Reply to discussion “absolute grain boundary energies in copper”

LETTERS

TO THE

Reply to Discussion “Absolute grain boundary energies in copper”* In his note,

Absolute

copper, Fleischero) theoretical

value

grain boundary

contends

energies

that, in computing

in the

of E, for [OOIJ tilt boundaries

in

copper, sufficient allowance was not made by Gjostein and Rhines’2) for the decrease in elastic moduli that occurs over the temperature range from room temperaFleischer claims that the Young’s ture to 1065%.

EDITOR

modulus

need not be the same as that for the par-

ticular function of C,,, C,, and C44, that occurs in the equation for E,. Concerning the first point, Zener’s) has discussed Koster’s work, pointing out that the marked deviations from a linear decrease in modulus with temperature

result from the relaxation

stress by viscous grain boundaries.

has pointed out that, in the case of zinc, his measurements on single crystals decrease in a linear manner

Modulus data of Koster(3) indicate that the reduction

with temperature,

in moduli

polycrystalline zinc show large linearity at high temperatures.

should be 59 per cent, as opposed

15 per cent correction From this he concludes

that was originally that the theoretical

to the applied.

E, agrees

with the slope of the large angle portion of a plot of E/B vs. In 6. To explain the change in slope that such a plot shows when 8 <

that the presence of a small twist component boundary decreases. originally

given

by

Gjostein

and

found that the Read-Shockley the

in the

may as much as double the slope as 8 This interpretation is opposed to the one

experimental

values

Rhinesc2).

equation

only

when

They

agreed with 0 <

5’,

and

whereas Koster’s

Other work also supports Kamentsky(7) has determined copper

single

25”G750°C.

5”, he attempts to demonstrate

of shear

Moreover, Alers’e)

linearly. linearity

crystals

over

measurements deviations

on

from

this point of view. Young’s Modulus for the temperature

range

Up to about 450°C the modulus decreases

Above this temperature, deviations from occur and vary exponentially with tempera-

ture, while the ratio of Q-i to AEIE remains constant. From this Kamentsky are linear

functions

temperature

concludes that the true moduli of temperature,

and the high

drop is caused by dislocation

effects-

suggested that the deviations observed at larger angles resulted from the fact that as 0 increases the strain

possibly thermal unpinning. He also points out that the theoretical work of Bradburn@) and Ludloff’a)

energy becomes

predicts

confined to regions in which Hookes

a

linear

Law no longer is obeyed. It is the purpose of this note to show that, (i) the

temperature.

reduction

state of knowledge

in elastic

constants

used by Fleischer

is

It would

variation

seem, therefore, bearing

of

modulus

with

in view of the present on this subject,

that a

unrealistically large and results in erroneous conclusions, and (ii) that while it is true that the presence

linear relationship

of a small twist component

C,,, (712 and C,, are extrapolated from room temperature to 1065”C, the following values are obtained:

will affect the magnitude

of E,, his analysis is based on a fallacious assumption, which leads to a magnification of this effect.

C,, = 13.7 x 1011 dynes/cm2 C,, = 11.0

For an [OOl] tilt boundary,

E, is a function

of C,,,

C,, and C,, (equation (5) of Ref. (2)) and, ideally, values for these constants at 1065OC should be used to these data

E,, at this were not

temperature. available

method, the room temperature values were reduced 15 per cent, as was suggested by Readc4). To use as a substitute for this latter procedure, an extrapolation of Young’s Modulus data of Koster(3), which were taken on polycrystalline specimens, may be criticized on two points: modulus measurements made on

METALLURGICA,

VOL.

8, APRIL

1960

10n dynes/cm2

These constants give a value of 1200 ergs/cm2 for E, at 1065°C.

This value differs significantly

angle experimental

value of 820 f

provides no support for Fleischer’s

from the large

15 ergs/cm2 and claim that experi-

ment and theory agree at large angles. Effect of a twist component In this section it will be shown that the discrepancy between the experimental small-angle-E,,, 1520 f 100

polycrystalline specimens may give erroneously low values at high temperature and, less importantly, the magnitude of the temperature variation of Young’s ACTA

x

C,, = 4.98 x 1On dynes/cm2.

Unfortunately,

and in lieu of this

If the data of Alerst6) for

is a good approximation.

Elastic moduli

calculate

between modulus and temperature

ergs/cm2, and theoretical value of 1200 ergs/cm2 may be accounted for by the presence of a small twist component in the boundaries having 0 < 5”. The 363

264

ACTA

treatment

METALLURGICA,

given here is a development

FrankoO) and Read and Shockleyol) grain boundary,

of theory

and it differs in concept

put forth by Fleischer. The basic difference

seems

to

of

for a complex from that

lie in Fleischer’s

VOL.

8,

1960

case, taking d = [OOI]; their result was used by Gjostein and Rhines(2) to evaluate the theoretical E, at small angles. The effect of a small twist component may be studied by relaxing the requirement that 4 be in the

assumption that the total energy of the grain boundary

plane of the grain boundary.

E may be broken into the sum of two components: a tilt energy E,, which varies with the tilt angle 8,

component

according

For such a grain boundary,

to Read-Shockley

relationship,

and a twist

energy E,, which is taken to be independent Using this assumption,

-

as 0, decreases the slope becomes However,

it should

increasingly

be noted

is vector having

and

a value corresponding

E, for a twist boundary.

But according

above,

singularity

the slope of

treatment Frank’s

this

should

type

become

does

not

given below. method for determining

densities in a general grain boundary are proportional

to

to relation infinite.

appear

equation

Noting discussion

and ei is the energy of the ith set.

This

e)

(24

(2b), 7oi is dependent

of the material

(2b) upon

the elastic

and the direction

of the

line. under that 0 E O,, for the boundaries (the twist component is <0.2”), an exami-

The effect of small twist component

can be evaluated,

however, from equation

on E,

(2b), which

shows that E, is composed of a sum of energy terms, Toibi2, each contributing to the total E, in proportion to the density per unit 8 of the ith set dislocations, N,. In an f.c.c. lattice a general grain boundary may be composed of only three sets of dislocations, and in the special case where ii lies in the plane of the grain boundary, i.e. it is a pure tilt boundary, only two sets are needed. Read and Shockley(ll) have derived an expression

for E, from equation

(2b), for this latter

and

6, = a/2[101],

fi = [0, cos $, sin 41,

results are obtained;

N,2 = l/a2{sin2 y + [sin 4 cos y + cos 4 (cos y + sin y)12)

(5)

N,2 = l/a2{sin2 y + [sin C$cos y -

nation of equations (2) shows that it cannot be placed in the form E/O, = E,(A - In 6) + Es/e,, as Fleischer assumes.

h2 = a/2[OTl],

U = [cos y, sin y, 0]

may be written in the form

In equation dislocation

6, = a/2[011],

the following

(1)

In

(4)

directions of one the crystals, and the definitions of the various vectors are taken to be

is given by

-

, and so forth.

When the X, Y and Z axes are taken along the (100)

E, = $C~~~N~bi2. i constants

62 x 6,

b, - 6, x 6,

per unit 8 of the ith set of

E = Eo&4

of the subscripts

the dislocation

E = t9~N,ei i dislocations,

C is the normal

and 6,* may be found from the

equation by rotation

6,* = _

of the

shows that they

Read(a) has shown that the total energy per unit area

where Ni is the density

following

(3)

Ni and a direction

normal to the ith set of dislocations, to the grain boundary

in the

to 8, where 0 is defined now as the

grain boundary

which

a magnitude

A

magnitude of the relative rotation, about a common axis ii, that brings the two grains into coincidence. of the complex

lies in the plane

where fli boundary,

N, and

given by Readc4)

fi(fi - 6,* x Q)

more

8, = 0, there remains only a pure twist boundary

the densities N,,

mi = 6,* x s -

that when

neces-

of a third set of dislocations.

Na may be found from a relationship

Es/e,, so that

the slope should approach given

sitating the introduction

he finds that the slope of the

E/B, vs. In 8, plot is given by -E, negative.

of Be.

In this case, a twist

will be present in the boundary,

cos 4 (cos y -

sin y)12)

Na2 = 4/a2(sin2 y)(cos2 c$)(1 + sin2 $). When y = 0, the boundary

is pure tilt,

d (sin C$+ cos 4), N, = i (sin C$which

are

the

results

(6) (7)

and N, =

cos 4) and Na = 0,

obtained

by

Read

and

Shockley(ll).

To evaluate the effect for a symmetrical

tilt boundary

that is of interest,

provided

e/2 <

1, equations

let I$ = 012. Then,

(5), (6) and (7) become

N, = l/o 1/(1 + 2 sin y cos y + sin2 y)

(3)

N, = l/u d/(1 -

(9)

2 sin y cos y + sin2 y)

N, = 2/a(sin y).

(10)

On substituting the relationships given above into equation (2b) and assuming that the 70i are all equal to the value used by Read and Shockley(ll) for dislocations lying along (100) directions, the result is

E, = y + (1 -

((1 + 2 sin y cos y + sin2 y)+ 2 sin y co9 y + sin2 y)$ + 2 sin yj.

(11)

LETTERS Table

E0 ergs/cm2

8,

8,

sin y

I

I-

i.04 : 0.6

..___

v O 2.3” 5.1” 18”

Since in this work y < 20”, the sum of the first two terms enclosed in the bracket may be replaced by a constant, having a value of about two, thus allowing E, to be written as (1 + sin y),

for

y < 20”.

For small values of the tilt and twist components the magnitude

(12) of 19,

It is clear then from equations (12) and (13) that if 8, has an average value of the order 0.2”, y will have its largest value when 8, is small. Table 1 illustrates that for f3, = 0.6”, the smallest tilt angle studied, E,, may be as large as 1560t ergs/cm2, and that presence of a component

when ee < 5“. The variation experimental 100 ergs/cm2. experimental This

result

only

E, significantly

affects

of E,, from

1560 to 1320 ergs/cm2,

from 0.6” to 2’, agrees well with the value in this range, In the neighbourhood

namely, 1520 f of 0 = 3-7”, the

slope changes from 1520 to 820 ergs/cm2. is compatible

with

the

corresponding

theoretical value of 1200 ergs/cm2 for 0 = 5O. It should be noted that theory and experiment do not agree when 0 > 5“. This is true also in the case of [OOl] twist boundaries, where the theoretical value of 1100 ergs/cm2 differs significantly

from the large angle

slope of 770 ergs/cm2. It is believed that the arguments

presented

above

show that the Read-Shockley equation agrees with experiment only when 0 is small as is to be expected in view of the fact that its derivation is subject to the limitation that spacing of the dislocations in the boundary region, encloses

should be much larger than diameter of the surrounding material

the

dislocation

experiencing

line,

non-Hookean

Metallurgy Dept. Scienti$c Laboratories Ford Motor Company

which strains.

N. A. GJOSTEIN

7. L. A. KAMENTSKY, Thesis, Cornell Universit,y, Dept. of Engrg. Phys.; available in Air Force Off. Sci. Res. Rept. No. 1, Contr. No. A.F. 18(600)1000, Sept. 1, 1956. 8. M. BRADBURN, Proc. Camb. Phil. Sot. 39, 113 (1943). 9. H. F. LUDLOFF, J. Accoust. Sot. Amer. 12, 193 (1940). 10. F. C. FRANK, Carnegie Institute of Technology Symposium on the Plastic Deformation of Crystalline Solids p. 150. U.S. Dept. of Commerce, Office of Technical Services Washington 25, D.C. 11. W. T. READ and W. SHOCKLEYin Imperfections in Nearly Perfect Crystals, p. 352. Wiley, New York (1952). * Received November 16, 1959.

t This value may be even larger, evaluate when 8. is small.

No satisfactory

interpretation

lished for the forming

has yet been estab-

mechanism

of the fine sub-

structure revealed by means of electron microscope the surface of electropolished aluminium. The surfaces

“selfstructure” discovered

of with

brightened replica

since 8. is difficult

to

on

aluminium

methods

first

by

Bucknell and Geachcl), and Brownt2), studied later by Nutting and Cossle&J3), examined in detail by Welsh(4) in different electrolytes, Buss~(~) in connection solution

ability

has been interpreted last by with the preferential dis-

depending

on

the

orientation

of

crystal faces. Successful

studies

of the

formation mechanism coatings on aluminium

cell structure

and the

of the porous anodic oxide were made by Keller et d.(G),

Booker et aZ.(‘), Paganelli(*) and others. In spite of all these investigations explanation

is yet given

great similarity

no satisfactory

for the repeatedly

of size and arrangement

stated

between the

hexagonal cell structure of anodic oxide coatings and the so-called “selfstructure” for electropolished surface layers. The reason for these uncertainties attributed to the fact that electropolished were examined

hitherto

merely

by different

can be surfaces replica

methods. investigation of Systematic baths led to a suitable electrolyte

electrobrightening to develop

a solid,

porous oxide layer, thin enough to be employed for transmission electron microscopy, without the aid of replication. Rolled, annealed and cold worked high-purity (99.99 per cent) A- and

Dearborn, Michigan

4412 PP.)

1. R. L. FLEISCEER, Acta Met. 7, 817 (1959). 2. N. A. GJOSTEIN and F. N. RRINES, Acta Met. 7, 319 (1959). 3. W. KOSTER, 2. MetaUk. 89, 1 (1948). 4. W. T. READ, Dislocations in cryetuls, p. 195. McGraw-Hill, New York (1953). 5. C. ZENER, in Imperfections in Nearly Perfect Crystals, p. 289. Wiley, New York (1952). 6. G. A. Alers. Submitted to J. Appl. Phys.

A new interpretation of the substructure of electropolished aluminium surfaces*

. e

as 8, increases

265

of y may be found from the relation tan y E :

small twist

EDITOR

References

~~___

0 0.2 0.2 0.2

E, E T

THE

1

___.

1200 1250 1320 1560

TO

(99.5 per cent) called NAUTAL,

samples of commercial

B-aluminium and an Al-Mg alloy N, were polished anodically using an