A calculation of the elastic constant c33 of a graphite crystal as a function of temperature

A CALCULATION OF THE ELASTIC CONSTANT Cl,., OF A GRAPHITE CRYSTAL AS A FUNCTION OF TEMPERATURE B. T. KELLY UKAEA Reactor Fuel Laboratories,

Springfields Works, Salwick, Nr Preston, Lancashire, England

(Received

1973)

5 November

Abstract-A calculation is presented of the elastic constant CM of a graphite crystal as a function of temperature up to 2500 K, taking into account the anharmonic contribution and the changes in interlayer interactions due to the large lattice thermal expansion. Parametric variations in the theory show that the anharmonic contribution to G depends principally on the parameter (a’C,,/~e~,). Comparison of theoretical results with the experimental data, which is mainly from neutron scattering experiments, shows that the data can be accounted for if (a*CJae~,) lies in the range 7-10 X 10” dynes/cm*. A theoretical estimate of (a’C,,/ae?,) based on Lennard-Jones potentials between atoms in adjacent basal planes gives a value of 9.07 X lOI dynes/cm?. 1. INTRODUCTION Measurements highly

of the

oriented

natural

which

by

Shirane[5]

theoretical

up

variation

of the

if we know e,, as a function

variation

inter-

of C& as a function A

number

of the interlayer published[lO-131

forces and

has been

shown [12, 131 that pair potentials

vided

of

value

of C,,[12,

that

of in

tuted

it

can be used to calculate

the

of temperature,

in

anharmonic

of

effects

in

these

are small. If the values of Cs, and obtained in this way are substithe

thermal

Kelly and Walker

expansion

500°K [17]. The

theory

of

[ 151, much too large a varia-

tion of cr, with temperature neutron

is obtained scattering

above

data [4-91

indicate that Css does not decrease as rapidly with increasing temperature as the simple

first derivative with respect to strain e,, parallel to the hexagonal axis. These potentials the

ac-

of C& and (a C,,/a eLZ)with strain e,,;

parameters (aC,,/ae,,)

the Lennard-Jones and Buckingham types readily account for the value of CJJ and its

for

to

be accurately

then CQ3 and (aC,,/ae,,) can also be estimated as a function of temperature, [16, 171, pro-

and

have been

2200°K.

potentials[lO-131

of neutron

have

account

could

showed

(Y, parallel

counted for using the theoretical values of CQJ and (G,/ae,,) at 0°K. The interlayer

ap-

graphite

cannot

axis

zero.

Brockhouse

been

hexagonal

meas-

Roy [4],

to

the

coefficient

at absolute

studies

studies

values

paper [ 151 the author

the expansion

to be any direct

and others[6-91

temperature

that

tempera-

However

to yield values

In a previous

and

to be good

of the temperature

constants.

scattering preted

at ambient

to the values

and

Seldin good

and suggestions have recently been made that C4 is largely electronic in origin [14].

of

graphite

produced

constants

do not appear

urements elastic

have

constants

of Some,

are also believed

proximations There

crystals

[l-3]

for the elastic ture

pyrolytic

graphite

co-workers

elastic

131

theory of the change of interlayer forces with strain indicates, and thus either the constants

.535

536

B. T. KELLY

in the pair potentials are actually temperature dependent or the anharmonic contribution to Css is significant. There may also be anharmonic effects in (aC,,/ae,,), but the only experimental data is at ambient temperature. The purpose of this paper is to calculate the variation of Css with temperature, including both the effect of strain on the interlayer forces and the anharmonic contribution.

and, therefore,

from (3) (5)

Substituting Equation spectively gives

(1) in (3) and (5) re-

4. THEORY Consider a unit cube of graphite crystal with the c-axis parallel to a cube edge which is also the z-axis of a Cartesian coordinate system. The free energy/unit volume of the cube is written as [18]: F = E + kTx

i

In (1 - exp [- hu, /kT]),

(1)

where E k h vi

is the lattice energy/unit volume is Boltzmann’s constant, is Planck’s constant, is the i’th lattice vibrational frequency. The summation is to be taken over all lattice vibrational modes in unit volume. The thermodynamic relation between pressure P and free energy F is p=_

( > _g

*

(2)

T

The volumetric strain of a graphite crystal is essentially the same as the interlayer strain e PI, so that to a good approximation, P is equivalent to a stress a,, parallel to the hexagonal axis, i.e., (3) By definition, (4)

and

exp (-h/kT) [l - exp (-hvi/kT)]*

’

(7)

In order to evaluate Equations (7), we need the values of the lattice frequencies pi and their first and second derivatives with respect to stram eZZ. The acoustic mode lattice vibration frequencies are given in the Komatsu [19] semicontinuum model of the lattice dynamics of graphite by analytic expressions. The theory of the thermal expansion coefficients of graphite crystals developed by Kelly and Walker[15] and refined by Kelly[ZO] shows that only one vibration mode, the out-ofplane acoustic mode, is significantly modified by strain eZZ_The dispersion relation of the mode is:-

where p is the density, CT~and a, are the wave number components parallel and perpendicular to the basal planes, and 6 is the bond bending coefficient as defined by Komatsu [19]. The differentials of the dispersion relation (8) are:

GRAPHITE

The

calculations

cient

CRYSTAL

of thermal

(Y, [20] indicate

AS A FUNCTION

expansion

that (&Y/ae,,)

tively small effect compared

method

coeffi-

has a rela-

to (8 G/a

ezL_)and

537

OF TEMPERATURE

due to Crowe11[23]

riate

pair potentials

tion

over

in which approp-

are summed

a uniform

atomic

(a&/de,,). The parameter S is a property of the intra-layer covalent bonds which cannot

layer. We assume

a Lennard-Jones

be much

tween

in adjacent

thus

affected

we assume

lected

that

e%,). With

strain

(a’s/ae?,)

by comparison

(a%,,/8 tions

by interlayer with

these

e,, and

assumptions

(9) and (10) reduce

rated

may be neg-

(a’C,,/ae%)

atoms

by a distance

by integra-

density

in each

potential

layers

only,

besepa-

r

and Equa-

(13)

to:

I

+$’ P

a (12)

The values of (aC,,/ae,,) and (aC,,/ae,,) are known from the work of Spain et al. [21] but it

is a constant.

is necessary

over a uniform

mate at the

in order

(a*E/de:,),

to evaluate

(d’C,,/de:,),

interlayer

spacing

(7), to esti-

and (a’C,,/aeZ,) corresponding

to

where

plane

from

Thomas-Fermi to provide

calculations

estimates,

[22].

If Equation

following

energy/unit

0°K. Each of the above values may be estimated from atomic pair potentials between atoms in different layer planes[lO-131 or order

A is the Van Der Waals constant density

(T of atoms

Crowe11[23],

volume

and rll

(13) is integrated the

in a basal interlayer

is

E =v[;

(2)“-

11,

(14)

In

we will follow the

where

No is the number

of atoms/unit

volume

538

B. T. KELLY

and d the interlayer spacing. The equilibrium condition for the crystal at O”K, with interlayer spacing d = do, is C3E

(ad>d=4=0)

(15)

and by using d, = 3.3535 x lo-@ cm C&d,) = 3.67 X 10” dynes/cm*. In the same way we obtain

and

($?)d_d..

(20)

= d,“(j$)d;d”

= -l%s(do),

which yields r,,== 2d:.

(16)

The energy E at a spacing d different is readily found to be

from do (%)d=do

E(d) =+GV#$-+I. The elastic constant given by

1d4

($=d’(s) =!+_)[1~(!%)6_5](~)Y

(19)

The value of Equation (19) at a temperature T”K is obtained by substituting the value of d for natural graphite at that temperature [24],

0 exp {(h/k~)(4~~3*~,4 1- exp {- (h/k~)(4T’~‘u,4

((

= 251Css(d,).

-1,4d,

+ (C,,/pm2d,2) sin’ 7rd,uZ + (C~lp)c~?“*} + (Css/p,r*d2) sin’ rdOgz + (C+JP)~~)~‘~I I

C44 CQ3 sin’ rr d,u, + a,’ 47r262u,4 + 2 pn d, P C44 sin’ r d,c+, + - ua2 P (I-

exp {-- (hlkT)(4r 26 ‘aa4 + (Css/p,r2d02)sin2 Td,u, f (Cti/P)Ua’)“‘} exp {- (h/kT)(4,r*62u2+ (CQS/pr2d:) sin’ Ird,uZ + (cdPb~)“2))*

x 1 [(sin’ ~d,a,lp~*d,‘)(aCQQ/aell) + (~~21~)(acdae=)12 4 [4T262~,4 + (Css/pr2d,*) sin’ rrd,u, + G/P)u~I

which is our principal

result.

(21)

Experimentally [21] (a CM/ae,,) = - 6C44 but it is not possible to estimate (aC,/ae,,) or (d*CJae%) from the Crowell model because the assumption of a uniform atomic density in the basal planes reduces the shear resistance, and hence CM, to zero. However we note that the numerical coefficient of CSs in the equation for (#C,,/aeL) is roughly the square of the numerical coefficient of C& in the equation for (aC,,/?Je,,) and, making the same assumption for the parameters associated with shear, choose (L!‘C,,/deE,) = 35& for initial calculations. The final equation for CSs on conversion of the second term of Equation (7) from a summation to an integration over the first Brillouin zone is

0 (W

= = 12V;:No.

The first term in Equation (7) at a spacing d different from d, corresponding to temperature T”K is

’

= d,‘(j$)d_dQ

(17)

C&3at 0°K (i.e. d = d, is

C&d,) = d,2 $j [

which is to be compared with the experimental value[21] of - 15Css, and

da, I)

. due.

GRAPHITE 3. NUMERICAL

Equation

AS A FUNCTION

CRYSTAL

OF TEMPERATURE

EVALUATION

(22) is of the form

(23) where we refer to A(T) as the anharmonic contribution. The expression for A(T) in Equation tion

(23) was programmed

on a computer

Central

Technical

using

the following

by Mr

for

evalua-

M. Toes

Services

of the

numerical

of the UKAEA

values.

Numerical value

Parameter

6.623 x 10mz7 Ergs/set 2.46 X IO7 cm-’ 3.3535 X lo-” cm 2.26 gmicc 1.380 x 10-‘h Ergs/deg 6.11 X lo-” cm2/se6’ 3.67 X 10” dynes/cm’ 4.00 x 10” dynes/cm’ - 15G5 = - 5.5 x 10” dynes/cm’ - 6Cs4 = - 2.4 X 10” dynes/cm* 251C,, = 9.0’7 X 10’” dynes/cm’ 36C,, = 1.44 X 10” dynes/cm*

h (T,,,

d,,

P k

6

G C14 (aCk/de,,) (aG/de,J (dYk/deb)

(a2C.dae~,)

Fig. 1. Anharmonic

ature

value of

A (T) as a function

is shown in Fig. 1. The calculated

of Cs, as a function pared

of temperature

with the experimental

The (a’C,Ja

effect

of variation

e:,) and (a’C,,/a

was examined retaining The

of temper-

(a’C,,/ae~,)

values in Fig. 2. of the

their values while

numbers

examined

parameters

ez,) on the term A (T)

by changing

the other

values

values

are com-

as in the table.

were

= 6-O X 10” dynes/cm*

results

12.0 X

10’” dynes/cm2 ef,) = 1-O X 10” dynes/cm*

over

most

linear

and

2.0 x

with

numerical

values

of A (7)

case calculated

as a comparison.

the

as a function

initial

required

contribution

of

A(T)

lattice

tion

(22).

the

The

from

(8’C,,/a

upon in Fig.

Fig.

of

likely,

d-

representa-

shown

in Fig. 3

(a’C,,/ae:,)

that

the latter

The

The

precise

values

two experimental

highest

estimate

from

in

indicate

7 and 10 x our simple

of 9 X lOI dynes/cm*. deduced

to

of these points

temperature

lies between compared to

as has been

at a

depen-

experimental

sufficiently

correct

and

the magnitude

upon

former.

us to choose

theoretical

with

while it is strongly

2 are not

2 at the

a good data.

of A(T)

values

the

in Fig. 2 and

the first term of Equa-

is only weakly dependent

data

that

between

varying

e:,) demonstrate

dent

is

The

the sum of the two terms

calculations varying

and

500°K.

difference shown

shown in Fig. 2 is quite tion of the experimental with

range

above

points

In fact

that

to C&, is positive

contribution

obtained

of

1 show

is approximately

to explain

the experimental spacing

calculation

in Fig.

temperature

that (a’C,,/aef,) lOI dynes/cm’

of temperature for these parameters are shown in Fig. 3 which also includes the first

A (T) to C,i

of the temperature

magnitude

allow

10” dynes/cm2 The

of

the anharmonic

parameters. (a ‘G/a

contribution

A(T) which are shown

given temperature and

“K

4. DISCUSSION

The

the The

Temperature,

analysis

It is of

specific heat data[l9], that Cs, is sensitive to the layer stacking order. The experimental

540

B. T. KELLY

Lattice + AnharmonIc contribution

Lattice

I

I

IO00

2000 Temperature,

OK

Fig. 2. Calculated and experimental

case 1 2 3 4 5

Symbol(aC33/&) v 0

l 0 +

points

denoted

by the solid inverted

Case 5 Case 4

1.44x10'2 I.00 2.00 1.00 2.00

Temperature,

Fig. 3. Variation of anharmonic

values of C&.

(a2C44/ae&)

9.07x10'= 6.00 6Go 12.00 12.00

contribution

contribution

triangles

OK A (T) to CSSwith (62C,3/Se2z7).

in Fig. 2 correspond to a less well stacked material (P = 0.2)so that the differences may

cally two-dimensional with frequencies very strongly dependent upon the bond bending coefficient S and to an insignificant extent on

well be real (although

CSs and C,.

Ross [9] comments

that

his point at 300°K is not reliable). It must be noted that we have neglected any contribution to CSs due to the out-ofplane optical modes. previous publications,

As we have described in these modes are practi-

very

The

insensitive

bond bending to e,, and thus

coefficient is the approxi-

mation is likely to be a good one. The large magnitude of the anharmonic contribution A (T) at high temperatures has consequences for other studies which have

GRAPHITE

previously

relied

on the first term

CRYSTAL

on calculations in Equation

AS A FUNCTION

of CS5 based (22) alone.

OF TEMPERATURE

.i41

REFERENCES

1. Soule D. E. and Nezbeda C. W., J. A$$. Phys. 39, 5122 (1968). 1. It has been shown [17] that a calculation 2. Blakslee 0. Z., Proctor D. G., Seldin E. J., of (Y, as a function of temperature in which Spence G. B. and Weng T., .J. A@l. Phys. 41, CJ, and (aC,,/ae,,) are allowed to vary with 3373 (1970). the interlayer spacing alone, and not includPhys. 3. Seldin E. J. and Nezbeda C. W., J. A@/. 41, 3389 (1970). ing an anharmonic contribution, leads to seri4. Roy A. P., Can. J. Phys. 49, 277 (1970). ous disagreement with experiment at high 5. Brockhouse B. N. and Shirane G.. Bull. Amer. temperature (> 500°K). The experimental Phys. Sot. 17, 123 (1972). data and the calculations presented here for 6. Dolling G. and Brockhouse B. N., Phys. Rev. CS3 show that it varies much less than the 128, 1120 (1962). B. and Smith H., if the variation of 7. Nicklow R., Wakahabayashi lattice term alone; Phys. Rev. B5, 4951 (1972). (aC,,/a e,,) is also much less then expected 8. Ross D. K., Sanalan Y. and Szabo F. P., Therfrom the lattice term alone, due to an anharmal Neutralisation and Reactor Spectra Vol. 1, p. manic contribution, then the behaviour of (Y, 343. IAEA Vienna (1968). can be understood. 9. Ross D. K., J. Phys. C. Solid State Physics 6, 3235 2. Weertman et al. [16] have proposed a (1973). 10. Girifalco L. A. and Lad R. A., J. Chem. Phys. 25, theory of the thermal creep of polycrystalline 693 (1956). graphite in which it is necessary to estimate 11. Agranovich V. M. and Semenov L. P., J. NW/. CaJ and C44 at 2773°K and also dfi/dT the Energy 18, 141 (1961). variation of an average elastic constant p with 12. Kelly B. T. and Duff M. J., Carbon 8, 77 (1970). 13. Green J. F. and Spain I. L., To be published. temperature. These parameters were esti14. Spain I. L., Presented at the Eleventh Biennial mated from their room temperature values Carbon Conference. Gatlinbere USA (19731. and the variation with interlayer spacing as ,r Kelly B. T: and Walker P. “L., Carbon 8: 211 predicted by the Crowell me;hod.-TheexI” (1970). perimental data and the calculations pre16. Weertman J., Green W. V. and Zukas E. G., Mater. Sci. Engng 6, 199 (1970). sented here show that the use of the Crowell 17. Kelly B. T., High Temperature-High Pressure 5, method overestimates the changes considera133 (1973). bly, so that and the theory of creep needs 18 Mott N. F. and Jones H., The Theory of the re-examination. Properties of Metals and Alloys. Oxford UniverThe theory presented in this paper can sities Press (1936). 19. Komatsu K., J. Phys. Sot. Jafian 10, 346 (1955); only be tested in detail when direct measureJ. Phys. Chem. Solids 6, 380 (1958); J. Phys. Chem. ments of CS5 vs temperature become Solids 25, 707 (1963). available. 20. Kelly B. T., Carbon 11, 379 (1973). 21. Green J. F., Bolsaitis P. and Spain I. L., J. Phys. Chem. Solids 34, 1927 (1973). Acknowledgements-I am deeply indebted to Mr M. 22. Santos E. and Villagra A., Ph,ys. Rev. B6 3134 (1972). Toes of the Central Technical Services of the UKAEA for the programming of the numerical 23. Crowell D. J., Chem. Phys. 16, 1407 (1957). calculations. The UKAEA Member for Reactors, 24. Reynolds W. N., Physical Properties of Graphite. Mr R. V. Moore gave permission for this work to Elsevier Press (1968). be published. 25. Bonjour E., Private Communication.