J. P. L A U R I A T ET P. P E R I O 183

foy (Physique Nuclraire). Nous remercions Mme Garin (D.Ph.N/M.E., C.E.A., Saclay), MM Bfihler (Physique Nuclraire, Orsay) et Hugnot (S.A.I.P. Schlumberger) pour leurs conseils.

Rffrrences DREWER, J. I. & FITZGERALD, R. W. (1970). Mat. Res. Bull.

5, 101. FRIANT, A. (1970). C.E.A./Conf. no 1541, Saclay.

GIESSEN, G. & GORDON, G. E. (1965). Science, 159, 973. GOtJLDINO, F. S., WALTON, J. T. & PEIaL, R. H. (1970).

LE.E.E. Trans. Nucl. Sci. 17, 218. KERN, H. F. & MCKENZXE, J. M. (1970). LE.E.E. Trans.

Nucl. Sci. 17, 260. MARTINI, M., MCMOTH, T. A. & FOWLER, I. L. (1970).

I.E.E.E. Trans. NucL Sci. 17, 139. Notice Technique S.A.I.P. (1969): Pr~amplificateur de

charge refroidi pour d~tecteurs semi-conducteurs.

J. Appl. Cryst. (1972). 5, 183

X-ray Small-Angle Scattering of Glassy Carbon

BY R. PERRET* AND W. RULAND]"

Union Carbide European Research Associates, S.A., Rue Gatti de Gamond 95, 1180 Bruxelles, Belgium

(Received 10 December 1971; accepted 22 December 1971)

The micropore system of glassy carbons has been characterized by a number of structural parameters obtained from X-ray small-angle scattering. The pore sizes and pore shapes are very similar to those already observed in other non-graphitizing carbons, notably carbon fibres. The results suggest that the pore shape is needle-like with sharp edges and that the pore structure stems from a felt-like entangle- ment of stacks of ribbon-shaped carbon layers.

1. Introduction

Glassy carbons (also named vitreous carbons or poly- mer carbons) are compact non-graphitizing carbons with considerable mechanical strength and chemical resistance. They are produced by controlled pyrolysis of thermosetting resins. The name emphasizes the glass-like appearance and the conchoidal fracture of the material. In spite of its relatively low density (1-35-1.55 g.cm -3 as compared with 2.25 for graphite) the material is extremely impermeable to gases even at high temperatures. The microporosity which causes the low density is thus, in general, inaccessible.

It has already been emphasized by the work of Franklin (1951) that the microporosity is one of the main characteristics of non-graphitizable carbons. Rothwell (1968) has studied the microporosity of glassy carbons by X-ray small-angle scattering but his method of evaluation does not permit the deter- mination of general structural parameters. A detailed discussion of this point is found in an earlier small- angle study of non-graphitizing carbons (Perret & Ruland, 1968). The evaluation procedure used in the present study is essentially the same as that used in this earlier work and in our studies on carbon fibres (Perret & Ruland, 1969, 1970). This will enable us to

make a quantitative comparison of the pore structures of these materials.

2. Experimental and results

The small-angle scattering of various types of glassy carbon was measured with a Kratky camera using monochromated Cu radiation and a xenon-filled proportional counter with pulse-height discrimination. The samples were in the form of fiat platelets with a thickness of a few millimeters. The scattering curves were normalized using the method of calibrated filters (Luzzati, 1960).

Fig. 1 shows an example of the curves obtained in a log J-log s plot. Except for a few curves which show a rather steep increase towards very small angles, presumably due to the presence of large holes, Guinier's law is followed over a considerable angular range (Fig. 2). At large angles the intensity is composed of the scattering of the pores following Porod's law and a scattering due to the anisotropic density fluctuation in the carbon material. This shows up clearly in s3j-s 2 plots in the form of a linear relationship at large s values (Fig. 3). This relationship, together with the normalization of the scattering intensity, permits the determination of the correct value of lp

* Present address: Centre de Recherches de K16ber-Colom- bes, 49 rue Jean Jaurrs, 95-Bezons, France.

t Present address: University of Marburg, Fachbereich physikalische Chemie, Lahnberge, Block H, Marburg, Ger- many (BRD).

I °°sJ(s)d s 0

21r lim s3J(s) S---+oO

184 X - R A Y S M A L L - A N G L E S C A T T E R I N G O F G L A S S Y C A R B O N

the only size parameter which is not influenced by interference effects due to the distribution of pore centres, and of the absolute value of the density fluc- tuations which are a measure of the perfection of the parallel stacking. The latter parameter, which has been designated

(A2a3) (A2L) (a3)----q- + (L)2

in the earlier work (Perret & Ruland, 1968, 1969) emphasizing its relationship to the fluctuation of the interlayer spacing and the layer size in a given stack of layers, will be called Fll in agreement with the more general definition developed recently (Ruland, 1971). After subtraction of the background due to Fll as described in the earlier work, the correct value of the length of coherence

I ~J(s)ds 0 it=

rc sJ(s)ds 0

is obtained and the parameter P(1 -p)02 is determined from the normalized integral intensity

2n ITsJ(s)ds= VP(1- p)o2.

l ~'r3))(r)dr 0

2 fo r ? ( r )d r

which is the most general definition of the Guinier radius in the case of infinite slit height.

Finally, the intersect distribution function g(r) is computed from the corrected J(s) values in the way already described (Perret & Ruland, 1968). Examples

.103

10 2

Assuming the electron density Q of the carbon material to be defined by the a v e r a g e in ter layer spacing as ob- ta ined f rom the pos i t ion of the 002 band the vo lume f rac t ion P o f the pores con t r ibu t ing to the observed 1 smal l -angle scat ter ing is de te rmined and can be com- pared wi th the poros i ty calculated f rom Q and the bulk densi ty measured by a f lo ta t ion method .

For the eva lua t ion o f RG, the Guin ie r radius, the app rox ima t ion

J(s) = exp ( - 2rc2R2s 2) lg

is cons idered to be val id for small values of s. This means tha t RG is given by

Jn(S)

30000C • • • re°me•m%•••_ ~_3

2000OC ~'~- .,,: ...........

+I000 c ",,~ • • • • ooooeeeoeooeee°

• . . . % & ,,..

\ eo ~°•

g. \ o \: \

I I I0 10 2

s [163k 1 ]

Fig. 1. log J-log s plots for a heat-treatment series of glassy carbons (LMSC).

Table 1. Structural parameters.for glassy carbons obtained by small-angle scattering

l~ lc Porosity P from from from from from from ~s) glr) ~s) g~) SAS bulk

density V10 7.22 7.25 11.16 11.16 0.176 0.253 V 25 , 12.25 1 2 . 3 1 20.72 20.84 0 " 2 9 1 0.336 GC 10 6"63 6"66 10.00 10.06 0.176 0"254 GC 20 9"01 9"06 1 4 " 6 9 1 4 " 8 0 0 " 2 5 7 0.298 LMSC 1000 7"44 7"44 1 1 " 4 2 11"55 0 " 2 0 9 0"237 LMSC 2000 10"10 10.07 1 7 " 0 8 1 6 " 8 0 0"274 0"283 LMSC 3000 20-76 20.96 32.48 33.44 0.392 0"389 ERA 10 5"57 5"57 6"71 6"48 0.103 0'326 ERA 15 6"74 6"77 1 0 " 3 8 10'33 0.232 0.328 ERA 1000 5"40 5"38 6"95 6.79 0 " 1 2 6 0.304 ERA 1500 7"14 7.13 10.76 10.82 0.228 0.315 ERA 2000 10.32 1 0 . 3 1 16.68 16.62 0.292 0-323

Volume fraction

of micro- Ssp (lp~(e.~) (/m~te~)l~,g(O) RG Fll pores m2.g-1 A

0.696 523' 8.78 41"12 0.655 8.8 0.0510 0.866 465 17"32 42.20 0"718 1 2 - 6 0.0053 0"693 570 8.06 37.82 0.642 5-7 0.0432 0.862 567 12.16 3 5 " 1 6 0.684 7"8 0.0167 0'884 569 9"41 3 5 " 6 0 0.648 6"9 0"046 0.967 519 13"88 3 6 " 7 9 0"694 1 0 " 8 0.0100 1"000 341 3 4 " 1 4 52'96 0.594 19"4 ~--0 0"316 480 6.21 5 4 " 0 8 0.570 5"9 0.0624 0"707 754 8"79 29.10 0"646 5"9 0"0416 0"414 574 6.17 42.78 0.596 5.8 0.0541 0.724 691 9.24 31.27 0.633 7-0 0.0358 0.'904 556 10"58 35.34 0.672 8-7 0.0120

R. P E R R E T A N D W. R U L A N D 185

of such curves are shown in Fig. 4. The g(r) curves are rather similar for all glassy carbons studied. The most important feature is the non-zero value of g(0), which indicates the presence of sharp edges in the shape of the pores.

The results of the measurements are given in Table 1. The size parameters lp and lc obtained by direct evalua- tion from J(s) and from the moments of g(r) are in good agreement, which represents a check of the con- sistency of the method.

The porosity obtained from the absolute intensity of the scattering is in general smaller than the porosity

J ( s )

10 2 1000"C --o-o ~ - - - - - o - - - - ~ -

~ - . . . ~ 0 O 0 *C "-----o._

I I I I I I I 1 , I ~ _ _

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 s 2 [10"~,~, -2]

Fig. 2. Guinier plots (log J-s 2) for a heat-treatment series of glassy carbons (LMSC).

°o • ° . ' °

1000'C o ~ .

.+ . . . . + ' "

+°.

• , , , ' " "

, ,-

_61 ....

....:.': . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ° . .

. . . . i _ .J . .................... ~ . . . . . . . . . . . . . __._.t 20 40 60 80

2000"C.

3000*C

s~[~o,k'2] 1oo

Fig. 3. s3J-s a plots for a heat-treatment series of glassy car- bons (LMSC).

'~ g(r)

0.70

' ~ e e e e 0 .60 '

e •

e • • 0.50

eo •

0.40 • • • 3 0 0 0 " C

0.30 "ee •

o 2 0 0 0 . C • e

0.20 1000.C • • • •

0.10 • • o • • • • • °Co

° ° 0 o ° • ° 6 • • • 0 I I ° ° q ' o a [ I '~ • 4, I --

0 10 2 0 30 40 50 60 70 80 90 r" -[/~]

Fig. 4. Intersect distribution functions for a heat-treatment series of glassy carbons (LMSC).

calculated from the bulk density except for the 3000 °C sample of LMSC glassy carbon where the values are equal within the limits of error. The difference is ob- viously due to pores with sizes outside the range meas- urable with the experimental conditions chosen. The volume fraction of micropores increases with in- creasing heat-treatment temperature for all series studied, the structure becomes thus more homoge- neous.

The specific internal surface related to the micro- pores obtained from lp, P and the bulk density is very high. As shown in Fig. 5 it has a maximum around 1500 °C HTT which is due to the fact that both lp and P increase with increasing HTT.

The average pore size as given by the average length of segments

(,Zpores)- I - P

increases continuously with HTT as can be seen in Fig. 6. The values are similar to those obtained for other non-graphitizing carbons (Perret & Ruland, 1968) and carbon fibres of various precursor materials (Perret & Ruland, 1969, 1970; Fourdeux, Perret & Ruland, 1971a). This seems to indicate that the same type of micropores is present in these carbons. The comparison with carbon fibres is particularly inter- esting since we know from the anisotropy of the small- angle scattering of these fibres (Ruland, 1969) that these pores have a needle-like shape and are oriented with their long axis parallel to the carbon layers. The values of (/pores) for the carbon fibres are related to the cross-section of the pores perpendicular to the fibre axis whereas the values for the is•tropic material re- present an average over all directions in space. In the is•tropic case the exact expression for the averaging over the distribution of the lengths l of straight lines intersected by the pore walls is given by

186 X-RAY S M A L L - A N G L E S C A T T E R I N G OF GLASSY C A R B O N

1 - 1

where () , , stands for the number average in a given direction in space and ( )~ for the averaging over all directions. The evaluation of the small-angle scattering of carbon fibres measured perpendicular to the fibre axis with an 'infinite' slit parallel to the fibre axis gives

1 -~

where ( )o stands for the average over all directions perpendicular to the fibre axis. In the extreme case of perfect alignment of the long axis of the pores parallel to the fibre axis and infinite length of the pores along this axis, the parameter (/pores)is smaller by a factor re/4 than the value obtained for the same size and shape of the pores in an isotropic distribution of pore axes. The fact that the values of (/pores) for carbon fibres and isotropic non-graphitizing carbons are found in the same size range is thus not at variance with the assump- tion of a simularity of the pore shapes of these materials.

The values of /,

(Zm.,ter)-- p ,

a measure of the average distance between pore walls within the dense carbon material, are in the same size range as those of the apparent stack size Lc which can -be obtained from the width of the 002 band. This indicates that the micropores are separated by a single stack of layers, a feature which has also been observed in carbon fibres (Fourdeux, Perret & Ruland, 1971a).

It can be shown that the value of l~g(O) is a measure of the angularity of the pore shapes independent of the pore size. The values obtained are in the range 0.6-0.7, which corresponds to the theoretical values one can compute for needle-shaped pores with sharp edges. This proves unambiguously that the shape of the pores can by no means be considered spherical as Rothwell (1968) has assumed. It is also for this reason that the elimination of the effect of the distribution of pore centres on the value of RG was not considered feasible with an isotropic short-range order approximation like the one proposed by Rothwell (!968) for this purpose. One can certainly expect that the RG values observed are, to a large extent, affected by the interference pro-

- duced by the distribution of pore centres and that the Guinier radii of the individual pores are much larger than the RG values observed, but this distribution cannot possibly be of the type which one observes in colloidal solutions of spherical particles. The existence of a well-defined and relatively large range of s values in which Guinier's law is valid implies that the distribu- tion of pore sizes and of inter-pore distances is relative- ly narrow.

The anisotropic density fluctuations Fll decrease with increasing HTT (Fig. 7) in nearly the same wayas

observed in PAN-carbons in bulk (Perret & Ruland, 1968) and in unstretched carbon fibres (Perret & Ru- land, 1969, 1970; Fourdeux, Perret & Ruland, 1971a). This indicates that the improvement of the parallel stacking of carbon layers without three-dimensional ordering is occurring in nearly the same way in iso- tropic and anisotropic non-graphitizing carbons pro- vided that no external forces are applied.

3. Discussion

The results of the present study show that the micro- pore structure of glassy carbons is very similar to that of other non-graphitizing carbons and, in particular, to that of carbon fibres. The pore size and the angul- arity of the pore shapes suggest the needle shape found in carbon fibres as the most probaNe pore shape in glassy carbons as well. Since the needle shape of the

2/g , s

800

7~

600

500

/.00

3O0

20O

100 _

* ° I t o V

I I I HTT

1000 . 2000 3000 "C

Fig. 5. Specific internal surface due to micropores as a func- tion of HTT • Tokay Electrode Co.; © Carbone Lorraine; V Lockheed; • own preparations.

k ,(~po res)

10 g

!

I000

I I HTT

2000 3000 *C

Fig. 6. The pore size parameter (/pores)'as a function of HTT • Tokay Electrode Co. ; © Carbone Lorraine; V Lockheed; • own preparations.

R. P E R R E T AND W. R U L A N D 187

007

006

005

OOZ.

003

002

001

FL 1

A \ * \ v

\ • "¢

\ \

~ o

V

~ o --..___~ . . . . HTT

1010 I 2000 30~0 "C

Fig. 7. Anisotropic density fluctuations as a function of HTT • Tokay Electrode Co; © Carbone Lorraine; V Lockheed; • own preparations.

pores is conditioned by the ribbon shape of the carbon layers, it seems to be reasonable to assume that the latter are also present in glassy carbons. Further evidence for this is given in recent electron microscope

studies (Fourdeux, Perret & Ruland, 1971 b; Fo urdeux & Ruland, 1971). We can thus conclude that the ribbon model, proposed originally for carbon fibres only, offers in fact a consistent interpretation of various structural features common to all non-graphitizing carbons.

References

FOURDEUX, A., PERRET, R. & RULAND, W. (1971a). Inter- national Conference on Carbon Fibres, their Composites and Applications, London 1971, Preprint No. 9.

FOURDEUX, A., PERRET, R. & RULAND, W. (1971b). Tenth Biennial Conference on Carbon, Lehigh University, Beth- lehem, June-July, 1971, Paper FC-22.

FOURDEUX, A. t~ RULAND, W. (1971). C. R. Acad. Sci. Paris. In the press.

FRANKLIN, R. (1951). Acta Cryst. 4, 352. LUZZATI, V. (1960). Acta Cryst. 13, 939. PERRET, R. & RULAND, W. (1968). J. Appl. Cryst. 1, 308. PERRET, R. & RULAND, W. (1969). J. Appl. Cryst. 2, 209. PERRET, R. & RULAND, W. (1970). J. Appl. Cryst. 3, 525. ROTHWELL, W. S. (1968). J. Appl. Phys. 39, 1840. RULAND, W. (1969). J. Polymer Sci. C28, 143. RULAND, W. (1971). J. Appl. Cryst. 4, 70.

3". Appl. Cryst. (1972). 5, 187

The Precision Determination of the Lattice Parameters and the Coefficients of Thermal Expansion of BiFeO3*

BY J. D. BuccI, B.K. ROBERTSON AND W. J. JAMES

Department of Chemistry and Graduate Center for Materials Research, University of Missouri-Rolla, Missouri 65401, U.S.A.

(Received 26 July 1971 ; accepted 15 November 1971)

The lattice parameters of BiFeO3 were determined with the Straumanis method. At 25-13 + 0.02 °C, the hexagonal parameters are ah=5"5799+0"0003, and ch= 13.8670+0"0005/1~. The temperature depen- dence of the lattice parameters i n the range 20-325 °C is given by the equations: ak = 5" 5764 A + 6-06 x 10- st, and ch = 13.8620/~ + 2.10 x 10 -4 t. In the range of 344--838 °C, the lattice parameters obey the following equations: an = 5.5946 + 6"83 x 10 -5 t, and ch = 13.7251 + 9.05 x 10 -4 t - 12.503 x 10 -7 t 2 + 9"40 x 10-10 t 3 -3"57 x 10 -13 t 4. By extrapolation of the angular separation of the 11.0 and the 10.4 reflections, the electrical Curie temperature was determined to be 845 + 5 °C.

Introduction

During the past decade much attention has been de- voted to the perovskite-like substance, BiFeOa; how- ever, there seems to be considerable disagreement amongst the various investigators concerning the lat- tice parameters, their temperature dependence, and the number and nature of the phase transitions.

Filip'ev, Smolyaninov, Fesenko & Velyaev (1960) indexed the X-ray pattern of BiFeOa on the basis of a

* Contribution No. 138 from the Graduate Center for Materials Research, University of Missouri-Rolla.

rhombohedrally distorted unit cell, aRh=3"965 A and agh = 89°28'. The measurements were made on samples synthesized from the oxides at 600-725 °C. Subsequent investigators (Tomashpol'skii, Venevtsev, & Zhdanov, 1964; Ismailzade, 1965; Tomashpol'skii, Skorikov, Ve- nevtsev & Speranskaya, 1966) arrived at approximately the same parameters. Zaslavskii & Tutov (1960) re- ported the hexagonal parameters, ah= 5.581 A and ch= 6.934 A.

On the basis of single crystal and neutron diffraction data, Michel, Moreau, Achenbach, James & Gerson (1969) concluded, however, that in order to explain the existence of observed superstructure, a double cell had