Title: Ordering and the K-state in nickel-molybdenum alloys
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Title: Ordering and the K-state in nickel-molybdenum alloys
Alternate Title: K-state in nickel-molybdenum alloys
Physical Description: xiv, 224 leaves. : illus. ; 28 cm.
Language: English
Creator: LeFevre, Bruce Gordon, 1937-
Publication Date: 1966
Copyright Date: 1966
 Subjects
Subject: Nickel-molybdenum alloys   ( lcsh )
Electric resistance   ( lcsh )
Metallurgical and Materials Engineering thesis Ph. D
Dissertations, Academic -- Metallurgical and Materials Engineering -- UF
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
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Thesis: Thesis -- University of Florida.
Bibliography: Bibliography: leaves 213-222.
Additional Physical Form: Also available on World Wide Web
General Note: Manuscript copy.
General Note: Vita.
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Bibliographic ID: UF00097863
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: alephbibnum - 000570658
oclc - 13717044
notis - ACZ7637

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ORDERING AND THE K-STATE IN

NICK(EL-MOILYBDENUM ALLOYS











By
BRUCE GORDON LEFEVRE


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILLMlENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY













UNIVERSITY' OF FLORIDA


December, 1966












ACKNOWLEDGEMENTS


The author would like to'acknowledge the as-

sistance of the following people who contributed their

time and effort toward a successful conclusion of this

work: Professor A. G. Guy (Cormm-ittee Chairman), Pro-

fessor J. J. Hren, Graduate Assistant R. W. NewJman,

Professor R. W. Gould, Professor E. .. Sta ec, Professor

J. E. Spruiell, anr Mr. E. J. Jan ins. Pro e- or Guy

suggested the topic for this re-a .rc~ anu ?oviued many

helpful suggestions duria - the course of the experiments

and the preparation of the manuscript.

Professor Hren and Ri. W. Newvman are responsible

for the success of the field-ion microscopy partion of

this work. It was they who built the microsco 3 and de-

veloped the techniques used in this study. They also

contributed heavily in the interpretation of the micro-

graphs.

Professor R. W. Gould and Professor E. A. Starke

(Georgia Institute of Technology) provided many helpful

suggestions concerning the x-ray diffraction experiments

and the interpretation of the data. Professor J. E.

Spruiell and his associates at the University of Tennessee

provided the alloys that were used in this research as










well as many helpful suggestions concerning experimental

techniques. It was the work of Professor Spruiell and his

associates in the Ni-Mv~o system that stimulated interest in

the present research.

Mr. E. J. Jenkins who is the electron microscopist

in the metallurgy department of the University of Florida

contributed unselfishly of his own time in connection with

that portion of the work.

The author is also grateful to the Graduate School

of the University of Florida and the Nazional Science

Foundation for providing financial support for this research.

Last, but by no means least, the author would like

to acknowledge his wife Elaine for her moral support and

constant encouragement during the many periods when things

"just weren't going right."


iii1

















TABLE OF CONTENTS


Page


. . . . . . i


ACKNOWLEDGMENTS ...


LIST OF FIGURES ...............


ABSTRACT . . . . . . . . .


INTRODUCTION .. .. .. .. .. ... .


. .v


. .Xii


. .1


CHAPTER


I. REVIEW OF THEORY ..........


II. REVIEW OF PREVIOUS WORK ON NICKEL-
MOLYBDENUM ALLOYS .........


III. EXPERIMENTAL EQUIPMENT AND PROCEDURES


IV. EXPERIMENTAL RESULTS AND DISCUSSION .


V. CONCLUSIONS AND RECOMMENDATIONS FOR
FURTHER STUDY ...........



APPENDIX . . . . . . . . .


REFERENCES . . . . . . . .


BIOGRAPHICAL SKETCH .............


. . .12



. . .63


. . .89


. . 127



. . 201



. . 210


. . .213


. . .223













LIST OF FIGURES


Figure Page

1. Schematic resistivity-temperature curve
characteristic of K-state alloys (after
Thomas) .. .. .. .. .. .. .. .. .. 3

2. Curves of resistivity vs. temperature for
Cu-Ni alloys (after Dekker) .. .. .. .. 17

3. Values of a/Me2 vs. atomic number obtained
from conductivity measurements at 00,
(after Mott and Jones) .. ... .. .. 19

4. Schematic view of wide s band overlapping
narrow d band in transition metals (after
Bardeen) .. .. .. .. .. .. .. .. 21

5. Resistivity vs. composition for Cu-Au alloys
in ordered and disordered states (after
Bardeen) .. .. ... .. .. . . 22

6. Plots of resistivity vs. composition for
alloys Pd with Cu, Ag, and Au (after
Svensson) .. .. .. .. .. . . . 26

7. Typical x-ray diffuse powder patterns
corresponding to clustering, randomness,
and SRO in metallic solid solutions
(after Warren et at.) .. .. .. . . 34

8. Schematic illustration of the field-ion
microscope (after Brandon) .. . .... 54

9. A field-ion micrograph of tungsten
(courtesy of R. W. Newman) .. .. . 56

10. A field-ion micrograph of CoPt showing a
domain of SRO above the critical
temperature (after Southworth and Ralph) . 58

11. A binary phase diagram of the Ni-Mo system
(after Hansen) . . . . . . . 64










LIST OF FIGURES--Continued


Figure Page

12. A partial binary phase diagram of the Ni-Mo
system (after Guthrie and Stansbury) .... 65

13. Schematic representation of the ordered
Ni4Mo (9) phase (after Guthrie and
Stansbury) .. .. .. . . . . . 70

14. Time-temperature-transformation curve for
the a to B transformation in Ni-29.1
w/o Mo (after Lampe and Stansbury) .. .. 76

15. Spruiell's model of the SRO existing in a
Ni-10.7 a/o Mo alloy quenched from 10000C
into iced brine (after Spruiell) . ... 80

16. Spruiell's model for the SRO existing in
Ni4Mo quenched from 10000C into iced
brine (after Spruiell) . .. .. ... 81

17. Curves of specific heat vs. temperature for
Ni, Ni-Cr, and Ni-Mo alloys (after Stans-
bury, Brooks, and Arledge) .. .. .. .. 86

18. Plot of electrical resistance vs. annealing
temperature for a Ni-10 a/o Mo alloy
quenched from 9500C (after Sukhovarov
et al.) . . . . . . . . . 88

19. Photograph of the pumping station used to
seal specimens in evacuated Vycor capsules
for heat treatment . ... .. ... 96

20. Schematic view of the precision Kelvin
bridge .. .. .. .. .. .. . . . 100

21. Schematic illustration of experimental
arrangements used for x-ray diffuse-
scattering measurements . .. .. .. 104

22. Schematic drawing of the x-ray diffuse-
scattering apparatus used in the present
research .. .. .. .. .. .. . . 106

23. Photographs of the diffuse-scattering
apparatus .. .. .. .. . . . 107










LIST OF FIGURES--Continued


Figure

24. Measurements of the effect of air scattering
on x-ray diffuse-scattering data .....


Page


109


25. X-ray diffuse-scattering curves of pure Ni
in the cold-rolled and in the annealed
conditions . . . . . . . .

26. X-ray diffuse-scattering caused by {lll}
peak of pure Ni in the cold-rolled and
in the annealed condition .......

27. Schematic illustration of the x-ray small-
angle-scattering apparatus (after Gould
and Gerold) . . . . . . .

28. Effect of cold rolling on the resistivity
of the Ni-10.5 a/o Mo alloy and the
Ni-14.0 a/o Mo alloy ..........

29. Change in resistivity of a cold-rolled
Ni-10.5 a/o Mo alloy (specimen R-1) on
annealing at 5000C ...........

30. Change in resistivity of a cold-worked
Ni-14.0 a/o Mo alloy (specimen R-2) on
annealing . . . . . . . .

31. Change in resistivity of quenched Ni-Mo
alloys on annealing at 5000C ......


.. 118


ll9


122


.. 128


130


133


137


32. Change in resistivity of a cold-worked
Ni-14.0 a/o Mo alloy (specimen F-4) on
annealing . . . . . . .

33. Change in resistivity of Ni-14.0 a/o Mo
alloys on annealing .........

34. Electron transmission micrograph of
precipitated platelets of the R phase
in Ni-14.0 a/o Mo (specimen F-3) cold-
rolled to 20% reduction of area and
annealed for I hour at 6500C. Magni-
fication x 57,000 ..........


. . 138


. . 140







. . 143


Vll










LIST OF FIGURES--Continued


Figure Page

35. Electron transmission micrograph of pinned
dislocations in the Ni-14.0 a/o Mo alloy
(specimen F-3) cold-rolled to 20% re-
duction of area and annealed for 1 hour
at 6500C .. .. .. .. .. .. .. .. 144

36. Change in resistivity of x-ray specimen
X-1 and X-2 on annealing .. .. .. ... 146

37. Diffuse-scattering curve for specimen X-1
(10.5 a/o Mo) after solution-heat-
treating . .. .. .. ... .. .. .. 148

38. Diffuse-scattering curve for specimen X-2
(14.0 a/o Mo) after solution-heat-
treating .. ...... .. .. .. .. 149

39. Corrected diffuse scattering curve for
specimen X-1 (10.5 a/o Mo) after
solution-heat-treating ... .. . 150

40. Corrected diffuse-scattering curve for
specimen X-2 (14.0 a/o Mo) after
solution-heat-treating .. .. . .. 151

41. Diffuse-scattering curve for specimen X-1
(10.5 a/o Mo) after cold-rolling to 70%
reduction of area .... .. . ... 154

42. Diffuse-scattering curve for specimen X-2
(14.0 a/o M~o) after cold rolling to 65%
reduction of area .. .. .. .. .. .. 155

43. Corrected diffuse-scattering curve for
specimen X-1 (10.5 a/o Mo) after cold
rolling to 70% reduction of area .. .. 156

44. Corrected diffuse-scattering curve for
specimen X-2 (14.0 a/o Mo) after cold
rolling to 65% reduction of area .. .. 157

45. Diffuse-scattering curve for specimen X-1
(10.5 a/o Mo) after cold rolling and
annealing for 30 minutes at 5000C .. .. 160


V111









LIST OF FIGURES--Continued


Figure

46. Diffuse-scattering curve for specimen X-2
(14.0 a/o Mo) after cold rolling and
annealing for 30 minutes at 5000C ...

47.' Corrected diffuse-scattering curve for
specimen X-1 (10.5 a/o Mo) after cold
rolling and annealing for 30 minutes
at 5000C . . . . . . . .

48. Corrected diffuse-scattering curve for
specimen X-2 (14.0 a/o Mo) after cold
rolling and annealing for 30 minutes
at 5000C . . . . . . . .

49. Diffuse-scattering curve for specimen X-1
(10.5 a/o Mo) after cold rolling and
annealing for 10 hours at 5000C ....

50. Diffuse-scattering curve for specimen X-2
(14.0 a/o Mo) after cold rolling and
annealing for 10 hours at 5000C ....

51. Corrected diffuse-scattering curve for
specimen X-1 (10.5 a/o Mo) after cold
rolling and annealing for 10 hours at
50000 . . . . .

52. Corrected diffuse-scattering curve for
specimen X-2 (14.0 a/o Mo) after cold
rolling and annealing for 10 hours at
50000 . . . .

53. Diffuse-scattering curve for specimen X-1
(10.5 a/o Mo) after cold rolling and
annealing for 201 hours at 500oc ...

54. Corrected diffuse-scattering curve for
specimen X-1 (10.5 a/o Mo) after cold
rolling and annealing for 201 hours at
50000 . . . . .

55. Diffuse-scattering curve for specimen X-2
(14.0 a/o Mo) after cold rolling and
annealing for 246 hours at 500oC ...


Page


. .161




. .162


163


164


. .165




. .166




. .167



. .168




. .169


170










LIST OF FIGURES--Conts'nued


Figure


Page


56. Corrected diffuse-scattering curve for
specimen X-2 (14.0 a/o Mo) after cold
rolling and annealing for 246 hours at
5000C . . . . . . . . .

57. Diffuse-scattering curve for specimen X-2
(14.0 a/o Mo) after cold rolling and
ann i;;ng for 3.75 hours at 6300C and
I oi at 6500C . . . . . . .

58. Diffuse-scattering curve for specimen X-2
(14.0 a/o Mo) after cold rolling and
annealing for 3.75 hours at 6300C and
5 hours at 6500C ... .........

59. Diffuse-scattering curve for specimen X-2
(14.0 a/o Mo) after cold rolling and an-
nealing for 3.75 hours at 6300C, 5 hours
at 6500C,and 1.5 hours at 7500C .....

60. Field-ion micrograph of the Ni-14.0 a/o Mo
alloy (specimen W-1) quenched from 8500C

61. Field-ion micrograph of the Ni-14.0 a/o Mo
alloy (specimen W-1) quenched from 8500C

62. Field-ion micrograph of the Ni-14.0 a/o Mo
alloy (specimen W-1) quenched from 8500C
and photographed during the process of
field evaporation ............

63. Field-ion micrograph of the Ni-14.0 a/o Mo
alloy (specimen W-1) quenched from 8500C
and photographed during the process of
field evaporation ............


.. 171




.. 172




.. 173




.. 174


.. 178


.. 179




. .180




.. 181


64. Field-ion micrograph of fully-ordered
Ni4Mo (specimen W-3) showing a grain
boundary . .. ... ... .. .. .. 182

65. Field-ion micrograph of fully-ordered
Ni4Mo (specimen W-3) showing grain
boundaries and at antiphase domain
boundary .. .. .. .. .. ... .. 183










LIST OF FIGURES--Continued


Figure Page

66. Field-ion micrograph of fully-ordered Ni4Mo
(specimen W-3) showing grain boundaries
and an antiphase domain boundary . .. .. 184

67. Field-ion micrograph of fully-ordered Ni4Mo
(specimen W-3) showing a translational
antiphase domain boundary .. .. .. .. 185

68. Field-ion micrograph of fully-ordered Ni4Mo
(specimen W-3) showing a translational
antiphase domain boundary .. ... .. 186

69. Field-ion micrograph of fully-ordered Ni4Mo
(specimen W-3) photographed during the
process of field evaporation .. .. ... 187

70. Field-ion micrograph of the Ni-14.0 a/o Mo
alloy (specimen W-2) annealed for 2.5
hours at 700oc .. .. .. .. .. .. .. 191

71. Electron-transmission micrograph of 3-phase
platelets in the Ni-14.0 a/o Mo alloy
(specimen F-5), cold-rolled and annealed
in the hot stage. Magnification x 21,500 .194

72. Electron-transmission micrograph of B-phase
platelets in the Ni-14.0 a/o Mo alloy
(specimen F-6) quenched from 10000C and
annealed in the hot stage. Magnification
x 6,500 . . ..... .. . .. 195

73. Electron-transmission micrograph of B-phase
platelets in the Ni-14.0 a/o Mo alloy
(specimen F-3) cold-rolled to 20%
reduction of area and annealed for 5
hours at 6500c. Magnification x 31,500 . 196

74. Small-angle-scattering curve for the Ni-14.0
a/o Mo alloy (specimen F-3) after annealing
for 5 hours at 6500C .. .. .. .. .. 198










Abstract of Dissertation Presented to the Graduate Council in
Partial Fulfillment of the Requirements for the Degree of
Doctor of Philosophy

ORDERING AND THE K-STATE IN
NICKEL-MOLYBDENUM ALLOYS

By

Bruce Gordon LeFevre

December, 1966

Chairman: Dr. A. G. Guy

Major Department: Metallurgical and Materials Engineering


The K-state is a phenomenon that is marked by

anomalous property changes in certain alloys without ob-

servable macroscopic structural changes. One of these

anomalies is a decrease in resistivity with cold work and

an increase in resistivity with annealing, an effect which

is opposite to that found in normal alloys. Short-range

order has frequently been suggested as a possible cause of

the K-state. In this work studies were made on Ni-Mo to

determine the role of short-range order on the formation

of the K-stLate. Polycrystalline specimens containing 10.5,

14.0, and 20.0 a/o Mo were studied by means of electrical

resistivity, x-ray diffuse scattering, electron transmission

microscopy, field-ion microscopy, and x-ray small-angle

scattering. These specimens were examined after a variety

of mechanical and thermal treatments.


x11









Previous studies have shown that the K-state can

be formed in Ni-Mo alloys and that short-range order exists

in the a-phase region; however, no direct correlation has

been made between short-range order and the K-state in

this system. The results of the x-ray diffuse-scattering

measurements made in this study show that there is a one-

to-one correlation between the degree of short-range order

and the resistivity changes. It was found that when speci-

mens are treated in such a way that the resistivity in-

creases, there is also a significant increase in the degree

of short-range order, and conversely.

A possible explanation of how short-range order

increases the resistivity in Ni-Mo alloys was provided by

field-ion micrographs of specimens in the ordered and

disordered conditions. Ni-14.0 a/o Mo specimens were

quenched from the a-phase region and compared with Ni-20.0

a/o Mo alloys which had been annealed to develop the fully-

ordered Ni4Mo structure (B phase). From a comparison of

the two it was concluded that short-range order in dilute

Ni-Mo alloys consists of small, imperfect domains of

Ni4Mo-type long-range order, approximately 50 A in diameter.

It is suggested that the resistivity increase is caused by

a decrease in the number of effective electrons as the

degree of short-range increases. A rough calculation


X111










shows that as a result of the NiLq~o-type of ordering, the

Fermi surface lies closer to a Brillouin-zone boundary

than for the case of a random solid solution. This can

be attributed to the fact that ordering changes the

symmetry of the solid solution and provides new reflecting

planes which give rise to superlattice lines.

The early stages of precipitation of the B phase

was studied in the Ni-14.0 a/o Mo alloys by means of

resistivity, electron microscopy, and small-angle scattering.

It was concluded that the small domains which make up the

short-range order in the a phase provide nuclei for the

precipitation of the B phase at lower temperatures. By

direct observation of the precipitation process in an

electron-microscope hot stage it was found that the B phase

grows directly into platelets without the occurrence of

any intermediate metastable phases. The habit plane of

the platelets was found to be the {lll}.


X1V













INTRODUCTION


The effects of solute elements on such mechanical

properties of metals as yield strength and hardness have

long been known in an empirical way. The science of

metallurgy, however, is fast approaching the stage where

these effects are considered in a more fundamental way and

are explained in terms of structural features such as

detailed local atomic arrangements, density and types of

lattice defects, early stages of precipitation, vibrational

modes of the lattice, and even the density of electronic

states and their effect upon the type of atomic bonding.

These structures can be inferred from new and sophisticated

tools such as x-ray and neutron diffraction, electron

microscopy, field-ion microscopy, anelasticity, magnetic

suscept~ibility, calorimetry, thermoelectric power, the Hall

constant, the Mossbauer effect, nuclear magnetic resonance,

and others. A pro arty of certain alloys that has generated

considerable interest in the past few years is the so-called

K-state (more properly called the K-offect). Interest in

this phenomenon stems from the fact that it is marked by

an anomalous change in certain physical properties (notably

resistivity) without any observable macroscopic structural

changes. At one time or another all of the above mentioned


- 1





- 2


tools, with the exception of nuclear magnetic resonance

and Mossbaur effect have been used in investigating various

aspects of the K\-state.

The first definitive work done on the K-state was

by Thomas in 1951 Ill. In this study he noted certain

anomalies in the physical properties of several alloy systems

in which at least one component was a transition element.

The alloys that he studied were Ni-Cr, Ni-Cu-Zn, Fe-Al,
Cu-M, A-Mn an NiCu.Upon thermally annea ing specimens

which has been previously quenched from an elevated tempera-

ture or cold-worked he found an unusual increase in resitivity

in a certain range of temperature as shown in Figure 1.

Such an effect has since been found in a number of other

alloys that will be mentioned later. In almost every case

one of the components is a transition element, a fact which

is important when considering the cause of the resistivity

anomaly.

Although this resistivity anomaly has been rather

arbitrarily selected as the "definition" of the K-state,

primarily because of the ease with which it can be detected,

there are other significant property changes which accompany

the change in resistivity and these might well be used to

monitor the K-state. For example, a 20% increase in

hardness was found by Nordheim and Grant (2] and Lifshitz {3]














































Temperature

Fig. 1--Schematic resistivity-temperature curve
characteristic of K-state alloys. Curve (a), ob-
tained on heating samples quenched from 8000C.
Curve (b) characteristic of normal alloys. Curve (c) ,
contribution of K-state (after Thomas).


-3


(a)





-4


in a Ni-20a/o Cr alloy upon formation of the K-state.

Lifshitz also reported that for a Nichrome alloy in which

the K-state has been formed, no increase in hardness was

observed for specimens cold-worked in the range 20-80%

reduction of area. After 80% reduction of area the K-state

was completely destroyed and the normal work-hardening ef-

fects were observed. In Ni-8.3a/o Al Kurdyumov et aZ. [4]

found a change in elastic modulus upon quenching and an-

nealing. Davies 15! used dilatometry and x-ray diffuse

scattering to study the K-state in Fe-Al alloys in the

composition range 16-25a/o Al. He observed as much as a

0.03% contraction and a 25% increase in flow stress upon

formation of the K-state. In a recent study of K-state

phenomena in Ag-Pd alloys, Chen and Nicholson [6] used

resistivity, Hall constant, thermoelectric power, and mag-

netic susceptibility measurements. They found that cold

working up to about 20%" reduction of area reduced the

resistivity of these alloys. Thereafter the resistivity

increased as it would under normal conditions. Negative

deviations were found in the thermoelectric power upon

cold working up to 90% reduction of area. In other recent

work in the Ni-Al system by Starke, Gerold and Guy [7] and

by Hornbogen and Kreye [8], electron microscopy was used as

well as resistivity and x-ray diffuse scattering. In this

system the formation of ordered Guinier-Preston (G.P.) zones




- 5 -


of Ni3Al was found to play an important part. Starke et aZ.

also used small-angle x-ray scattering to verify the ex-

istence of these zones. Stansbury et aZ, 19] found an

anomalous increase in the specific heat of Ni-Cr and Ni-Mo

alloys which are known to form the K-state.

The previous examples show the many ways in which

the K-state can manifest itself. Muller and Muth (10] have

summarized many of these effects. There is a heat evolution,

a lattice contraction, a shift in the Hall constant to more

positive values, an increase in the effective mass of the

charge carriers with a decrease in their mobility, a lowering

of the magnetic susceptibility, and an increase in hardness

and elastic modulus. And of course, one can add to the

list: an increase in specific heat, an increase in thermo-

electric power (cold working decreases it and destroys the

K-state), and an increase in strength.

As for the nature of the structure which characterizes

the K-state, some form of ordering, clustering, or G.P.

zones has been suggested by many investigators depending

upon the particular system under consideration. Actually

there has been little direct observation of such structures

by x-ray techniques since the time when ordering was first

suggested by Yano Ill] in 1940 for the Ni-Cr system. The

whole problem has been complicated to some extent by two





- 6


factors: C1) Most of the work has been done on Ni-Cr alloys,

in which a large effect occurs, but in which xrray measure-

ments of ordering are extremely difficult because of the

small difference In the atomic scattering factors of Ni and

Cr [1,2,3,10-23]. C2) Much of the work has been done on

alloys in which a proper distinction was not, or could not

be, made between effects associated with the early stages

of precipitation of a second phase and those effects truly

associated with changes in a single-phase solid solution.

A large part of the previously mentioned work on Ni-Cr

alloys falls into this category as well as some of the

earlier work on the Ni-Al system. It should not be sur-

prising to find measurable property changes associated with

the precipitation of a second phase, in fact, this is a

basis for the determination of equilibrium phase diagrams.

Hence a strict definition of the K-effect should not include

such systems. However, there is good reason to believe

that in many systems the structure of an as-quenched,

supersaturated solid solution bears some relationship to

the final precipitate. From this standpoint, therefore,

the study of two-phase systems in connection with the

K-state is of considerable interest.

Davies 15] summarized the theories proposed up to

that time for the structural effects responsible for the




- 7 -


K-state. They are: (1) short-range order II,11,20];

(2) some form of long-range order [12,2]; (3) clustering

or G.P. zones [3]; and (4) clustering of some atoms followed

by ordering of the matrix [24]. In his own work on Fe-Al

alloys, Davies attributed the K-state to an ordered structure

with a very small domain size. Since his x-ray patterns

indicated the coexistence of short-range and long-range

order, it is difficult to determine just how much of a role

each of these plays or just what the relationship is between

the two forms of order.

In references [7] and 18), mentioned previously, a

detailed study was made on Ni-Al alloys both in the single-

phase region (the face-centered cubic [FCC) a phase) and in

a two-phase region (the a phase + the ordered FCC a' phase

with the composition Ni3A1). It was found that short-range

order was responsible for a small but measurable K-effect

in the single-phase alloys whereas the formation of coherent

G.P. zones of the ordered a' phase was responsible for a

much larger K-effect in the two-phase alloys. A maximum in

resistivity for the two-phase alloys was found to occur

when the G.P. zones had a radius of about 15-20 A. It

appears that the situation here is completely analogous to

the maximum of resistivity in Al-Zn alloys [25] where the

critical size is about 9 A. It was assumed by Starke et

aZ. that the resistivity maximum is caused by scattering of





- 8


the conduction electrons near the Fermi surface by the G.P.

zones as suggested by Mott 126]. The maximum presumably

corresponds to a size of the zones that is about the wave-

length of the conduction electrons and is caused by an

increased probability for scattering of these electrons.

It could not be definitely concluded whether it was this

effect or the accompanying coherency strains in the matrix

that plays the larger role. This research on the Ni-Al

system was important because a clear distinction has been

made between the effects of ordering of the matrix and the

effects of heterogenieties in the early stages of precipi-

tation.

Recent work by Baer [27] on Ni-W and Ni-Mo alloys

shows a quite larger K-effect present in the single-phase

a field. The kinetics of the formation of the K-state in

Ni-Mo has also been studied by several Russian workers [28,

29,30] who proceeded on the assumption that the K-effect

in this system was in fact due to a form of short-range

ordering and therefore involved diffusion-controlled re-

distribution of atoms. Baer was able to show that short-

range order does in fact exist in Ni-W and can be destroyed

by subsequent cold work. Although no data were given for

Ni-Mo alloys, Baer stated that this system showed a similar

behavior. The presence of short-range order in the a phase




- 9 -


in the Ni-Mo system has been shown conclusively by

Spruiell [31] from x-ray studies of single crystals and

also by McManus 132] from similar studies on polycrystalline

specimens. No adequate study has been made in the Ni-Mo

system of the correlation between short-range order and

resistivity.

The Ni-Mo and Ni-W alloys apparently represent

systems in which some condition in the single-phase, solid-

solution matrix causes the K-state. This is apparently

true also in the Ag-Pd alloys investigated by Chen and

Nicholson [16, and by Westerlund and Nicholson [33]. It

must be emphasized, however, that even though short-range

order has been suggested many times there are other distinct

possibilities which will be discussed in more detail later.

Guy [341, for example, suggested the possibility of solute

concentration at dislocations, the migration of which is

enhanced by the presence of excess vacancies in the freshly-

quenched specimen.

In summary, the K-state has been attributed to one

or more of three factors: C11 the early stages of precipi-

tation of a second phase; (2) departures from randomness in

the solid-solution matrix due either to solute concentration

at defects or to some form of order, either local or long-

range; (3) changes in electronic configuration due to local





-10


lattice distortions. In cases where no second phase is

involved, most investigators agree that there is a certain

fundamental change in electronic structure of the K-state

alloys with appropriate cold working and annealing. Whether

this change involves ordering or simply static displacements

due to cold work could only be speculated, since there has

been little correlation of x-ray analysis with the electronic

property changes such as resistivity, Hall constant, and

magnetic susceptibility. Much of the speculation could

have been eliminated by qualitative diffuse-scattering

x-ray measurements on polycrystalline samples. The im-

portance of diffuse x-ray scattering measurements was also

emphasized by Baer [27] who attributed the K-state to

short-range order and recommended determining the detailed

atomic arrangements involved in each alloy system. The

determination of the exact nature of the local atomic

arrangements appears all the more important in view of

recent evidence of a K-effect in the systems Au-Cu (35, 36,

37], Cu-Al [38], and Cu-Zn [381, systems in which no tran-

sition element is present.

The present research was undertaken to learn as

much as possible about the structural effects which ac-

company the K-effect in the Ni-Mo system. In particular,

the role of short-range order was investigated directly




-11 -


by diffuse x-ray scattering and was correlated with resis-

tivity measurements made after appropriate mechanical and

thermal treatments. These measurements were supplemented

by electron-microscopic and field-ion microscopic studies.

The Ni-Mo system is particularly convenient for such an

investigation because it shows a reasonably large K-effect

and is amenable to x-ray studies of ordering because of

the large difference in atomic scattering factors. The

primary objective of the research was to obtain detailed

experimental evidence which would either support or refute

the proposal that local order is responsible for the

K-effect in the Ni-Mo system. An additional objective was

to determine the nature of the local order that is known

to exist in dilute alloys of molybdenum in nickel.












CHAPTER I

REVIEW OF THEORY


1.1 Resistivity of AL~oys

Excellent summaries of the theories of electrical

conductivity in metals are given in Chapter 11 in Dekker [39]

and in an early review article by Bardeen [40]. These

reviews do not go into very much detail concerning the

effect of ordering upon resistivity; however, a very

detailed treatment of the effect of ordering is given by

Krivoglaz and Smirnov [41]. In their treatment, it is

assumed that the number of conduction electrons and the

density of states into which transitions may be made are

unaffected by ordering; thus only the effect of the re-

distribution of scattering centers is considered. On the

other hand, as has been discussed in the Introduction,

changes in electronic structure may be extremely important

in the case of K-state alloys.

The early theories of Drude and Lorentz [42] were

developed before the introduction of the band theory of

solids, hence they were severely restricted in much the

same way as the early theories of specific heats. These

theories contained the essential ideas that the current

is carried by electrons which move about more or less


-12




- 13-


freely and are subject to collisions with the crystalline

lattice. Associated with each collision is a mean time -r

(relaxation time), and the electron is assumed to lose all

the energy gained by the imposed electric field with each

collision. This results in an expression for the conduc-

tivity given by:

= ne2(1


where a = conductivity

n = number of electrons per unit volume

7 = relaxation time or mean time between collisions

m = mass of electron

e = charge on electron.

In this derivation use is made of classical statistics.

The theory was later revised by Sommerfeld [43] using

Fermi-Dirac statistics with a free-electron approach.

This led to the expression:


a = ne27/ 2


where rp = relaxation time of electrons at the Fermi level.

Finally, a treatment by Mott and Jones [44] using the band

approximation gave:


0 2 (3)
a effe 'F


where neff is the number of "effective" electrons.





- 14 -


The importance of this equation is that one can now

distinguish between metals and insulators as well as between

good metallic conductors and poor metallic conductors. For

example, the number of effective electrons in a filled band

is zero, as in the case of insulators. Furthermore, n ,f

even for the case of an unfilled band, depends, among other

things, on the width of the band, or in other words, on the

tightness of binding. nef also depends upon the level to
which the band is filled. Once the band structure for the

various metals is known, one can understand the difference

between good and poor conductors. The application of

equation (3) is based entirely on the terms nef and 'F'

In other words, the conductivity of a material is determined

by the number of effective electrons and by the relaxation

time between collisions with the ionic cores for those

electrons near the Fermi level.

Several important facts concerning the resistivity

of metals are worth reviewing at this point.

1. Although all the electrons of a metal take
part in the conduction process, it is the
relaxation time (or collision probability) of
only those near the Fermi surface that affects
the conductivity.

2. A perfectly periodic crystal, i.e., one with
no imperfections, at absolute zero would have
zero resistance. Deviations from perfect
periodicity are caused by:

a. Lattice vibrations




- 15


b. Lattice defects (vacancies, dislocation,
grain boundaries, etc.)

c. Foreign impurity atoms.

3. According to Matthiessen's Rule, the resistivity
of an alloy can be written as: p = p_ + p(T)
where p is a temperature-independent term that
includes effects of 2b and 2c above, and p (T)
is a temperature-dependent term determined by
the interaction between electron waves (photons)
and lattice vibrations (phonons). p~ is often
called the "residual" resistivity. Since it
is temperature-independent, resistivity measure-
ments in the study of lattice defects are
frequently made at low temperatures to remove
the p(T) term and increase the precision.

Most of the theory of resistivity is concerned with

the evaluation of the terms p(T) and p The p(T) term can

only be completely understood in terms of photon-phonon

interactions as originally developed by Bloch [45]. In

this treatment each lattice vibration is considered as a

wave (phonon) having a certain propagation vector q which

designates its momentum. It also has one of a number of

quantized energies in complete analogy with the propogation

of electron waves. A complete treatment involves the use

of time-dependent perturbation theory. Out of such a

treatment comes certain selection rules for photon-phonon

interactions, and each interaction of a photon with the

lattice results in the absorption or emission of a

vibrational quantum. At low temperatures the higher-

frequency modes of oscillation are not excited and therefore




- 16 -


the collision of a photon with a low-energy phonon results

in only a small deviation of the wave vector k and a small

loss of energy for the electron. Hence the resistivity

decreases with temperature.

The fact that p (T) and p are ordinarily independent

of one another is demonstrated by the work of Linde [46]

shown in Figure 2. The temperature-coefficient of resis-

tivity is affected very little by alloying copper with

nickel since the primary effect of alloying is to increase

po, i.e., to simply add a constant to the function p vs. T.
There is, of course, a slight effect of alloying on the

p (T) term because of the change in the Debye temperature

of the crystal with alloying.

Matthiessen's rule, however, does assume that there

are no significant changes in the electronic structure as

a result of alloying. It must be borne in mind that,

although the principle influence on resistivity is ordinarily

the distribution of the scattering centers themselves, the

electronic effects accompanying alloying, ordering, intro-

duction of defects, etc. may be extremely important. In

fact, as previously mentioned, it has been suggested that

this is the basis of the K-state phenomenon. The importance

of the electronic properties of a metal can be demonstrated

as follows. Equation (3) can be rewritten in terms of the




- 17 -


6-








5-















2-J






1-





-2 0- 0
Teurtro


Fi.2-uvso rssiiyv.tmprtr o uN
alos(fe ekr





-18


mean free path of electrons at the Fermi surface Ap:


a 2A (4)
a effe "F F-

where V, = velocity of electrons having the Fermi energy.

At high temperatures, Ap is directly proportional to 92

the Debye temperature of the solid, hence:


a = KM(62/T) (5)


where K is a constant containing the terms nef and VF,

and M is the mass of the atom. A plot of a/M62 versus

atomic number at room temperature is given in Figure 3

for various metals. This plot reflects the electronic

properties of these metals since the ordinate is a measure

of the conductivity per unit of vibrational displacement.

Especially noteworthy are the differences between values

for monovalent metals and the neighboring divalent metals

such as K and Ca, Cu and Zn, Rb and Sr, Ag and Cd, Cs and

Ba. The poorer conductivities of the divalent metals is

due to the fact that they have fewer effective electrons;

1.e., neff is smaller. Another interesting feature is the

low~ conductivity of the transition elements. This can be

seen by comparing the neighboring pairs Ni and Cu, Pd and

Ag, Pt and Au. An explanation can be given in terms of

the band structure of the transition elements as shown in




- 19 -


20-





16-





12-









Zrr

old




I~G I "crP


0 10 20 30 40 0 S
Atomic number

Fig. 3--Values of q/MO82 vs. atomic number obtained from
conductivity measurements at 00C (after Mott and Jones).





-20


Figure 4. The transition elements have unfilled overlapping

s and d energy bands as shown. The d band is extremely

narrow compared to the s band and hence nef for the d band

is very small. Therefore, most of the conduction electrons

are contributed by the wide s band. However, transitions

after scattering with phonons can now be made from the s

band to states in the d band since the Fermi level passes

through both bands. The high density of states near the

Fermi level in the d band means that there is a relatively

high rate of transition of electrons from the s band to

lower states in the d band. In other words the narrow d

band contributes a large number of possible transitions

for the propagating electrons in the s band and hence

increases the probability of interactions of the electrons

with the scattering centers. In ordinary metals the d band

is filled and such transitions are not possible.

The term p for disordered alloys has been calcu-

lated by Nordheim [47] using the Bloch theory. The result

is that pO is proportional to x(1-x) where x is the concen-

tration of solute or solvent. The normal effect of long-

range ordering is to reduce p0 simply because the structure
becomes more periodic. This is illustrated by the work of

Johansson and Linde [48] shown in Figure 5. Curve (a)

shows the effect of alloying without ordering and the





















3d band


















O













(1 ,i l ,u

Fi -Shmtcviwo i sbn oelpignro
8 adi rasto rstl ate ade)


- 21





-22


- --Quenched from 6500C


Annealed at 2000C


0 25 50 75 100
a/o Au

Fig. 5--Resistivity vs. composition for Cu-Au alloys
in ordered and disordered states (after Bardeen).




-23


result is in agreement with the prediction of Nordheim.

Curve (10) shows the effects of long-range order at the

compositions Cu3Au and CuAu. As expected, there is a

sharp decrease in pO. An expression predicting the effect
of long-range order on p has been derived by Krivoglaz

and Smirnov [41] using the many-electron quantum theory.

Their approach was to determine the probability of the

transition of a system of electrons from one state to

another as a result of the perturbation produced by an

imperfectly-ordered arrangement of atoms. One very im-

portant assumption is that the number of conduction

electrons is independent of composition and of the order

parameters; consequently, this treatment determines only

the effect or rearrangement of scattering centers. The

result is:


p = const. [x(1-x) 3/16 12] (6)


where n is the long-range-order parameter. Since n in-

creases as the degree of order increases (it varies between

0 and 1 for the stochiometric composition), this expression

correctly predicts that long-range order decreases the

resistivity. For a random solution, i.e., n = 0, this

equation reduces to that of Nordheim.

Krivoglaz and Smirnov have also developed an

expression for the effect of short-range order on p, again





-24


based on the assumption that the number of conduction

electrons remains unchanged. They found that when only

the first coordination shell is considered, short-range

order should decrease p just as does long-range order.

There are some important restrictions on this derivation

how~ever. The treatment of Krivoglaz and Smirnov assumes

that there is "correlation" and nothing more is said

concerning its nature. Correlation simply means that there

is a statistically greater than random preference for

unlike neighbors in the first shell. The various possible

forms that this correlation might take could have sig-

nificantly different effects. These will be discussed in

a later section. The total effect of short-range order

becomes more confusing when correlation is considered

beyond the first coordination shell. This point is

illustrated by the data of Damask [37] in which it is

found that the residual resistivity of CuzAu above the

critical temperature decreases up to 4850C and then begins

to increase. It is not clear whether this complex; behavior

is due to electronic effects or whether it can be attri-

buted purely to the complexity of "correlation" between

neighboring scattering centers.

One further example should serve to point out the

differences between electronic and lattice contributions




- 25 -


to the total resistivity. In some alloy systems involving

a transition element with a non-transition element it is

found that a plot of vs. Ex is assvmetric as shown in

Figure 6. The resistivity does not follow the carabolic

shape predicted by Nordheim with a maximum .at 50 a/o solute.

Instead the maximum is displaced towJard the Pd-rich side.

Mott offered the explanation that in alloys above 40 a/o

Pd the d shell is not complete. In accordance writh the

previous discussion of the transition elements this leads

to a resistivity that is higher than normal.

The effect of precipitates on resistivity is

actually a combination of several factors, as mi ';; be

inferred from the previous discussion. In the very early

stages of the precipitation process there is the enhanced

scattering of electrons at the Fermi surface because of

the presence of small heterogenieties having the same

size as the wavelength of the conducting electrons [7,25,

263. A crantitative expression for the change is resis-

tivity, b caused by the presence of zones has been

calculated : Matyas [50]. In addition, coherency strains

between the precipitate and the matrix also tend to increase

the resistivity. On the ot'. hand, the process of -

cipitation removes solute atoms from the matrix an .ts

them into a more periodic arrangement in the preciolesate.




























































0 20 40 GC 10C

a/o Pd!


Fig. 6--Plots of resistivity vs. composition for alloys
PC with Cu, Ag, and Au (after Svensson [49:).


----~- ----------- -r 7


1 7 / _/


- 26 -


40i


=1
s
r~
r.
ii
2C
I
-r,
L:
U
FI


1





- 27 -


This effect tends to re.::ce the resistivity since the

combined resistivity rcay be considered as a weighted

average of that of the matrix and that of the precioitate.

The effect of depleting the matrix usually predominates,

and one sees an abrupt Cdecrease of resistivity upon

precipitation. This has been a classical tool in the

determination of equilibrium phase diagrams.

A co: alete theory of resistivity of metals is, of

course, much more detailed and complicated than the simple

version presented here. The purpose of this section is

s'.; ply to give a brief review of the i-- ortant factors

that will be considered in discussing the anomaly of

resistivity in the K-state alloys. An attempt has been

made to emphasize toe di:~ rence between the electronic

and the lattice contributions to the resistivity. Failure

to do so has led to some widely held misconceptions such

as the idea that ordering always decreases the resistivity.




It has long been knowi~n that the di -...e scatter..1

of x-rays can give much information r: the structure

of the crustal1ine solids. This can be t..l. :stood without

going in-to the math<..atics of the prob 3m by simply re-

calll;. the fact that co..olete int -cference, which is

necessary to give sharp spots at the reciprocal lattice






- 28


nodes and zero intensity between the nodes, depends upon

rect crystallinity within the specimen. Any c: arture

from this perfect periodicity results in a broadenina of

the diffraction peaks an? a general increase in the

"bac`:-rou`cnd" or diffuse scattering between Teakcs. A good

review~ article dealirne with the ex:;-_rimental measurement

and the interpretation of diffuse scattering is that of

~arren and Averbach [51).

As pointed out in this article, the background

intensity can be attributed to one of the following:

1. Scattering by air or other ext:L sous material
in the primary beam

2. Scatteri:-- jy other wavelengths if the beam is
not monochromatic

3. Cc.T;ion-modified scattering

4. Ca _erature-diffuse scattering

5. Diffuse scatter:'r- due to a structural disorder
within the specimen.

By eacuti: 'i air near the specimen and by proper

arran~-ement of slits and baffles the scattering from

source 1 can be eliminated. By using a monochromator the

scattering from source 2 can be eliminated. The Compton

rmod~ified scattering, which is the "inelastic" component

of the scattering process, can be computed and subtracted

once the sources 1 and 2 have been accounted for and the

intensity put in absolute or electron units. The




-29


temperature-diffuse scattering can be determined experi-

mentally by making measurements at various temperatures

and then extrapolating to absolute zero. Alternately, a

calculation may be used in some instances to obtain a

satisfactory correction for this factor (52]. More de-

tailed treatments of the effects of temperature-diffuse

scattering are given by Walker and Keating [53], and by

Paskin [54]. The remaining intensity is truly character-

istic of the structure of the specimen. One of the most

useful applications of diffuse scattering is to determine

local order in alloys. For this purpose, the intensity

must be converted to electron units by comparison with a

standard substance whose intensity in electron units can

be calculated. An electron unit is the intensity scattered

by a single free electron in the given direction, 5, of

reciprocal space.

One of the early pieces of work done on the in-

terpretation of local order from diffuse x-ray scattering

was by Cowley [56] in a study of Cu3Au using single crystals.

His formulation was later extended by Warren, Averbach,


Following the convention of Flinn [55] the term
local order will be used to mean any local departure from
randomness in the solid solution. This includes clustering,
which is a preference for like nearest neighbors, and short-
range order (abbreviated SRO) which is a preference for
unlike nearest neighbors.





-30 -


and Roberts [57] to include the contribution to diffuse

scattering due to difference in atomic sizes. This

development is outlined here.

Consider an alloy containing A and B atoms arranged

in some fashion. The total scattered intensity in electron

units is given by:


I = CC fmf ,exp~i ii*( Am) 7
mm

whee =2n(s -sg), s and so being unit vectors in the

direction of the scattered and incident beams. f and

fm, are the atomic scattering factors of atoms located at

the tips of vectors Rm and R from some arbitrary origin

within the crystal. The following substitutions can be

made:

r. = average distance between two lattice points
where one is in the ith coordination shell

with respect to the other

rAni = r.(1 + c ~), where r ~i= distance between

two A atoms where one is in the ith shell with

respect to the other

r -i = r.(1 + c ~i)


rABi = r (1 + CABiZ




- 31


". = C1 -P./. where P. probability of finding

an A atom at a distance r. from B atom and mA
is the mole fraction of A atoms


i -1ii .. 1 B mB

where~ = /f tomic scattering factor of A
n B atomic scattering factor of B3

The intensity can then be written as:

I = I ,: ( fA) 2 Colaexp(ik~r.)

+ 'r .1xr..exp~ik-r..)

+ G2m (n jfA + mB B) 2,expZ ik"-(Rm m;! 8


where N is the total number of atoms in the primary beam.

The last term in the bracket gives that part of the spectrum

resulting in sharp diffraction peaks. When this term is

dropped, the remaining diffuse intensity from a single

crystal can be written as:


,"B B fA 1 ~nCamncO2(h


mh2 + nh3) C~~m2n(1hi +
mnn

mh2 nh) sn 2(1h mh + h3)(9)


where hi, h2, and h3 are the fractional reciprocal lattice

coordinates. The corresponding equation for polycrystalline

specimens is,





-32 -


I = m m (f ffl~z1cia.(sin Sr.1/Sr -

iEc.B.1(sin Sr.)/Sr. cosSr.] (10)

where S = 4x sin /X and c. is the coordination number
thI
of the it shell.

The actual experimental data consist of a set of

intensities (in counts per second) as a function of S.

These data are converted to electron units per atom by

comparison with a suitable standard as described above.

The Compton-modified and temperature-diffuse scattering

components, also in electron units (eu) per atom, are

determined and subtracted. The remaining intensity-

distribution function is that given by equation (9) or

(10). It represents the diffuse intensity of the specimen

on a per atom basis attributable only to the particular

way in which the A and B atoms are arranged in the solid

solution. To solve the problem of what that arrangement

is, one must first invert equation (9) or (10), depending

upon whether the single crystal or powder method is used,

and then determine the parameters a and B.

The further analysis involves assuming various

models to determine which atomic arrangement best fits the

derived parameters. For cases in which the size-effect

terms are negligible the a's can be obtained directly from





-33


single-crystal data by Fourier inversion. The powder data

can be put into a form which also makes a Fourier inversion

possible. The problem becomes more involved when the B's

cannot be neglected. A suitable technique has been developed

by Borie and Sparks [58] for separating and determining

the a and a coefficients from single-crystal data. For

the powder method a technique has been developed by Flinn,

Averbach and Rudman [59] for separation of the a and B

terms; however, this method is subject to a certain amount

of false detail and a complete separation is difficult to

obtain. The powder method also suffers from the fact that

there is inherently less detail in these data. The intensity

at each 6 setting is an integrated measurement over the

surface of a sphere in reciprocal space with a radius of

sin 6/X. It is the general consensus of opinion among

experts in the field that data obtained by the powder

method should be used only for a qualitative picture of

the local order; any quantitative interpretations should

be based upon single-crystal data.

Before discussing some of the limitations of the

Warren-Cowley development, a good insight into the effects

of local order on the diffuse intensity can be gained by

looking at the predicted curves of equation (10) for powder

samples. These effects are summarized in Figure 7. For a





































- I i l l l I ll I


-34 -


a >0, clustering


a =0, random



at <0, SRO


10 I


1,6


2,4


3.2


Fig 7 -Typical x-ray diffuse powder patterns corresponding
to clustering, randomness, and SRO in metallic solid so-
lutions (after Warren et al.).




- 35-


specimen exhibiting a completely random distribution of

the solute atoms, the ai are equal to zero and the intensity

decreases monotomically (Laue monotonic scattering). For

alloys in which a >0 we find maxima near the origin and at

other reciprocal-lattice nodes. This behavior is to be

expected when one considers the fact that a >0 implies a

preference for like neighbors, i.e., clustering, which in

turn would be expected to produce the well known small-

angle-scattering effect. Good examples of systems in which

such a diffuse-scattering distribution has been obtained

are Al-Zn r51], and Al-Ag [60]. In these cases analysis

of the data does show that "1>0 as expected. For u1>0 the

curve typically shows a maximum at Sri-4.5. This represents

a condition in which there is a greater-than-random oc-

currence of unlike nearest neighbors. The exact shape of

the curve will depend on the higher order a. parameters as

well as on the values of the 6's, but in general a specimen

exhibiting a preference for unlike nearest neighbors will

exhibit a curve, with a maximum at Srl 4.5.

Several points should be noted in connection with

the Warren-Cowley development. First, the equations are

not restricted to the description of local order but also

reflect the effects of long-range order (abbreviated LRO).

As pointed out by Spruiell [31], if LRO is present the a's





- 36


will have definite values which depend upon the degree of

this type of order. The convergence of the series will

depend upon the size of the antiphase domains, and the

series itself will describe the sharp superstructure

reflections characteristic of LRO. One may visualize the

diffuse scattering of SRO as representing the extreme

broadening of the superlattice peaks. As the degree of

order increases, these peaks increase in sharpness so

that there is a general transition from the diffuse maxi-

mum to the sharp superlattice lines. This view aids in

visualizing the distribution of diffuse intensity from

non-random alloys, although, as Warren points out, one

cannot in general regard SRO as simply small domains of

LRO. In the derivation of equations (9) and (10) the

preference for like or unlike neighbors is regarded only

in a statistical sense. This will be discussed in greater

detail later.

Another important feature of the Warren-Cowley

development is that the distance between a given pair of

atoms is assumed to be only a function of the species

involved and not a function of the distribution of atoms

separating the pairin question. One would expect, however,

that not all tenth-nearest A-B pairs, for example, would

have the same separation. Huang (61] used a different




-37


approach in analyzing the size-effect contribution. His

model assumes that the lattice is an elastic medium with

centers of distortion at the sites of the solute atoms.

His theory predicts that because of the size effect, there

will be a reduction of the integrated intensity of the

primary reflections and a concentration of diffuse scattering

in the vicinity of these peaks. Borie has shown [62] that

both the effects predicted by Warren and those predicted

by Huang ought to be observed. The predictions of his more

general approach can be summarized as follows:

1. The diffuse scattering caused by local order
and by size effect are both present, as pre-
dicted by Warren.

2. The diffuse scattering predicted by H~uang is
also present. It is concentrated near the
Bragg maxima and along a line from each
maximum through the origin of reciprocal space.
Temperature-diffuse scattering also increases
in the vicinity of the Bragg peaks as shown
by Warren [52].

3. The integrated intensities of the Bragg peaks
are reduced as a result of the lattice distortions
produced by the size effect. This effect is in
complete analogy with the reduction in intensity
caused by the dynamic displacement of atoms as
a result of thermal vibrations. The size-
effect displacements are, of course, static,
but the effect is similar.

4. The size-effect distortions cause a broadening
in the diffuse short-range order peaks and a


The treatment by Walker and Keating [53] shows
that the effect of thermal vibration is also to
broaden the short-range order peaks and to
reduce the a('s.





- 38


The interpretation of the diffuse-scattering

intensity in terms of the best model for the type of local

order present has to a large extent been speculative. The

reason is that no unique solution can be obtained. It is

well known in x-ray structure determination that the exact

atomic arrangement within a crystal cannot be determined

from the intensity distribution alone. The situation is

no different when considering the diffuse intensity. The

usual approach is to assume a reasonable model based upon

some knowledge of the phase diagram, and then to modify

this model in a trial and error fashion until the best fit

for the observed a's is obtained. There may, however, be

many models which will fit the same set of a coefficients.

It is significant that almost every instance of SRO

can be explained by assuming that it consists of small and

imperfect domains of some form of LRO existing in the given

alloy system. This coincidence is surprising since Warren

has pointed out that there is nothing in his development

that calls for such a model. In his treatment SRO is

considered only as a statistical departure from the random

probability of unlike neighbors in the first, second,

third, and higher coordination shells and no domain structure

is assumed.

A good example of SRO that can be interpreted as

small domains of LRO is found in the alloy CuAu [63].





-39


CuAu orders by forming alternate layers of copper and gold

atoms on {100} planes with a slight accompanying tetragonal

shrinkage. This shrinkage should result in a slight shift

of the superlattice reflections such as (001) and (003)

outward from the origin. In specimens quenched from above

the critical temperature, Tc, the diffuse SRO peaks occur

approximately as though they were produced by a broadening

of the (100) and (300) reflections, and they are actually

shifted outward as expected. The primary reflections,

however, are not shifted since the overall symmetry of the

lattice is cubic as a result of the averaging of small

slightly tetragonal domains of ordered Cunu. The domains

are so small that diffuse maxima are produced rather than

sharp superlattice lines. The diffuse maxima are shifted

outward in accordance with the slight tetragonal shrinkage

of the lattice. Sato et aZ. [64) used electron diffraction

to observe very diffuse superlattice spots in CuAu as much

as 500C above T.

CuPt is another example of an alloy in which the

SRO appears to be best explained by assuming it to be

composed of small domains of LRO. Analysis of powder data

by Walker [65] on specimens held above Tc showed the un-
usual result that al and a3 are zero whereas a2 is negative

and aq is positive. This result can be understood by





- 40-


examining the structure of fully-ordered CuPt, which consists

of alternate {111} layers of Cu and Pt atoms. For such a

structure there are six Cu and six Pt atoms in the first

coordination shell of any given atom, hence al is zero.

The Same is true for all the odd-numbered shells. For

"2 all the neighbors are unlike atoms, and for aq all the

neighbors are like atoms, hence ar is negative and a( is

positive. In view of the similarity of the a coefficients

calculated from the data obtained, Tc, it is reasonable to

assume that SRO in CuPt consists of small domains of

CuPt-type LRO.

A recent article by Gehlen and Cohen 166] con-

cerning the determination of SRO models by a Monte Carlo

computer technique offers further evidence for the small-

domain concept, at least for the case of Cu3Au. The

technique is one of obtaining the best fit to the first

three a coefficients by a trial-and-error computer calcu-

lation of the rearrangement of a three-dimensional array

of atoms. For Cu3Au above Tc, the calculated structure

consists of small ordered domains embedded in a random

matrix. It was found that the same final configuration

was reached regardless of whether the starting matrix was

completely ordered or completely disordered. This result

gave the authors confidence in the method although they

could not prove analytically that this method is unique.




-41


It is common to expect clustering (a >0) for those

solid solutions that ultimately decompose at a lower

temperature (or at a slightly enriched composition) into

two terminal phases and to expect SRO (a <0) for those

solid solutions that transform into an ordered phase.

Although this idea is to some extent intuitive, it is

supported by the observed behavior in a number of systems.

For example the CuAu, Cu3Au, and CuPt systems mentioned

above exhibit SRO at an elevated temperature and LRO at a

lower temperature. Examples of clustering in systems

which later decompose into G.P. zones have been previously

cited [51,60]. In the Au-Ni system, on the other hand,

it is doubtful that this type of behavior occurs. The

miscibility gap in this system indicates that one should

find clustering at temperatures above those at which

decomposition takes place. According to the quasi-

chemical approach [67] a positive heat of mixing, charac-

teristic of Au-N~i, should be observed for a solution

exhibiting clustering. However the surprising result of

Flinn et at. 168] from analysis of powder data is that

a <0, indicating a preference for unlike neighbors. This

system has been re-studied by Munster and Sagel [69) who

find a positive al in agreement with the quasi-chemical

theory. In this case the disagreement might be resolved

if precise single-crystal data were available.





-42


It appears that SRO (and/or clustering) is the

rule rather than the exception in solid solutions. Some

insight into the driving force for these departures from

randomness are provided by the theories of Friedel 170],

Flinn [55], and Rudman 171]. The theories of Friedel and

Flinn are based on electronic structure whereas in the

more intuitively appealing theory of Rudman, elastic strain

energy is visualized as the driving force that produces a

local rearrangement of atoms. So far all the alloys

mentioned have been face-centered cubic. Evidence has

also been found for short-range order in body-centered

cubic and close-packed hexagonal systems. What may be

even more surprising at first glance, is the fact that

SRO has even been found in systems which exhibit complete

solid solubility, e.g., Au-Pd [72].

In summary, it is clear that the usefulness of

diffuse scattering of x-rays for studying local order in

solid solutions has been adequately demonstrated. Although

data from polycrystalline specimens may be used for a

qualitative picture of the nature of the local order,

single-crystal data are definitely desirable for quanti-

tative results. It is a popular concept that SRO is

simply small domains of LRO. While there is evidence that

this is a good model in some systems, there is nothing in




-43


the scattering theory or in the thermodynamic theory of

solutions that justifies this model. Until convincing

evidence for such a model can be presented for the par-

ticular system in question, it is best to consider SRO

as simply a greater-than-random preference for unlike

nearest neighbors. Since the particular configuration

assumed by the atoms represents the state of minimum

energy, it is reasonable to assume that it should reflect

the decompositional tendencies of the system at lower

temperatures, but it has not yet been proven that this

is universally true. This point can be further emphasized

by using the Au-Pd system as an example. This system is

completely isomorphous according to current data, and yet

it shows a definite SRO. The fact that a unique solution

to the diffuse-intensity data cannot be obtained means that

other experimental data are needed to support the presumed

model. Field-ion microscopy offers some hope for a means

of direct observation since there is evidence that it

can be used to detect a small region of a domain structure.


1.3 Explanations of K-state Phenomena

Several papers have dealt with possible explanations

of the K-state in various systems. Now that a brief dis-

cussion has been given on the theory of resistivity, the

peculiarities of the transition elements, and the concepts





- 44 -


of local order, this background can be employed in an

analysis of some of these theories.

In the previous discussion of resistivity of metals

emphasis was placed on the distinction between purely

electronic effects and those effects caused by a change

in scattering because of a redistribution of ion cores

and defects within the lattice. It has been suggested

that subtle changes in the electronic structure of a solid

solution can accompany SRO. This view has been supported

by Koster and Rocholl (16] in their work on Ni-Cr. They

concluded that the K-state is related to a change in the

density and mobility of electrons and holes in the upper

energy bands as a result of SRO. This conclusion was based

on measurements of resistivity, Hall constant, thermoelectric

power and magnetic susceptibility. No particular model for

the SRO was assumed. The assumption was made that SRO has

the same effect as adding more solute atoms because in both

cases the number of unlike nearest neighbors increases.

Even though a certain amount of correlation will exist

between scattering centers as a result of SRO and may

decrease the resistivity according to Krivoglaz as dis-

cussed before, it was assumed that the change in the

electronic structure of the alloy upon ordering has the

larger effect.




-45 -


The significance of the particular model assumed

for SRO is seen when one considers the theory of Gibson [73].

In this treatment it is assumed that SRO consists of small

domains of LRO. The relative position of the Fermi surface

with respect to the first Brillouin-zone boundary of the

random lattice and the superlattice determines whether SRO

should produce an increase or decrease in the resistivity.

This idea has been discussed by Muto [74], Slater 175], and

Nicholas [76] in terms of the "superzone" concept. Upon

forming a superstructure from a random alloy, the symmetry

of the lattice is changed; e.g., in the transformation from

B to f3' brass the Bravais lattice goes from body-centered

cubic to simple cubic merely as a result of ordering.

With changing symmetry the Brillouin-zone construction in

K-space changes and in turn produces a change in the

effective number of conduction electrons. In particular,

if the Fermi surface lies well inside the Brillouin-zone

boundary of the disordered alloy and close to the Brillouin-

zone boundary of the ordered alloy, then SRO will decrease

the number of conduction electrons and thereby increase

the resistivity. This of course, corresponds to a K-effect.

It is important to bear in mind that for this theory the

idea that SRO consists of small domain is important, but

the presence of a transition element is not. In fact, in





-46


certain Ag-Pd alloys which show a K-effect 19.5 a/o and

41.3 a/o Pd) Chen and Nicholson 16] found that neither

cold work nor heating changes the magnetic susceptibility.

They concluded therefore, that although the transition

element Pd is present, the 4d shell is completely filled.

Logie et aZ. [77] attributed the K-state in the

Au-Pd system to local lattice distortions rather than to

SRO. It will be recalled that the high resistivity of the

transition elements is caused by the high probability of

transitions of the conduction electrons from the s to the

d band upon scattering. According to Logie's theory the

s-d scattering is enhanced by local lattice distortions

around dislocations. The argument is supported by a

rough estimate of the expected change in resistivity based

on the approximate density-of-states curve for the 4d

shell of Pd. The authors were able to predict from a

simple model that the d shell of Pd becomes filled at

about 55 a/o Au and that the maximum decrease in resis-

tivity due to deformation should occur at about 50 a/o Au.

The maximum decrease in resistivity does in fact occur at

about 50 a/o Au. At least two arguments can be used

against this theory. First of all, a decrease in resis-

tivity with deformation is still found in alloys as rich

as 80 a/o Au, well beyond the concentration at which the





- 47 -


d shell is completely filled. Furthermore, the authors

have assumed that since no superlattices have been found

in Au-Pd, SRO or clustering does not occur. As we have

already noted, Copeland and Nicholson 172] have found SRO

in Au 40 a/o Pd by x-ray diffuse scattering. Further

support, however, for the idea that s-d scattering is the

prime cause of the K-state is the fact that the K-state

has even been reported in pure Cr (78). On the other hand,

the research on Cu-Au (35,36,37], Cu-Al (38], and Cu-Zn [38]

indicates that such effects can be produced in non-tran-

sition element alloys.

The idea that local lattice distortions, rather

than SRO is responsible for the K-state is supported also

by Jaumot and Sawatzky (79] and by Aarts and Houston-

MacMillan [80]. The reasons are not quite the same as

those of Logie et al., however, Jaumot and Sawatzky in

their study of Cu-Pd assumed that the number of conduction

electrons is increased by cold work. This is presumably

accomplished by a shift in the centers of gravity of the

s, p, and d bands as a result of local dilations of the

lattice. A change in s-d scattering was deemed unlikely

largely on the basis that the alloys are diamagnetic,

indicating a filled d shell. A similar conclusion was

reached by Chen and Nicholson [6] regarding the filled d





-48


shell in Ag-Pd. They agreed that a change in s-d scattering

is not the primary consideration. On the other hand, they

felt that SRO is a more likely explanation than local

distortions. The explanation of Aarts and Houston-

MacMillan for Ag-Pd is similar to that of Jaumot and

Sawatzky. A more recent paper on Ag-Pd by Westerlund and

Nicholson 181) lends support to the suggestion that SRO

rather than local lattice distortions causes the K-state.

This conclusion is based on the fact that there is a

relatively large decrease in the Hall constant upon cold

work. The authors pointed out that this decrease in the

Hall constant reflects a large increase in the number of

effective electrons. It is difficult to see how this

factor alone favors the idea of SRO over that of local

lattice distortions.

A treatment of the effect of SRO and of clustering

on resistivity has been given by Dehlinger (82]. He

considered only the effect of redistribution of solute

atoms as scattering centers and not the effect of any

changes in electronic structure. The surprising conclusion

is that SRO can cause an increase in resistivity over the

value characteristic of the random distribution. This

conclusion was reached upon considering Damask's data 137]

along with the calculations of Gibson 173]. Using known





-49 -


parameters of SRO for Cu3Au at 405 and 4600C Gibson's

calculations showed that p decreases with increasing

temperature in this range in agreement with the data of

Damask. Gibson's calculations also predicted that the

presence of SRO would increase the resistivity above the

value for a random solid solution. The equation for the

residual resistivity due to SRO is:

CACB tt)
p =K N VA VB1 ImnCCalme Imdfdtf' (11)
N A Imn


where + and +^ represent the wave vectors of an electron

before and after a transition is made because of the

perturbations of the potential from different atoms A and B.

The integration is taken over the Fermi surface, which is

assumed to be spherical and independent of the degree of

SRO. In this formulation SRO is considered only statisti-

cally regardless of whether or not it is truly a domain

structure. As pointed out by Krivoglaz, however, the

value obtained by a calculation of this type depends on

the range over which the almn parameters are considered.

Until definite models for SRO can be found this type of

calculation is at present reliable for quantitatively

predicting changes in p. It does, however, show that SRO

can cause an increase in p simply on the basis of the

redistribution of scattering centers, contrary to the





-50


generally accepted notion that SRO will always decrease p

because it represents a greater degree of periodicity.

Physically it is difficult to see why this should be

true except in the case of very small domains. It has

been argued by Bennet [83] that antiphase domains

boundaries will scatter conduction electrons. It is easy

therefore to see that this type of SRO could lead to a

maximum in p corresponding to a certain domain size.

This conclusion has in fact been implied by Davies [5] in

his explanation of the K-state. In general, however, it

would seem that the fundamental changes in electronic

structure which accompany SRO will have a larger effect

than scattering by domain boundaries.

Muller and Muth [10] have offered further support

for the theory that subtle changes in electronic structures

are responsible for the K-state. After summarizing the

properties which generally accompany the K-state, they

concluded that it is accompanied by a lowering of the

Fermi level with a transition of s electrons to d states.

They also proposed that for Ni-Cr and Ni-Cr-Fe alloys an

increase in covalent bonding of the d electrons is re-

sponsible for the lattice contraction, the increase in

hardness and elastic modulus and the decrease in magnetic

susceptibility. They assumed that SRO is responsible for

these effects.




-51


As far as the author is aware, only two other

explanations for the K-state have been offered. One is

the possibility of the effect of preferred orientation.

It would be extremely difficult to rule this out in all

cases. However, it is possible to produce a K-effect in

some alloys by simply quenching and annealing. Under

these conditions very little change in preferred orientation

would be expected. The final explanation to be considered,

due to Taylor and Hinton (12], is that the creation of an

ordered structure will create new reflecting planes for

the electrons at the Fermi level. They calculated that

for the Ni3Cr superlattice the d spacing of the {100}

planes is roughly the right magnitude for the Bragg

reflection of the valence electrons, assuming that each

Ni atom contributes 0.6 electrons and each Cr atom con-

tributes 1.0 electrons. Hence, if one assumes that SRO

consists of small domains of Ni3Cr, this idea can be

used to explain the increased resistivity. This explanation

is open to the following criticism. If the lattice were

perfectly ordered, the resistivity at absolute zero would

be zero. The reflecting superlattice planes would still

be present but the resistivity would be zero. In other

words it appears that the net effect of Bragg reflection

of electrons should be zero.




S52


In summary it appears that most of the current

suggestions as to the cause of the K-state phenomenon

favor the idea that certain fundamental changes in the

electronic structure of these alloys occur. Some of the

theories which have been previously cited attirubte these

changes in electronic structure to SRO, whereas others

attribute the changes to local lattice distortions. Since

most of the alloys which are capable of forming the K-state

contain a transition element, significance has frequently

been attached to the presence of a vacant d shell. There

is evidence, however, of the K-state in alloys that

presumably have a filled d shell. It has also been

suggested for some alloys that heterogenieties in the

solid solution such as clustering, G.P. zones, or con-

centration of solute atoms at dislocations are responsible

for the anomalous properties of the K-state.


1.4 Field-lon Microscopy

The field-ion microscope was used in this study in

an effort to determine the nature of the SRO that occurs

in the Ni-Mo system since, as will be shown later, this

has a direct bearing on the explanation of the K-state.

Field-ion microscopy is the only technique available at

present by which individual atoms can be resolved and

photographed. Although the problem of interpreting field-






S53


ion micrographs is in its very early stages, there is

much promise in this method for studying local order.

The field-ion microscope was first introduced to

the public in 1951 184] by Muller. A general review of

the principles and techniques of field-ion microscopy are

given in a later review article 185], also by Muller.

Individual atoms are imaged by the technique shown in

Figure 8. The specimen is polished to a small-radius tip

of a few hundred atom diameters and is placed at a high

positive potential (3-30 KV) with respect to a fluorescent

screen below. The high potential and the small radius of

the tip cause an extremely high field at the specimen

surface. This intense field, which may be as high as

5 x 108 volts per cm, polarizes certain atoms at the

specimen surface, by causing the electron cloud to be

pulled back, thus exposing the positive ion cores. When a

precooled imaging gas such as He or Ne is allowed to flow

over the tip at a very low pressure, the gas atoms are at

first polarized and attracted to the surface of the tip by

the high positive potential. Once at the surface and in

the vicinity of the polarized metal atoms, the gas atoms

are ionized by electronic tunneling and are accelerated

radially to the screen. On hitting the screen, the gas

atoms produce images of polarized atoms which caused the

ionization.




- 54 -


Specimen tip


Phosphor screen


Fig. 8--Schematic illustration of the field-ion microscope
(after Brandon).


































































n; 't ";: c.0~ ~ lca~ t~


-55 -


The rcso012o o0: c~ field-ion ;:.croscop is such

tCa in:..icidial cros :, ::: tir. 1 .:1 -e as points of

light. The r u,- Lion i1 oves as the tip is cooled to

loo: r tempera"-ures with a correslsond~i:. crease in the

~aItude of the ato.. 1 vljrations. ~ y certain ator~s

.a-o~ ce an i1.1 e, I~c:.-v :. T2 1:3 Is jiiue on an atomic

scale the curvalur o:1 Ch2 tip i3 c~ ~,oiac; t; h:

suirfacer is actually ec:. cool of Inyer c;L pianes of atomrs.

..crit rion for i: ai.. 111s not ye ce colualately

es -lished, u it is clear that the nbility of a given

ato- to Sc fie i. >o f .ctor, and


th-is i;; ti~~ i ez. to low~ tic~l. i- is bo1~ c? the


,S3 n Coor swi:lick occ . co e"




v 2 a~,orn ad t,: he :r~nev of

is cuite ap ..;< t. An ex mle cl

..om t-. :.0e is cive in Ficare 9.

a5 :ic. -ic.. microse "o the


suFr e.


for :0


ernOnSi;;Ci;i;e


a fel-lca aire p1.

:iplic tion


r, [SG,


e71 ir~ a C


:o 50 u/


a;rib', `S tii~i;~ i:;

irar;c;c;n array o~




-56 -


Fig. 9--A field-ion micrograph of tungsten
(courtesy of R. W. Newman) .




S57


This behavior was attributed to the fact that in a dis-

ordered alloy the atoms possess a random distribution of

binding energies. This leads to an irregular field evapo-

ration of the various atoms, and causes an irregular image.

The greater the degree of order, therefore, the more

regular and complete is the corresponding field-ion

image.

Short-range order has also been studied in the

Co-Pt system. This work was done by Southworth and Ralph

[88] employing the principle discussed above. These

investigators observed small domains of LRO embedded in a

disordered matrix above the critical temperature in CoPt.

On the micrograph a domain of LRO appeared as a small,

comparatively regular region against a random background.

This micrograph is reproduced in Figure 10 where the domain

is designated by the letter A. The interpretation by

Southworth and Ralph was that SRO in CoPt consists of

small domains of LRO. As far as the author is aware,

this is the only direct experimental evidence supporting

the small-domain concept of SRO. The present investigation

was undertaken to determine whether similar ordered domains

could be found in Ni-Mo alloys containing SRO.

In the study of ordering with field-ion microscopy

the ability to distinguish antiphase boundaries is important.




- 58 -


Fig. 10--A field-ion micrograph of CoPt
showing a domain of SRO above the critical
temperature (after Southworth and Ralph).




- 59


There has been very little work, done on the Interpretation

of crystallographic features from field-ion micrographs,

since this part of the science is in its early stages.

Consequently it Is not known exactly how an antiphase

boundary should appear on such a micrograph. The appearance

depends sensitively on the particular system and on the

type of boundary in question. A good example is again the

system Co-Pt (88]. This system orders by forming alternate

layers of Co and Pt atoms on (002} planes. An antiphase

boundary of the type in which the "C" axes of the adjacent

domains are parallel can be distinguished as a line of

bright atoms and vacant sites cutting across certain poles.

For such a boundary, since the "C" axes of the two domains

are parallel there is no misfit between adjacent domains

even though ordering is accompanied by a tetragonal dis-

tortion. Apparently the boundary is visible because of

preferential field evaporation along the line of its

intersection with the surface. A rotational type of domain

boundary occurs when the two domains have their tetragonal

"C" axes at right angles. Such a boundary is clearly

visible as a line along which the rings of the crystallo-

graphic poles are sheared slightly. This shearing is due

to the fact that a certain amount of misfit occurs along

this type of boundary.





- 60


There are two other features of a domain boundary

that might contribute to its visibility in a field-ion

micrograph. Even though the boundary is distorted, it is

an interface, i.e., a region in which the stacking sequence

has been interrupted. Consequently the Imaging properties

in the region of this interface should be affected to some

extent since the interface represents a region of pertur-

bation of the lattice potential. On this basis alone one

would expect the boundary to appear as a bright or dark

line. If a slight distortion is present within the boundary

its visibility should be enhanced since the distortion

represents an additional perturbation of the lattice.

Regardless of which of the above factors contribute

to the visibility of a domain boundary it should appear as

a line across which there is no rotation of the crystallo-

graphic poles. This is the criterion that distinguishes a

domain boundary from a grain boundary, a subgrain boundary,

or a twin boundary.


1.5 Smatt-Angle X-ray Soattering

Small-angle x-ray scattering measurements were used

in the present work as a means for examining the early

stages of precipitation of the B phase. This work was

complimented by electron-transmission microscopic studies.

The small-angle scattering technique can be used to study




-61


the growth of particles which are in the size range of

approximately 10 to 1,000 A; hence, it effectively compli-

ments electron-microscopic techniques in many cases. With

proper analysis one can determine such quantities as the

size of the particles, the surface area per particle, and

the density of particles in the system.

The theory of small-angle scattering is discussed

in great detail in a book by Guinier and Fournet (89]. The

phenomenon is based on the fact that any dispersion of

particles or voids within a matrix will produce a measurable

scattering intensity near 26 = 0 since all particles,

whether they are crystalline or not, will produce a {000}

diffraction node. This node, in the ideal case, is a very

sharp peak, though it cannot be observed because it cannot

be separated from the primary beam. If the particles are

in the size range 10-1,000 A, however, this {000}node will

broaden sufficiently to be detected without interference

from the primary beam, provided that a suitable experimental

technique is used. Hence the small-angle scattering

phenomenon is actually a special case of line broadening

due to particle size. Measurements are generally made in

the range 26 = 0.5-60

The most common application of small-angle scattering

measurements is to determine particle sizes through a




-62 -


quantity known as the radius of gyration. The square of

the radius of gyration is defined as the second moment of

the mass of the particle divided by its volume. For spheri-

cal particles the true radius and the radius of gyration

are equivalent. Guinier (116] has shown that in a system

in which the distribution of precipitates is random and

dilute, the small-angle scattering intensity can be written

as:


I = Kexp 4223


(12)


radius of gyration of the particle

26 expressed in radians

wavelength of incident radiation

a constant which contains geometric factors,

incident beam intensity, density of particles,


where R =


etc.

This equation is known as the Guinier approximation. Within

the range for which it is valid a plot of log I VS. E2

yields a straight line. From the slope of this line the

radius of gyration of the particles can be determined.












CHAPTER II

REVIEWS OF PREVIOUS WORK ON NICKEL-MOLYBDENUM ALLOYS


2.1 .The Nicket-Molybdenum Phase Diagram

Most of the basic features of the Ni-Mo binary phase

diagram have been established. In Figure 11 the general

form of the complete diagram is shown. It is taken from

Hansen's Constitution of Binary AZZoys 190]. The region

between 12 and 50 a/o Mo has recently been reinvestigated

[91,92]. This part of the diagram is shown in Figure 12.

The main differences between this and the previous diagram

are an increase in solubility of Mo in Ni at the peritectoid

reaction, a +t y*B, and a decrease in the solubility of Mo

in Ni below this reaction.

The earliest published work on the Ni-Mo system was

by Grube and Schlecht 193] in which they reported the results

of Baar, Dreibholz, and Koster and Schmidt along with their

own. Baar and Dreibholz used thermal analysis and optical

microscopy in a study of the high-temperature region of the

diagram including the solidus and liquidus lines. Koster

and Schmidt found that alloys containing greater than 18 w/o

Mo showed an increase in hardness upon aging. According to

their work the aging began at 6000C and the maximum effect

was observed at approximately 8000C. Grube and Schlecht


-63















I I 1 I I I I I I


jAMo



"3~35QL
s


0 10 20 30 40 50 60 70 80 90


a/o Mo


- 64


>so,


NiqMo Ni3Mo ; N.


1500 /
1430C | / V

1400-y /


1300~- I 13150C


12 00j-I








-910*C !
~8600C



8 ~ Iii II



70 ~1 II I
600-I 1


isa hIl I fl


Mio


4001----LT--L--~lr


U

a,
k
7
c,
k
a,
c,
E
a,
E-(


Magn,
transt.-


2 0 C)-


0 5 10
a/o Mo


Fig. 11--A binary phase
Hansen).


diagramn of the Ng-Mo op~tem


Cafte~r




-65


1100





1000-









ayto


700~ -a




a+






600





500
16 20 24 28 32 36
w/o Mo

Fig. 12--A partial binary phase diagram of the Ni-Mo system
(after Gurhrie ana Stansbury).





- 66


used resistivity, metallography, hardness, and ~x-;cay dif-

fraction in an attempt to complete the phase diagram but

were only partially successful. They were able to identify

the B phase, which corresponds to the composition Ni4Mo, as

a tetragonal phase; however, there was considerable un-

certainty in the low-temperature region of their proposed

diagram between 31 and 62 w/o Mo.
The work of Grube and Schlect was continued by

Grube and Winkler 194]. They were able to confirm the

existence of the previously questionable y phase corre-

sponding to the composition Ni3Mo. They identified this

phase as having a close-packed hexagonal structure with a

c/a ratio of 1.65. The y phase has since been identified

by Saito and Beck [95] as an orthorhombic phase of the

Cu3Ti type, having the following lattice parameters:

ao = 5.064 A; b0 = 4.224 A; co = 4.448 A. According to

Saito and Beck there is reasonably good agreement between

the observed and the theoretical x-ray intensities for the

y phase if one assumes the orthorhombic structure. Despite

the disagreement on the structure of the y phase, the

complete diagram proposed by Grube and Winkler has proven

to be essentially correct. There have been changes in

solubility limits and reaction temperatures however,

The work of Grube and Winkler was largely confirmed

by E11inger 196] who used x-ray diffraction, metallography,





-67


and hardness data. E11inger proposed a complete phase

diagram which is very little different from that of Grube

and Winkler except for the reaction involving the evolution

of the B phase. According to E11inger the B phase pre-

cipitates by means of the peritectoid reaction: a + y*8B.

According to the diagram of Grube and Winkler the a phase

can transform directly to the B phase; however, this may

be regarded as a special case of the above peritectoid

reaction in which the compositions of the a and B phases

are the same. This apparent difference is therefore only

a question of the solubility of Mo in the a phase at the

peritectoid temperature. The crystal structure of the B

phase was also identified by E11inger as close-packed

hexagonal in agreement with Grube and Winkler; however,

Saito and Beck identified this phase as orthorhombic

as already noted. The most recent work in this area of

the phase diagram is by Guthrie and Stansbury [91] and

Stoffel and Stansbury [92]. Their work is shown in

Figure 12, and it is in agreement with that of Grube and

Winkler with regard to the evolution of the B phase.

The crystal structure of the B phase, which was

originally identified by Grube and Schlect as face-centered

tetragonal, has since been revised by Harker [97]. Harker

identified the Bravais lattice as body-centered tetragonal




- 68-


andthespce rou a C~ -I .This structure has since

been confirmed by Guthrie and Stansbury 191], who also

showed that the B structure can be derived from the a by

the process of ordering accompanied by a tetragonal dis-

tortion. The evolution of the B phase from the a phase

will be discussed in more detail later. The solubility

limits of Mo in Ni established by Guthrie and Stansbury

are in reasonable agreement with the work of Mikheev in

this region [98]. Riddle [99] made dilatometric measure-

ments and determined that the temperature of the peritectoid

reaction a + y*8B is 8650C. This result is also in good

agreement with the work of Guthrie and Stansbury.

The crystal structure of the phase NiMo has not

been completely established; however, it appears to have

tetragonal symmetry. This conclusion is based on the

work of Smiryagin et at. [100), Bagaryatskii and Ivanovskaya

[101], Shoemaker et at. [102], and Obrowski fl03].


2.2 IThe a to B Transformation

The crystal structure of the B phase, corresponding

to the composition Ni4Mo is shown in Figure 13. This

structure, which is body-centered tetragonal, was first

derived by Harker [97] and later confirmed by Guthrie and

Stansbury [91]. Guthrie and Stansbury also showed that

this structure can be derived from the FCC a structure by





-69


the process of ordering accompanied by a tetragonal dis-

tortion. The relationship between these two structures

is also shown in Figure 13. The nature of the ordering

is such that the Mo atoms are arranged as far apart as

possible within the lattice. The unit cell of the 8

phase contains 8 Ni and 2 Mo atoms. The coordinates of

the Mo atoms are 0,0,0; 1/2,1/2,1/2; and the coordinates

of the Ni atoms are x,y,0; x,y,0; y,x,0; y,x,0; x+1/2,

y+1/2,1/2; x+1/2,y,+1/2,1/2; y,+1/2,x+1/2,1/2; y+1/2,

x+1/2,1/2 where x = 0.2 and y = 0.4. Harker's values for

the lattice parameters of the unit cell are ao = 5.720 A,

co = 3.564 A, c/a = 0.6231. The volume of the cell is

2-1/2 times the volume of the distorted unit cell of the

FCC lattice from which it is derived. Accompanying the

transformation are shrinkages of approximately 0.5% and

1.0% in the tetragonal ao and co directions, respectively.

There is good evidence that the B phase actually

does form by ordering of the a phase. It can be shown

that for such a process there are 30 possible ways by

which the 8 phase may nucleate, but due to the multiplicity

of the lattice this results in only six different orien-

tations. Spruiell (31] has shown by oscillating single

crystal x-ray diffraction patterns that the predicted

orientation relationships between the a and 6 phases are




S70 -


[110 ] --+-
O O


O Ni atoms
O Mo atoms
', position of Mo atoms
in next (100) layer








Fig. 13--Schematic representation of the ordered Ni4Mo
Cg1 phase Cafter Guthrie and Stansbury).





S71, -


always found. Further evidence that the B phase is derived

by ordering of the a phase is provided by the x-ray powder

data of Guthrie and Stansbury [87]. During the transfor-

mation a to B, the primary a refections split into triplets,

with the exception of the {lll} and {222} planes. One of

the peaks of the triplet is a nearby superlattice reflection

of the B phase. The other two peaks are due to a splitting

of the original primary peak since the a and B phases have

slightly different lattice parameters. During the early

states of the a to B transformation the Debye pattern shows

the formation of a face-centered tetragonal lattice. As

the transformation proceeds, the weak superlattice lines

appear, indicating the presence of the fully-ordered

body-centered tetragonal B phase. This picture correlates

with the hardness data of Guthrie and Stansbury in the

following way: there is a rapid initial increase in

hardness during the a to B transformation followed by a

decrease and then another increase. Guthrie and Stansbury

interpret the initial increase as being due to local

tetragonal distortions of the a lattice at the beginning

of the transformation. The absence of the well-defined

superlattice lines indicates that the degree of order at

this stage of the transformation is very imperfect. As the

degree of order increases, coherency strains increase to a





-72


certain point and then coherency is lost. This loss of

coherency is accompanied by a decrease in hardness. As

the domains of the B phase continue to grow in size and

perfection and to impinge upon one another the hardness

again rises.

Metallographically the B phase is observed to

grow from the a phase in a variety of forms. It sometimes

takes the form of a very finely-dispersed second phase,

but quite often it develops as a pronounced Widmanstaten

structure. Both Ellinger 196] and Guthrie and Stansbury

191] found that the finely-divided precipitate occurred

frequently in two-phase alloys that are near the solvus

line of the a phase whereas the pronounced Widmanstaten

structure occurred predominantly in alloys near the Ni4~o

composition. Both forms of the precipitate formed

preferentially near, but not within, grain boundaries,

and very little coalescence of this dispersed phase was

observed even after several hundred hours of annealing.

With optical microscopy it is difficult if not impossible

to distinguish between a structure containing precipitated

B in primary a and one containing pure B that has trans-

formed from the a phase. This is because both structures

may form a highly developed Widmanstaten pattern. In the

case of the exact Ni4Mo composition this striated structure




-73


remains even after all the a phase has transformed. The

observed etching effects were attributed by Guthrie and

Stansbury to the high degree of strain that accompanies

the transformation and the mismatch, i.e. lattice strain,

at the interface between impinging domains. They also

assumed that the B phase precipitates as platelets.

Since only three directions were ever observed for the

striations which appeared in the primary a grains, they

further assumed that the platelets formed on (100} habit

planes within the a matrix. Spruiell [31] has since

shown that the observed striations were parallel to (111}

planes. There was some doubt, however, about whether

these were slip traces or actually B platelet;. No

confirmation of a {111} habit plane has yet been derived

from electron-microscopic data.

Alloys within the a and B region of the phase

diagram have been found by several investigators to

respond readily to age-hardening treatments. Studies of

this type have been performed by Ellinger [96], Guthrie

[104], Block [105], and Guthrie and Stansbury [911.

Although the agreement of the data is not particularly

good, the investigators all agree that largo hardness

increases can be produced within a few tenths of an hour

by annealing in the temperature range of approximately




-74


6000C to 8500C. Both E11inger and Guthrie and Stansbury

observed a definite two-stage hardening process during

isothermal annealing, and the latter even observed a peak

hardness during the first stage as previously noted. The

beginning of the second stage of hardening was found to

vary with the annealing temperature and the composition,

but in general it occurred after a time of several hours,

sometimes taking as much as 50-100 hours. The hardening

is assumed by Spruiell [31] to be mainly due to the strains

induced in the lattice as the B domains grow in size and

perfection.

Hardness measurements were used by Block [105] to

study the kinetics of the isothermal a to B transformation

in alloys containing 27, 28, and 29 w/o Mo. From these

data he constructed a time-temperature transformation

curve that had a "C" shape typical of many solid-state

phase transformations. The maximum rate of hardening was

found to occur at 8000C. Lampe and Stansbury [106] later

used resistivity measurements to study the kinetics of the

a to 3 transformation in an alloy with the Ni4~o composition.

They also constructed a TTT curve from their data and found

it to have a "C" shape. The "nose" of the curve was found

to lie within the temperature range 7100C-7750C, and in

this region the start of the transformation was too rapid





-75


to be observed. It was also found that above approximately

8600C and below approximately 6200C the transformation had

not begun even after 1,000 hours. This curve is reproduced

in Figure 14.

Since the a to 13 transformation proceeds isothermally,

it is assumed to occur by a nucleation-and-growth process

whereby uphill diffusion of Ni and Mo atoms is the rate-

determining factor. No evidence has yet been presented

that would indicate a shear-type transformation, nor is

there any evidence to indicate that an intermediate

metastable phase (or phases) forms during the transfor-

mation. If the a to B transformation is indeed a direct

homogeneous ordering process, as the orientation relation-

ship between the two phases indicates, then it follows that

at some stage of the process the G phase exists in a

coherent, or at least partially-coherent, "zone" state.

Due to the tetragonal lattice contraction upon ordering

and the high degree of strain that accompanies the growth

of these zones there will be a tendency for the zones to

lose coherency with the matrix. The B phase may therefore

prefer to nucleate at imperfections such as dislocations.

Evidence will be presented later in support of this idea.

The question of whether the transformation actually occurs

by a nucleation-and-growth process or by spinodal de-

composition cannot be answered from current data.






-76


O


i,C




*


OO
U)








OE






10


*O

13

=r-


uZ


30 'nPld a a





- 77


2.3 The X-state in NE-Mo AZZoys

Thomas (1] in his study of K-state alloys predicted

the occurrence of a K-effect in Ni-Mo alloys. This con-

clusion was based upon two facts: (1) Grube and Schlecht

[93] had observed an "S" shaped resistance versus tempera-

ture curve for 18-21 w/o Mo alloys, indicating the presence

or an anomaly. (2) The system Ni-Cr exhibits a strong

K-effect and Cr and Mo are both in group VI-B of the

periodic table. Anomalies were later found in the Young's

modulus of Ni-Mo alloys by Kritskaya et aZ. [~107] and

Polotskii and Benieva [108]. These investigators found that

alloys in the composition range 5-19.5 w/o Mo exhibited

an increased modulus after certain appropriate annealing

conditions. One of these conditions was annealing at

4000C after quenching from 1,0000C, corresponding to one

of the conditions under which the anomalous resistivity

rise can also be produced in K-state alloys. Sukhovarov

et aZ. [130] later verified the existence of a K-state in

Ni-10 a/o Mo alloys using electrical resistivity and

calorimetry measurements. Subsequent work by Kudryaytseva

et aZ. 128] and by Popov et aZ. r29] yielded similar results.

The experiments of references [28,29,30] were directed

primarily toward a study of the kinetics of formation of

the K-state in Ni-Mo alloys, and although it was assumed






S78


that SRO was responsible, no direct evidence from x-ray

diffuse scattering was obtained.

Spruiell (31] later did a study of SLO in the Ni-Mo

system by x-ray diffuse-scattering measurements on single

crystals containing 10.7 and 20 a/o Mc. he found a

definite SRO in both alloys after quenching from rempera-

tures in the range 900-1,2000C. He also found that an-

nealing the 10.7 a/o Mo a;;oy in the te perature range

4000C to 5000C sharpened .;;e di~ffue in,~~t yexima,

indicating an increase in the degree of SRO. Ci

temperature range corresponds to rhat ,r; wa;e ..~ 0-state

can be produced, although Spruiell did not correlate his

data with resistivity studies. Another sio .-.ficant study

made by Spruiell was the investigation of toa e4 ,e in

the diffuse intensity of a quenched 20 a/o as S: im~en

upon annealing at 6500C. The diffuse ca~_ used by

the SRO existing at the quenching tume~~r < o observed

to gradually split into doublets and to s..a:~c ...iderably.

The diffuse inten. ty appeared to be gradual/a ,. coachingg

the sharp intensity distribution of the fully ordered

Ni4Mo structure.

z..80I on the diffuse-scattering datac from the

10.7 a/o and 20 a/o Mo alloys, Spruiell concluded that SRO

in dilute Ni-Mo alloys is similar in structure to the





-79


NiqI\o type of LRO. In Figures 15 and 16 his models for

SRO in these alloys are shown. The several layers of

atoms in these figures are successive {002} planes with

planes 1, 2, and 3 lying above plane 0 and 1 2', and

3' lying below plane 0. The dark circles represent Mo

atoms, the light circles Ni atoms, and the crossed circles

"statistical-average atoms" which are 66.8% Ni. The

number of atoms included in each of the figures in the

minimum number required to obtain a reasonable fit to

the derived SRO parameters. In both the 10.7 and the

20 a/o Mo alloys, the SRO may be interpreted as small,

imperfect domains of Ni4Mo-type LRO.

Baer [109] did a similar x-ray diffuse scattering

study on single crystals of Ni4W, which has the same type

of LRO as Ni4Mo, and concluded that a domain structure of

Ni4Mo LRO is not the proper model for SRO in Ni-W. Baer's

model consists of double layers of {420} planes which are

alternately nickel-poor and nickel-rich as compared to

the random solid solution. He extended this idea to the

Ni-Mo system. Baer did an earlier study of the K-state

in Ni-W and Ni-Mo [27] in which he compared resistivity

measurements with diffuse x-ray scattering on Ni-W

polycrystals. He was able to correlate SRO with the

K-state in Ni-W but because of the low diffuse intensity




-80


Ol@7 O


-C plane 0








Plane 1 or 1'

000~


Plane 2 or 2'




Plane 3 or 3'


Fig 15--Spruiell 's model of the SRO existing in a Ni-10. 7
a/o Mo alloy quenched from 10000C into iced brine (after
Spruiell).


9



-$3-




-81


010


0 4
O O
OO <

O O
SOO
O


O
OO
O O

SOO
O O
O


Plane 0


OlO


O O Q

O O
9 0 0 ~


Plane 1 or l'








Plane 2 or 2 "






Plane 3 or 3'


010


010


OlO

OiO


Fig, 16--Spruiell's model for the SRO existing in NitMo
quenched from 10000C into Iced brine (after Spruiell).





- 82 -


of Ni-Mo alloys (due to the small difference in atomic

scattering factors) no such correlation was made.

McManus r32] has also demonstrated the presence

of SRO in Ni4Mo alloys quenched from above the critical

temperature. He used polycrystalline specimens and

derived the SRO parameter al from these powrder data. The

diffuse-scattering curve had the typical shape of a system

in which the atoms prefer unlike neighbors (Figure 7),

This result is in agreement with the work of Spruiell [31]

in which the proposed atomic model contained no Mo-Mo

nearest neighbors. The actual value of the a1 parameter

derived by McManus was -0.13 whereas the maximum possible

value is -0.250. Again no correlation was made between

the observed SRO and the K-state. Of importance, however,

is the fact that the size-effect parameter,31, was

essentially zero. This means that one can ignore the

scattering contribution due to atomic-size difference in

Ni-Mo alloys when correlating SRO with property changes.

The work of Popov et al. r29] on the kinetics of

K-state formation in Ni-Mo alloys gave interesting data on

the migration of atomic defects in FCC Ni and its alloys.

By isochronal annealing of 10 a/o wire specimens which

had been cold-worked or quenched from an elevated tempera-

ture, these investigators found two stages of K-state





S83


formation, one at approximately 1000C and the other at

about 3000C. These stages correspond to the two stages

of recovery of point defects in cold-worked and quenched

Ni, the activation energy for the lower stage being about

22 K-cal/mole and that of the second stage about 36 K-cal/

mole [110,111]. These two stages were attributed [112] to

the annealing out of interstitial atoms and vacancies.

Popov et al. also found that the activation energy for

the low-temperature stage of K-state formation in Ni-Mo

was about 22 K-cal/mole. They reasoned that if the K-state

were due to SRO, the low-temperature stage could not be

caused by the migration of interstitials for two reasons:

(1) In order for SRO to take place by interstitial

migration both Mo and Ni atoms would have to move by this

mechanism. Due to the large size difference between Ni

and Mo atoms, (Goldschmidt diameters of 1.24 A and 1.40 A)

it is unlikely that Mo atoms would move by an interstitial

mechanism. (2) If this stage were due to interstitial

migration then the motion of the larger Mo atoms would be

the rate-controlling step; consequently, the measured

activation energy should be much higher than the 22 K-cal/mole

since this is also the activation energy found in pure Ni.

For the high-temperature stage, they found an activation

energy of about 66 K-cal/mole as compared to 36 K-cal/mole





-84


for pure Ni. They assumed that this stage is indeed due

to the migration of vacancies as assumed by other in-

vestigators r110,111]. They assumed that the disagreement

in activation energies is caused by the binding energy

between Mo atoms and vacancies. By comparing the initial

rate of K-state formation for specimens quenched from

various temperatures and annealed in the range of the

high-temperature stage, they evaluated the energy of

formation of vacancies for Ni 10 a/o Mo. By comparison

with the activation energy for migration of vacancies in

the high-temperature stage, they evaluated the binding

energy between vacancies and Mo atoms as being approxi-

mately 20 K-cal/mole. These calculations were based on

the assumption that the K-state in Ni-Mo is caused by SRO.

As further evidence of the validity of this assumption it

was noted by the authors that these two stages are also

observed in K-state formation in Ni-Cr [ll3], and for that

system the activation energy for the high-temperature stage

is almost identical to that for pure Ni. Furthermore, the

atomic sizes of Ni and Cr are almost the same. This is,

at least, strong evidence that the K-state formation is

diffusion controlled in Ni-Mo and in Ni-Cr even though it

does not definitely prove that SRO is a necessary condition.




-85 -


Interesting anomalies in the specific heat of Ni-Mo

alloys were found by Stansbury, Brooks and Arledge 19].

They found that the specific heat versus temperature curve

for Ni 15 w/o Mo underwent a rapid rise in the temperature

range 500-6000C, indicating that changes in the structure

of the solid solution were taking place. By comparing the

integrated energies under the ideal and the real curves,

they found that the energy expended in the process corre-

sponding to the anomaly was about 5.5 joules/gm. The

authors attributed this effect to the partial destruction

of SRO as the temperature was increased. They found a

similar effect in Ni-Cr, which is another K-state alloy

thoughtto exhibit SRO, whereas pure Ni did not show this

effect (Figure 17). They concluded that the anomaly in

Ni-Mo and also in the Ni-Cr alloys was due to the de-

struction of SRO. Since the equilibrium degree of SRO

should decrease with increasing temperature one would

expect that SRO would be continuously destroyed as the

temperature increased. Therefore, one should see a higher

than normal value of the specific heat over the entire

temperature range studied. The fact that this effect is

not observed implies that the equilibrium state of SRO

does not keep pace with the changing temperature until

about 5000C is reached. This means that one should expect




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