Structural evolution of a glassy carbon as a result of thermal treatment between 1000 and 2700°c—I

Carbon, 1977, Vol. IS, pp. 5541

Pergamon Press.

Printed in Great Britu

STRUCTURAL EVOLUTION OF A GLASSY CARBON AS A RESULT OF THERMAL TREATMENT BETWEEN 1000 AND 27OO”C--I EVOLUTION OF THE LAYERS F. ROUSSEAUXand D. TCHOUBAR Centre de Recherche sur les Solides B Organisation Cristalhne Imparfaite, C.N.R.S., 45 Orleans la Source, France (Received

3 July 1976)

Abstract-The structural evolution of a glassy carbon is studied by analysis of the X-ray diffraction two-dimensional lines. A strong anisotropy is found in the growth of the carbon layers and in their lattice distortions. The results are correlated with the carbonization mechanism of the polymer precursor.

1. INTRODUCTION One of the crystallographic

methods

paper it is shown that we obtain a very good agreement between the observed and calculated curves provided the shape-function is derived from a rectangle. Such a model allowed us to precise the physical meaning of the scattering domain as deduced from the X-ray data.

available for the

analysis of X-ray diffraction diagrams of partially or totally disordered materials consists in comparing the experimental diffraction peaks with theoretical profiles calculated from a structural model. In some recent studies, Ergun[ 1,2], and later Fitzer and Braun[l3], have applied this indirect method to the study of a glassy carbon (in a large range of temperature, 200-3000°C) using a model derived from statistical considerations. Their structural model assumes[3] that in the layer plane the mean scattering domain is limited by defects distributed in a real random manner (defects consisting of various folds along the carbon sheets). The coherent domain is so called “defect-free distance”. This structural representation does not seem, a ptiori, entirely satisfactory, especially when we consider the origin of this type of carbon. A glassy carbon is always produced by the solid-phase pyrolysis of polymeric chains. According to the mechanism of carbonization proposed by Fitzer[4], the aromatic rings are formed at about 450-500°C by the coalescence of carbon skeletons produced during the pyrolysis. It is thus difficult to think, a ptioti, that the development of the carbon sheets might be carried on isotropically in all the directions of the plane and to assume a statistically isotropic model for the description of the layer. Moreover, the folds or the bends which appear along the carbon layers are certainly related to the presence of non-random sequences in the polymer chains and it is less probable that the defects which limit the “defect-free distance” would be distributed completely at random. These different arguments enabled us to try a possible refinement of the glassy carbon structure by using a structural model which is less random. For this work it is assumed that the domain which participates in the diffraction may be characterized by a definite shapefunction which is not necessarily isotropic. In the present

where I(s) represents the experimental intensity normalized to one carbon atom and corrected from polarisation and absorption, F commonis the Compton scattering factor of a covalent carbon corrected for the Breit-Dirac factor and f(s) is the coherent scattering factor for the same atom.

tThe authors are grateful to Dr. J. Maire of the Carbone Lorraine for furnishing the sample.

Figures 1 and 2 show the experimentally obtained interference functions for the samples heat treated

Results will be presented as part of the analysis of the two-dimensional reflections hk (Part I) and as part of the study of the different 001 reflections (Part II). This presentation does not correspond to the order in which the analysis has been carried out. However it enables us to separate the informations which are related to the internal structure of the sheet from those which describe the layer stackings. The full results led us to propose a three-dimensional structural model.

2. ExPERIMENTAL PROCEDURE The carbon which was studied resulted from the carbonization of polyfurfuryl alcoho1.t The study included 5 samples in the shape of flat pieces respectively heat treated at 1000, 1500, 2000, 2304 and 2700°C. No graphitized phase is found in this bulky material: the sample remains completely homogeneous up to 2700°C. The X-ray diffraction diagrams were performed by a stepscanning technique. A transmission geometry was used with a monochromator (Johann-Guinier) focusing the MO Ka, radiation. Sample thickness and slit size were chosen so that the effect of instrumental broadening on the experimental diffraction peaks could be neglected. Under these conditions, the only corrections to be made on the observed intensities were those of polarization and absorption. In order to eliminate the Compton scattering we have converted the corrected intensities to absolute values using the Warren and Gingrich’s method[Sl. The experimental diagrams have been analysed from the interference functions defined by the following relation: H(s)

55

=

J(s)- Fcm,o.b) f(s)

56

F. ROUSSEAUX and D. TCHOUBAR

(I I) Q

H (30) (22) (31) 121)

L

WV

(41) A

v

2.0 s. A-’ Fig. 1. CV 1000-Experimentalinterferencefunction H(s). 1.0

H

I l-

2.0 ..5,%-’ Fig. 2. CV 2700-Experimental interference function H(s). IO

respectively at 1000 and 2700°C. Such diagrams result from the superposition of two independent contributions; one corresponds to the 001 reflections resulting from the interferences between the layers and the other is due to the hk lines produced by the interferences within each layer. Because the hk lines have very asymmetric profiles which extend far in reciprocal space, there is partial overlap of their intensities which prevents the direct analysis of the complete profile of one line. We have thus adopted the following method for the interpretation of the observed diagrams: we look for the first peak in the diagram whose calculated profile agrees best with the experimental profile. Having found such a peak, we substract its contribution from the diagram which allows us to analyse the shape of the next peak and so on. This method enabled us to examine for each sample the first eight reflections, i.e. 002, 10, 004, 11, 20, 30 and 22. After this process of substraction we can restore a synthetic diagram calculated with all the values of the parameters deduced from the best fitting of each reflection. This synthetic diagram is then compared with the experimental interference function H(s).

scattered intensity has significant value only in the vicinity of straight rods parallel to one another and passing through the nodes of the biperiodic lattice reciprocal of the layer lattice. Along each straight rod the intensity varies as the square of the coherent scattering factor of carbon f while in the transverse direction the intensity distribution depends on the shape and the size of the layer. A random distribution of such layers produces the asymmetric interference hk lines. The intensity Z,,,(s) at any point of a hk line can be deduced from the corresponding hk rod structure. According to Brindley and Mering[6] the expression for I,&) can be described quite accurately by eqn (1):

where [A,,$ is the square of the geometrical structure factor of the carbon lattice. This factor can only take the two following values lAhk12= 1 for h - k# 3n; [A,,[* = 4 for h -k = 3n. p is the multiplicity factor of the hk reflection. fl is the area of the two-dimensional unit cell. f(s) is the square of the coherent scattering factor of carbon. s is the magnitude of the diffraction vector, s = 2 sin 8/A. X is a co-ordinate parallel to the direction of vector sbL[7]. This vector shlr lies in the plane perpendicular to the hk rod and joins the reciprocal space origin to the center of the hk node. T,(X) is the projection of the cross-section of the rod on the X direction. This function depends essentially on the size and shape of the scattering layers. The integration variable @is the angle between the vectors and the vector sbk. The expressions for T,(X) available for a rectangular shape were discussed in a previous paper[7]. We recall the essentials: -The carbon layer consists of an hexagonal arrangement of carbon atoms. If we consider that the scattering unit domain has the shape of a rectangle with the parameter Las the length and the parameter I as the width, the orientation of the length L may therefore only coincide with one of the two directions a and b shown in Fig. 3. These are indeed the only non-equivalent crystallographic directions. -If one chooses the elongation direction of the rectangle (a or b), the calculation shows that the expression for T,(X) varies depending on the values for h and k. Thus in the case of length along a, one has: lines ho: TN(Xl’30’

1 I

sin* (7rXL) 7+

’ *sin(F)+(F-+)sin’($Z$]] + [ E_47TxK.

(2)

3. METBOD FORCALCULATLNG TRR hk LINE PROFlLlLAPF%ICATION TO A

RJ%CTAh’GULAR DOMAIN 3.1 Carbon layer with perfect structure Let us consider that the scattering unit layer has a perfect two-dimensional lattice but a finite extent. The

Fig. 3. The length of a carbon ribbon can only be orientated along one of the two directions a or b.

51

Structural evolution of a glassy carbon as a result of thermal treatment between 1000and 27OOY-I lines hh:

+/3_63. L ~sin(47rXI)t [

(’7-1 Q3) sm . 2(2?rXl)I].

(3)

The comparison of (2) and (3) shows that the two functions differ essentially in the first term. The h0 line profiles are more strongly influenced by the parameter L than the hh line profiles. In other words, the effect of the rectangular shape on the profile is more pronounced for the h0 than for the hh lines. This effect consists of anomalies in the shape of the h0 profiles such that the curve narrows in the domain su~ounding the maximum. This n~rowing is more pronounced when L/l is iargep]. It can be easily shown that for elongation along b the expressions (2) and (3) are simply inverted, that is:

nature of the defects and the modifications to be introduced in the T,(X) function to improve the agreement. Figures 4 and 5 demonstrate how closely the experimental and calculated curves agree when &(X) (4) is replaced by the function TN(X) as defined for the rectangle. On Fig. 4, the experimental line profile 10 is represented, for the sample heat-treated at 2700°C (CV2700), as a solid line while the dotted line illustrates the best fit obtained when the profile is calculated from the rectangle model. The values of the two parameters Lo and Ilo used for this matching are listed in the Table 1. Further, on the same Fig. 4, the dashed line represents the theoretical profile calculated from the Warren’s model [8]

Model b Model a T,(X) for (lo)= TN(X) for (11) TN(X) for (11)= 7+,(X) for (10). This means that the deformation now appears in the hh profiles and not in the ho. This result is important because it shows that in carefully examining the shape of the first two profiles IO and 11, one will be able to determine if there is an effect due to anisometry and if so, to determine the direction of the elongation of the particle (a or b). 3.2 Carbon layer witk defects Consider that the sample is made up of waved and

folded layers. As far as diffraction is concerned a waved layer may be described by a series of planar zones which are mutually disoriented. When the disorientation is large enough, the X-rays diff~cted by the atoms ~Ion~ng to two successive zones of the same layer are no longer in phase. The function TN(X) which appears in the calculation of the line will then depend on the shape and dimension of the zones and no longer on the entire sheet. Further, when the structure contains defects such that the atoms are slightly displaced from their ideal positions the line profiles are broadened so that the apparent particie sizes are smaller than the actuai scattering domains. The term TN(X) must be therefore replaced by a function $&X) which depends both on the shape and dimensions of the scattering domains and on the lattice distortions. These distortions induce deformations of the unit cell which modify the theoretical intensity factor lAhkr. In order to account for all these imperfections, it is necessary to replace in (1) the function TN(X) by t,+,,.(X) and the term fA,,,j’ by a factor Qhk determined experimentally. Equation (1) must be replaced by:

The search for the best agreement between the calculation and the experiment depends therefore on the choice of the function $M(X). In the general case where the nature and the amplitude of the defects are indetermined, experiments have shown that it is more realistic, in first approximation, to replace the function $,k(X) by the term TN(X) derived from a geometric shape and to neglect the effects due to the lattice distortions. Then, it is the comparison between the experiment and this calculation which enabtes us to specify the physical

s, ii-’ Fig. 4. Line profile (10) of CV 2700sample. Comparison between

the experimental curve (solid line) and the calculated profiles assuming the “rectangle” model (dotted line) and the Warren’s model (dashed line).

4k

Fig. 5. CV 2700-Comparison between the experimental interference function H,&) (solid line) and the synthetic curves restored with the “rectangle” values (dotted line) and with the Warren’s values (dashed line). Table 1. Evolution of the apparent dimensions Ihk and L,,& (experimental values) of the rectangular scattering domain with the temperature treatment and the hk indexes of the twodimensional lines (hk)

(IO)

(11)

cv loOoO 1=19 L=31 cv 1500” 1=23 L=46 cv 2000” 1=27 L=68 CV 2700” I =32 L=78

I=15 L=30 f=18 L=40

1=23 L=6S

(20) f=12 L=Z

(21)

(30)

t=12

I=8 L-22 1=12 L=32

L=27

1=15 /=I4 L=37 L=36 I= 19 1=20 L=53 L=S2

(22)

f= 15

I=30

l-24

1=27

L=ii 1=20

L=75

L=68

L=68

L=S8

I=20

L=62

58

F. Rou~~ux and D. TCHOUBAR

which implies that the shape function is isotropic and can be approximated by a gaussian law. This gaussian depends only on one parameter which represents an average diameter for the scattering domain. Figure 5 anticipates the results, it makes the comparison between the experimental diagram without the 001reflectionst and two synthetic curves. One of them is deduced from the Warren’s model (dashed line), the other from the rectangle model (dotted line). 4. RESULTS FROMTHE RETANGLE

Extrapolation of these straight lines to zero abscissa shows, without introducing serious error, that the two l/Lr, straight lines intersect at the same point of the ordinate l/L0 = O:Ol while the pair of l/lhlr yields l/1,,= 0.022. It is thus possible to evaluate the length Lo and the width l0 which represent the actual mean coherent domain in the sample. CV2700”

MOLtEL

Table 1 lists the values of the parameters Lk and I (as determined for the best fit) for all the samples. The values of Lhlrand lhkare expressed in angstrSm units. Table 1 indicates that: when the heat-treatment temperature increases, the scattering domain dimensions increase; for one heat-treatment temperature, the dimensions decrease as the indexes h and k increase. This decrease makes the evidence of lattice distortions which prevent direct access to the actual size of the scattering domain.+

4.1.1 ~mens~ons of the mean scutte~ng domain and o~entation witk respect to the c~st~lograpkic uxes. If i/L,,, is plotted against s,& a straight line is obtained. Each straight line corresponds to a well-defined crystallographic direction. This is illustrated in Fig. 6(a). The l/L,,, data corresponding to the 10, 20, 30 reflections define a direction with slope (T,,whereas the l/L, data for the 11 and 22 reflections define another one with slope oz. A similar diagram is obtained for the lll,,k data (Fig. 6b).

It is also possible to define the orientation of this rectangle. A careful analysis of the two profiles 10 and 11 shows that the parameter Lo is directed along vector a as shown in Fig. 3. In other words the rectangle (LOx lo) as seen by the X-rays may be represented as in Fig. 7(a). 4.1.2 Physical meaning of the u parameter and nature of the distottions. Table 2 lists the values of the u and u’ slopes determined for different samples. From Fig. 6 a linear relation can be written for each crystallographic direction:

where l/Lht, is the width of the &k function. This Iatter represents by definition, the projection of the transverse section of a rod on the direction of vector s&7]. If Lo is quite large, the layer size effect becomes negligible on the profile of the I,$,~function, l/L,,k = a&k. The function y(x) which is the Fourier transform of +,,k(x) is given by the equation: y(X) =

(IO)

tll)(zot(21) (30)(22)

I

&k(x) eXp(hi%)

510

%o shk, a-1

dX.

Such a function would represent in the real space the section of the projection of the unit layer on its mean plane in the direction x ~rpendic~ar to the family of the atom rows (kk). y(x) denotes only the fluctuations of the distance between the rows of the kk family. Suppose no& that the scattering domains are limited to n rows. Let L0 be the length occupied by these n equidistant rows for an ideally planar sheet. Lo is equal to n . du if dMis the unit distance between these rows. If the Iayer lattice is distorted, that is, if all the rows are not in the same plane, one measures in the direction x a length LhLwhich is the projection of L, on the mean plane of the sheet. If AL = L - L,@,the relative variation AL/L,,t takes the form: AL -=

0

Lo= 100+58, 1,=45*5a

dd

Lhlr C7 > xn

5m

Fig. 6. CV 2700.(a) Plots l/L,,, vs sIk (rectangle model data); (b) Plots l/1,,, vs s,,~(rectanglemodel data).

Table 2. Experimentalvaluesof (Twhichrepresents the slope of the straight lines as those of Fig. 6

tH,,&) has been obtainedby subtractingfrom the full diagram H(s) the 001 contribution. SA similar result has been found in graphitable carbon study i9.101. $.shkmeasures the distance between the origin of the reciprocal space and a kk node.

cv 1000° cv 2000” cv 2700°

0.0180 0.0100 0.0055

0.056 0.031 0.020

0.0115 0.0066 0.0040

0.052 0.025 0.0148

Structural evolution of a glassy carbon as a result of thermal treatment between 1000and 27OO”C-I

59

where (Ad/d) represents the apparent “dilatation” of the unit distance between the rows, n equals Lo/dht. Expression (6) takes the form:

It appears that (7) is identical to eqn (5) as observed experimentally. This identity reveals the significance of the parameter u: (b)

From this result we can conclude that the linear relations of the type (5) mean that the layer is not planar and the hexagonal carbon rings are deformed by a series of contractions and expansions. The distortions are such that the average of the projected interatomic distances on the mean plane is close to the theoretical value for graphite. If d, represents the projection of the distance between two successive rows and pi its probability, experiment has shown that:

Fig. 7. (a) Plots of the rectangle lo= LO; (b) Illustration of the anisotropy of the lattice distortions. which only depends on the length L. The perpendiculars

to the other two families (10) are inclined at 30” to the direction 1.The correspondent terms T,(X) and T,(X) are such that T,(X) = 7’,(X) and depend on both I and L. In the same way, three families of crystallographically equivalent rows (11) contribute to the 11 line. The correspondent function TN(X) equals T;(X) + 27’S(X). However in this case, T:(X) depends only on I, TI = [(l/(nX)?) sin* (a x 1)], and is related to the family (11) 2 pid, = d = dhk. oriented perpendicular to the direction 1whereas the two functions T;(X) refer to the two families (11) whose At this step of the analysis, it can be concluded that the perpendiculars make an angle of 30” with L. These two linear relations of Fig. 6 are characteristic of an elastic functions depend on both L and L. deformation of the carbon layer and the function &(X) is When the l/Lk values for the lines ho, that is 10,20,30, given by TN(X) as it was defined. But, in that case, the L are plotted against s,,~, the terms T,(X) are only and L parameters are replaced by the projected values I,, concerned. Thus only the relative displacements of the and L,, such that: atoms in the two directions oriented at 30”to the direction 1are taken into account. The same, when the l/L,,, values g- = U&k for the hh lines, that is 11, 22, are plotted against shk,one hk Lo has to deal with the relative displacements in the two 1 1 directions oriented at 30” to the direction L. If it is - = i + (T’Shll. 1hk assumed that the slope u{ corresponds to the projection II of the relative displacement onto the direction 1 and 4.1.3 Evidence of the anisotropy of the distortions in the similarly for a; and the direction L (Fig. 6), it is possible layer. We can now interpret more quantitatively the data to evaluate the magnitude of the displacements A, and A, on Table 2. The slopes u, and u2 (Fig. 6a) are smaller than in the directions ON, and ON, (Fig. 7b) as follows: ai and u: in Fig. 6(b). This indicates that the displacements of atoms in the direction parallel to 1 are definitely 1 A,=ujxd,Ox much larger that in the direction parallel to L. The cos 30 distortions in the structure are thus anisotropic. 1 It is possible to describe the distribution of the atomic AZ= u2 x d,, x cos 30” displacements in the different crystallographic directions of the mean plane of the layer. Let us examine the with expressions (2) and (3) used for the principal directions 10 d,, = 2.13 ii UI = 0.02 and 11. For the 10 line, the function TN(X) is the sum of uz = 0.004 I d,, = 1.23A I three terms T,(X), ‘L’,(X),T,(X). These correspond to three equivalent reflections which are superimposed in the Al f 0.05 A powder diagram[7]. The three reflections correspond to the ( Azf 0.005 A. diffraction by the three families at atom rows (IO), rotated 120” with respect to each other (Fig. 7a). One of these A, is therefore ten times larger than AZ.This anisotropy of families is oriented perpendicular to the direction L and the distortions is illustrated in Fig. 7(b) where ON, and corresponding to this direction is the term ON, are vectors with lengths respectively proportional to A1and AZ.Hence, the displacements of the carbon atoms are confined within a rectangular area. This is simulated as T,(X) = (ri)ZL sin’ (?rXL) horizontal lines on Fig. 7(b). [

r+

1

60

F.

ROIJSSEAUX and D. TCHOUBAR

Having determined from the experiment the values of A, and A, it is now possible to define more precisely the mean unit cell of the carbon sheet. Indeed, if this unit cell remains hexagonal in projection and is defined by the graphite parameter a = 2.46 A, the positions of the two carbon atoms in the cell are certainly not exactly those of graphite, that is (l/3, 2/3) and (2/3, l/3). A slight displacement of the atoms may explain the differences between the experimental values of the coefficients Qhlr (Table 3) and the theoretical values IA,,,[’for graphite. If it is assumed as a first approximation that the symmetry remains hexagonal, the displacement A, is equivalent to a coordinate change as follows: (AZcan be neglected because it is ten times smaller than A,)

IflO

Y

(IO)

1 %3

1 A, tan30” x = --= 0.321 instead of l/3 3 a >

(30)

(20)

520

-50 O-1

%,.A

Fig. 8. Plots l/b,, vs skO(rectanglemodel data).

y = (1 - x) = 0.679 instead of 2/3. cv 1000’

Then the square of the geometrical structure factor [AMI is written as follows: lAhtl’ = 4 cos* [r x 0.321(h - k)].

(9)

Table 3 compares the coefficients calculated from eqn (9) with the experimentally obtained values Qh*for sample CV2700”. The calculated values are quite close to the experimental ones and thus explain the apparent anomaly of the coefficients Qlo and QZ, which are greater than 1. It should be pointed out that we have found from the experiment the same definition for the mean unit cell as that considered by Ergun [ 111.

cv 2000”

I//_,

CV27OOO

(20)

30

4.2 Evolution of the samples during thermal treatment In order to clarify the evolution of the scattering unit domain during the heat treatment, we have gathered in Fig. 8 all of the plots of l/b, as a function of s,,,,for the different temperatures and the same for l/L,,, in Fig. 9. It is clearly evident that between 1000and 2ooo”Cthe width I,, of the rectangle varies very little (40-45 A), while the length L,, increases from 45-100 A. There is therefore a real growth of the mean scattering domain between 1000 and 2OOO”C, but this growth is anisotropic since it does only in the direction a. Above 2ooO”Cthe domain remains unchanged. During the heat treatment, the amplitude of the distortions decreases as the slopes (T and o’ decrease. However, the anisotropy of the distortions which is measured by the ratio u,/u? has a tendency to increase.

Table 3. Comparisonbetween the observed Qh*values and the calculatedvalues of the square of geometricalstructure factor IA,,,\*when the atoms of the unit two-dimensional cell are slightly displaced (W QhLexperimental IA,12 calculated

(10)

(11) (20)

1.26 3.80 0.80 1.14 4.0 0.75

(21)

(30)

(22)

1.08 3.90 1.14 3.95

4.0 4.0

(301

%o she.

%

a-1

Fig. 9. Plots l/L,, vs s,,~(rectanglemodeldata). 5. DISCUS!SlON

The reasons for such an anisotropic evolution may be found if one refers to the carbonization mechanism described by Fitzer et a1.[4]. According to the authors, pyrolysis transforms, at about 45O”C,a chain of polyfurfury1 alcohol into a linear skeleton of carbon atoms (step IV, Fig. 10). Formation of the hexagonal rings can only be done by joining two parallel skeletons (step V). Thus, given the orientation of the rectangular domain which was defined by the X-ray diffraction, is seen that the length Lo of the rectangle necessarily corresponds to the direction of the polymerization of the linear carbon chains. In other words, the direction of elongation of the scattering domain is perpendicular to the axis of the carbon chains (Fig. 10). The structural evolution of the solid leads us to conclude that the polymerization between carbon chains continues until 2WC. This temperature probably corresponds to the end of the release of a gaseous phase. The width &,measured by X-rays therefore corresponds to the size of the coherent domain in the direction of the chain axis. This dimension remains constant during the heat treatment. It appears that the limitation of the

Structural evolution of a glassy carbon as a result of thermal treatment between 1000and UOO”C-I

61

parallel to the axis of the primly chains than in the perpendicular direction (direction of polymerization). Such an evolution, which maintains a certain anisotropic character (in size and distortions), must be ch~acte~stic of carbons produced in the solid phase from polymers and should allow one to have a better comprehension of the physical properties of a glassy carbon. In conclusion, this study shows that the choice of a model which is not implicitly isotropic provides evidence for a particular mode of evolution in carbons and allows in a precise way to join information about the growth of the domain with other phenomena which are taking place during the ~arboni~t~on. Polymerlzatlon Fig. 10. Scheme using results obtained by Fitzer et nl.2[4].

domain in the 4, direction may be imposed by the initial bends and folds of the primary carbon chains. To the extent that we have been able to define a shape-function for the sample, these bends are not distributed randomly but very likely follow significant sequences of the precursor polymer. Such a statement is not in contradiction with images obtained by high resolution microscopy[l2]. Indeed, the distribution of lengths of the two-dimensional domains visible in these micro~aphs should correspond to that of the chords of the rectangle (lox LO) defined by X-rays. An examination of the dimensions of the two-dimensional zones shows that they are included in the range 40-180 A, that is in a~eement with this work, The anisotropy of the distortions give the evidence that there must be much stronger strains in the direction

CAR Vut. IS, No. 2-B

Acknowledgemenrs-The authors are indebted to Prof. E. Fitzer

for stimulatingdiscussionsabout this work. REFERENCES 1. S. Ergun, Carbon 11, 221 (1973). 2. S. Ergun, Acfa Crysf. A29, 605 (1973). 3. S. Ergun. Phys. Rev. El@), 3371 (1970). 4. E. Fitzer, K. Muller and W. Schafer, In C~em~sf~ and Physics of Carbon (Edited by P. L. Walker, Jr.), Vol. 7, p. 237. Marcel Dekker, New York (1971). 5. B. E. Warren and N. S. Gingrich, Phys. Reu. 46, 368 (1934). 6. G. W. Brindley and J. Mering.Acta Cry&. 4, 441 (1951). 7. F. Rousseaux and D. Tchoubar, J. Appl. Cryst. 8,365 (1975). 8. B. E. Warren, Phys. Reo. 59, 693 (1941). 9. A. Bouraoui, Bull. Sot. Franc. Miner. Crisf. 88, 633 (1%5). 10. J. Meringand C. Schiller, Carbon 5, SO7(1967). 11. S. Ergun, Nafure Phys. Sci. 241,65 (1973). 12. L. L. Ban, G. M. Jenkins and K. Kawamura, Proc. Roy. Sot. Land. A327, 501 (1972). 13. W. Braun and E. Fitzer, Preprinf 12th Carbon Cot& Piffsburgh, Paper SP-1 I (1975).