at.% = 1.23 wt.% Cu was used. The solution treatment was the same as for the Fe-Au alloy in order to have conditions as similar as possible. Thin films were prepared from samples aged at 500, 600 and 700°C. Fig. 6 shows distribution and shape of the particles in the over-aged condition.

The shape of the f.c.c. e particles agrees with that determined by the extraction replica method,(r4) but there is a discrepancy in the size of the particles observed in the extraction and transmission micrographs. The extraction replicas show smaller particles. It could not be decided whether this is due to partial solution of the particles during preparation of the extraction replicas, or to enlargement of the image in the transmission picture. Fig. 6(a) shows the small particles that are present in large numbers after 25 hr

aging at 500°C; the appearance of these fine dispersed particles of high copper contentd4) is an indirect proof of the state of clustering that is believed to have existed in this alloy before nucleation occurred. The images of the particles in this condition are not quite sharp, which may be the result of distortion of the transmitted electron beam due to coherency stresses resulting from the phase transformation b.c.o. --f f.c.c. It is quite evident from Fig. 6 that dislocations are not the preferred sites for nucleation of copper particles. nucleation takes place both at dislocations and within the lattice. In some instances (Fig. 6b and c) particles grow larger if they are connected with a dislocation line. This may be due to an increased supply of copper atoms by pipe diffusion. Comparison of Fig. 6(a) and (b) shows also that the number of particles decreases with aging time. While the larger particles continue to grow, the smaller ones dissolve in the early stages of precipitation. The particles nucleated on dislocations have a higher chance to perish during growth than the particles nucleated inside the lattice

HORNROGEN: STRAIN ENERGY AND PREC.IPITATION 531

FIQ. 6. Distribution and shape of copper particles precipitated from an Fe--l.08 at.% Cu solid solution. Electron transmission x 21,000.

(Fig. fib and c), due probably to their greater supply of

copper. Nucleation seems to be as easy within the

perfect lattice as on the dislocations. Particles grown

to a size of more than 300 L% tend to take the shape of

rods (Fig. Gd). That indicates that above this size the

surface energy which led to the spherical shape is

overcome by the strain energy of the noncoherent

particles in spite of the small difference in atomic

volume of copper and iron.

DISCUSSION

The mechanism of nucleation is different for f.c.c.

gold-rich particles than for f.c.c. copper-rich particles,

and depends upon the strain energy needed for the

formation of the new phase. Nucleation of copper-rich particles requires little

strain energy, because the atomic volume of matrix

and of precipitate are about equal. Therefore, the

nucleation barrier is determined only by the surface

energy between the f.c.c. particle and the b.c.c.

matrix (equation 2). The observations show that in

this case there is no tendency toward preferred nuclea-

tion on lattice imperfections, Nucleation must origi-

nate from copper-rich clusters that have formed in the

b.c.c. matrix. Nucleation at high angle boundaries, at

fow angle boundaries, or at single dislocations, and

dislocation-free nucleation all take place with about

the same time dependence. By far the largest number

of nuclei are those that form without visible relation-

ship to imperfections. These nuclei, therefore, deter-

mine the precipitation behavior of the alloy. The

question of whether this represents homogeneous

nucleation, or nucleation on vacancies or clusters of a

small number of vacancies, can not be decided.

Nucleation theory assumes that a particle is a nucleus

if it is able to grow with a decrease in free energy.

This led to the assumption that all nuclei grow after

they have formed until the surrounding matrix is

depleted, but observations in the Fe-Cu and Fe-C

systemso’) show that growth of larger particles occurs

at the expense of smaller particles, even Iong before


the matrix is depleted. Particles connected with a

dislocation pipe tend to grow at the expense of those

in the matrix. This makes it difficult to relate observa-

tions of the growth of a single particle to the rate of

depletion of the matrix. A further complication occurs

whenever nucleation takes place at a variety of sites

with different time dependence.

In the iron-gold system, nucleation was almost

exclusively on dislocations after quenching from 840°C

and aging at 500, 600 and 7OO”C, the same treatment

given the iron-copper alloy. If solute atoms and

precipitate phase differ much in volume from solvent

atoms and the matrix, there will be a tendency for the

solute atom to segregate into a dislocation, but

additional strain energy is then required to form a

nucleus of critical size. The magnitude of the strain

energy term F, explains why general nucleation in the

matrix is so much easier in the iron-copper alloy than

in the iron-gold alloy. Small clusters of gold atoms

may grow in alpha iron, but they will not be able to

grow large enough to become nuclei. Therefore,

dislocations must be present in order to achieve

nucleation.

As possible explanation of the nucleation of gold

particles on dislocations, we have assumed the

following three steps which minimize the activation

barrier given by the surface energy and the strain

energy.

(1) Segregation of gold into the tension side of a

dislocation line with the “driving force” F, (equa-

tion 1).

(2) Formation of a stacking fault in the (001) plane

of alpha-iron (Fig. 7), which is kept stable and spreads

by segregation of gold atoms:

sessile + glissile + sessile

; [ill] 4 ;[I101 + $OOl]

(3) Formation of a two-dimensional nucleus of gold.

The stacking fault reduces the compressive stress

produced by clustered gold atoms and creates a

surface which is required for nucleation.

The a/2 [llO] partial dislocation of step 2 is able to

move as fast as gold atoms can segregate into the

stacking fault. Fig. 7 shows a schematic representa-

tion of the splitting of a dislocation that lies completely

in a (100) plane and of a dislocation that lies with only

a jog in a (100) plane. Fig. 7 may be compared with

the transmission electron micrographs of Fig. 3(b) and

(c). Growth of the edges of the particle can then

DISLOCATION IN (001)

/ DISLOCATION NOT IN (001)

;rtll + 5 [IYO] + q[oil]

DISSOCIATION UNDER INFLUENCE OF LARGE SUBSTITUTIONAL OR INTERSTITIAL ATOMS.

FIG. 7. Nucleation on dislocations in a-iron by formation of an +‘[llO] stacking fault.

HORNBOCEN: STRAIN ENERGY AND PRECIPITATION 533

continue by further movement of the stacking fault.

Growth in thickness will probably take place by

diffusion-limited addition of gold atoms to the flat

surfaces of the particle. The possibility of stacking

fault formation during precipitation has been

mentioned in f.c.c. lattices.(W) A corresponding

dislocation reaction in b.c.c. lattices could also occur

on (Oil),, planes, but the stacking fault in the (OOl),

plane may be more effective in reducing the com-

pressive strain energy of a plate in the b.c.c. lattice due

to the minimum in elastic modulus in [OOI],,.

NucIeation exclusively on dislocations leads to a

very uneven distribution of particles (see Fig. 30). In

this situation, it is impossible to relate the growth of

an individual particle to the gross rate of precipita-

tion. If the ma~itude of the chemical free energy

change (F,) is increased by higher under-cooling or

high supersaturation(i3) general nucleation in the

matrix, as in Fe-Cu, will be favored.

Nucleation in Fe-Cu and Fe-Au illustrate the effects

of a very low and a very high strain energy barrier to

nucleation. The nucleation behavior of most super-

saturated solid solutions will be intermedia.te between

these two with nucleation on imperfections and general

nucleation having a different time dependence. The

information presented illustrates the differences in

distribution, size and shape of dispersed particles that

can result from different mechanisms of nucleation and

growth. ACKNO~EDGM~NT

The author wishes to thank Mr. R. C. Glenn and

Mr. R. D. Schoone for their help in the experimental

work, especially the preparation of the electron

transmission samples of the aged alloys.

f : 3.

4.

ii. 7: a.

9.

10. 11.

12.

13. 14.

15.

16.

17. 18.

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the role of strain energy during precipitation of copper and gold from alpha iron

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