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Electron diffraction effects of conical, helically wound, graphite whiskers
Ultramicroscopy 49 (1993) 123-131 North-Holland
Electron diffraction effects of conical, helically wound, graphite whiskers W. L u y t e n , T. Krekels, S. A m e l i n c k x , G. V a n T e n d e l o o , D. V a n D y c k and J. V a n L a n d u y t UniL,ersity of Antwerp (RUCA), Groenenborgerlaan 171, 2020 Antwerpen, Belgium Received 25 June 1992; at Editorial Office 11 September 1992
Recently a new model for the structure of columnar graphite whiskers has been proposed; this model includes the presence of five-rings of carbon atoms, as in the fullerenes. The whiskers are assumed not to be single crystals but have geometrically well defined textures. Their unusual diffraction patterns exhibit rotation symmetry of high multiplicity. The wide variety of the observed multiplicities is explained by assuming small changes in the conical shape of the whiskers. We describe a series of electron diffraction experiments which have been performed on fracture cones of these graphite whiskers. These experiments confirm unambiguously the proposed microstructure.
I. Introduction Columnar growth of carbon was first reported by Tsuzuku et al. [1] and later by Haanstra et al. [2]. In a recent paper [3] the mechanism by which conically wound graphite whiskers grow was described and a possible relationship with the fivefold fullerene-like carbon rings was suggested. The proposed growth mechanism implies that the whisker is not a single crystal but has in fact a geometrically well defined texture. The electron diffraction effects associated with such a texture arc quite striking, giving rise to diffraction patterns with rotation symmetries of a very large multiplicity such as 30, 66, 126 and 222. It is the purpose of this paper to describe specific diffraction experiments which allow one to determine unambiguously the lattice geometry associated with the whiskers. We shall first describe briefly the formation mechanism of the graphite columns and their "lattice" geometry, and subsequently we discuss in detail the characteristic diffraction effects to be expected for such entities. Finally we present some diffraction experiments, which confirm the expected features.
2. Material morphology Under the given experimental conditions [2] needle-shaped carbon whiskers are formed. They are easily resolved under the optical microscope or in the scanning electron microscope (fig. la). They are up to 0.1 mm long and have a cross-section of about 3 mm. Most whiskers are broken and exhibit a shallow conical top on one side and a similar conical depression on the other side (fig. lb). During crushing in liquid nitrogen the needles break perpendicular to their length axis. The obtained thin flakes were deposited on a copper grid, which was covered with a carbon film and examined by transmission electron microscopy.
3. Electron diffraction patterns The diffraction patterns along the normal to thin circular cleavage slices of the whiskers are shown in fig. 2. The patterns exhibit different rotation symmetries with frequently observed multiplicities such as 30, 66, 126 and 222. The radii of the circles, on which the spot clusters are situated, correspond to the length of low-order
0304-3991/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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I,~ Luyten et al. / Conical, helically wound, graphite whiskers
Fig. 1. (a) Scanning electron micrograph of needle-shaped carbon whiskers. (b) Scanning electron micrograph of a cleavage fragment. Note the conical surfaces at both sides of the whisker.
W. Luyten et al. / Conical, helically wound, graphite whiskers
125
d Fig. 2. Different electron diffraction patterns along the zone normal to the cleavage flakes of conically wound graphite whiskers. The circles are divided in 18 (a), 30 (b), 42 (c) and 126 (d) segments.
diffraction vectors of graphite with I = 0 i.e. 1010, 1120 . . . . All such patterns have hexagonal symmetry, which simply reflects the hexagonal symmetry of graphite along [0001]. The segments or spot clusters in successive rings are in staggered positions. This is consistent with the assumption that the ring patterns are due to the superposition of many single-crystal diffraction patterns,
differing in orientation by discrete rotation angles about the [0001]* axis. Along zones inclined with respect to the foil normal the diffraction spot clusters are situated on ellipse-shaped curves; only spot clusters along two diametrically opposite arcs remain sharp, the larger part of the ellipses becoming continuously streaked (fig. 3).
126
W. Luyten et aL / Conical, helically wound, graphite whiskers
Along zones roughly perpendicular to the foil normal only rather broad contours of diffuse intensity are formed. In most patterns broadened and split 0001 reflections are visible. The separation of the split spots increases with increasing index I.
4. Whisker formation mechanism
The formation mechanism of such structures can be described by the following succession of imaginary operations. Suppose we start with a flat circular sheet of graphite with radius R (fig. 4a). We remove an angular sector/3 and we close the gap by making the point P coincide with the point Q, introducing simultaneously a relative offset of the two parts of the sheet over a distance equal to the sheet thickness (fig. 4b). We have now formed a helical surface situated on a cone. Along the junction line of the two parts steps are now present on each side of the sheet. Assuming that growth only occurs by the addition of growth units along the exposed ledges, a conically wound helical surface will be generated. Hereby the growth fronts adopt spiral shapes, similar to those associated with spiral growth around a screw dislocation, as proposed by Frank [41. However, whereas the usual spiral growth mechanism generates a succession of layers, all in the same crystallographic orientation, and therefore produces a single crystal, this is no longer the case here. The structures in successive sheets of the helical surface differ in orientation by the constant angles /3 and this orientation difference is cumulative. The semi-apex angle ~b of the cone is directly related to the angle /3; it is given by q5 = arcsin (1 - / 3 / 2 ~ - ) [3]. This apex angle is clearly seen in the scanning pictures of fig. lb.
Fig. 3. Electron diffraction patterns obtained by tilting the specimen off the zone normal to the cleavage flake. The tilt axis is chosen perpendicular to the length axis of the fracture cone and is represented by the straight line in (c). The tilt angle increases from (a) to (c).
W. Luyten et al. / Conical, helically wound, graphite whiskers
(b)
~ -
Fig. 4. Model illustrating the formation of a conical helix. In (a) a sector fl is removed from a disc. In (b) the angular gap is closed and a cone is formed.
The interface between successive sheets can be described as a pure twist boundary having a (0001) contact plane and a twist angle equal to /3. The magnitude of the angle/3 is determined by microscopic features of the model. In ref. [3] we assumed that the conical shape resulted from the initial introduction of a pentagon of carbon in a graphite sheet during the nucleation process. The presence of even a single pentagon may cause the graphite sheet to become conically deformed, as represented schematically in fig. 5. The angle of
Fig. 5. Twisted nucleus of a conical helix. The centre is a carbon five-ring. An orientation difference of approximately 20° is generated between the two sheets of the helix in the region of overlap.
127
overlap of the two parts of the same sheet is the angle /3 introduced in the macroscopic model. Taking into account the geometry of the pentagon and of the adjacent edge-sharing hexagons leads to a angle /3 of the order of 200-24 °, depending on the assumed extent of the deformation of the angles involved. An angle of the same magnitude emerges from considerations of the coincidence site lattice of two successive graphite sheets. Excellent fit of the superposed graphite structures is obtained for a twist angle /3 of this magnitude [3].
5. Texture of the whiskers
A detailed understanding of the diffraction effects requires a model of the diffraction grating. Since the whisker is not a single crystal but has a regular texture one can only define a local, or position-dependent lattice. We shall use polar coordinates r and 4) to refer to a site on the conical helix. The origin of this reference system is on the axis of the helix, a chosen reference direction being the origin of the azimuths 4~. The increase of 4~ by 27r (at constant r) is accompanied by a displacement over one layer thickness (i.e. c / 2 ) of graphite in the direction of the local normal and by a rotation over an angle /3 of an arbitrary reference direction defined with respect to the graphite structure. In successive sheets, n and n + 1, when viewed along the local normal to the graphite sheet, the orientation of the graphite structures differs by/3. Along a radial line, i.e. for constant 4) and varying r, the rotation of the graphite sheet remains the same in a given sheet. In a column parallel to the local normal, and centered on the point P0 with coordinates (r0, ~b0), the texture can be approximated by a stack of graphite sheets. Each sheet, with index n, is rotated over an angle /3 about the local normal with respect to the preceding sheet with index n - 1. When proceeding along a given sheet and describing a circuit around the axis of the helix the orientation of the graphite structure changes progressively, the orientation being /3 for a complete circuit, i.e. for an increase of 4) by 27r. As a result columns centered on points differing by A4,
128
W. Luyten et al. / Conical, helically wound, graphite whiskers
in azimuth contain textures which as a whole differ in orientation by an angle Aq,= ( Ad~/ZTr)/3,
(1)
independent of r.
The spotted ring patterns (of fig. 2) are the most informative ones. According to the model described above a simple algorithm allows one to generate the azimuths of the spot positions on a ring: ~bkj = j × 60 ° + k/3,
6. Interpretation of the diffraction patterns It is well known that the kinematical diffraction theory is sufficient to predict the positions of diffraction spots; the intensities are not correctly reproduced, however. In any case, the kinematical theory is applicable to a single sheet of graphite. The shape transform of a sheet is a line in reciprocal space, perpendicular to the plane of the sheet. The reciprocal space of a flat graphite layer thus consists of a set of parallel lines through the two-dimensional hexagonal array of nodes corresponding to the two-dimensional graphite lattice. In the present case the sheet is in fact a rather flat cone; the local orientation of the sheet changes and thus also the local orientation of these lines. The reciprocal space of a single conically wound graphite sheet consists of a hexagonal array of nodes, each one forming the apex of a narrow double cone with a semi-apex angle of about 20 °, having the local normals as generators. A plane (the Ewald " s p h e r e " ) normal to the axis of the cones intersects this configuration in a hexagonal array of points. Assuming the sheets to diffract independently, which is consistent with the kinematical approximation, the local reciprocal space of a volume element of the texture consists of the superposition of a large number of such double cone arrays, one for each sheet. These configurations of cones differ by/3 in orientation for successive sheets. The sets of cones situated on one diffraction ring overlap and thus produce a volume of diffuse intensity, which is roughly generated by having the apex of one of these narrow cones describe a circle keeping the axis of the cone parallel to the c* axis. A section of this reciprocal space parallel to the foil plane at l = 0 will produce the circular arrays of spot clusters, but oblique sections will lead to diffuse scattering along ellipse-shaped regions (fig. 2).
(2)
where j = 0 . . . . . 5 determines one of the six starting positions; k is an integer varying from 1 to n, where n is the number of sheets in the foil. We assume first the angle /3 to be a simple fraction 1/m of 60 °, i.e. m/3 = 60 °. The local structure centered on r 0, &0 then has a "unit cell" defined by the unit mesh of the coincidence site lattice in the basal plane and by a c-parameter equal to m times the sheet thickness normal to the basal plane. The texture has a 6m-fold screw axis as a symmetry element which will show up in the diffraction patterns as a 6m-fold rotation axis along c*. However, we must stress that this is essentially a local property limited to the volume elements centered on the line r, 4~0. In volume elements centered around r, ~h0+ A4~0 every sheet in the texture will have undergone the same rotation over Aq/= (/3/27r)A~b 0 with respect to its orientation in r, &0. The texture has still 6mfold symmetry, but it is rotated as a whole over k ~ and so will be its diffraction pattern. We thus arrive at the conclusion that the orientation of the diffraction pattern will depend on the azimuth of the selected area. Moreover, since the size of the selector aperture and its location r determine the angular range of O values admitted in the aperture, the length of the diffraction segments will increase with increasing aperture size. When assuming that the circular selector aperture contains a central angle ?, (fig. 6), the angular width A X of the diffraction segments will be given by A x = T(/3/2~-). This is a small effect since /3/2W is of the order of 1 / 1 8 and Y of the order of 20 °, for an aperture placed along the rim of the circular cleavage foil, i.e. the angular spread is of the order of 1°-2 ° . The crucial experiment that unambiguously identifies the described geometry of the texture is to demonstrate the rotation of the local lattice, i.e. of the diffraction pattern, with the azimuth of
IV.. Luyten et al. / Conical, helically wound, graphite whiskers
the aperture position with respect to the centre of a circular cleavage foil. Such an experiment is described below. The geometrical constraints, i.e. the formation of a line of pentagons along the axis of the cone and the formation of a low-energy coincidence site lattice, restrict the /3-values to the range 200-24 ° [3]. The only integer value of m is then 3, leading to 18-fold symmetry, which is the smallest observed multiplicity. Most multiplicities are much larger, however, e.g. 126. How can such a variety of multiplicities be reconciled with a very limited range of /3 angles? The reason is that a small change in the /3-value profoundly affects the geometry of the spot pattern generated by the algorithm. The algorithm (2) will generate a finite number of points if/3 is commensurate with 60 °. If/3 is incommensurate the points on the circle will form a dense set provided the number of sheets n ~ o0. In most actually observed cases one has
m/3 = 60 ° + E.
(3)
If now E is commensurate with 60 °, for instance IE = k × 60 ° with k and l integers, the generation of new spots will terminate again and a number
...--" .........
129
-.... f
/
f
",
i i
\
I
\ \
\
I
I
: i
j ',
',,.
j'
\
,f
I \
(a)
I
(b)
Fig. 7. (a) Simulated diffraction patterns for /3 = 22.842 and n = 84 (multiplicity = 126). The magnified part reveals the fine structure of the patterns; in an undeformed crystal all spots in a cluster are equally spaced. (b) Simulated diffraction pattern for /3 = 23.9 and n = 76 (multiplicity = 30). The magnified part clearly shows that the spots in each cluster are so closely spaced that the segments will be observed as being
continuous. of points on the circle given by 360°/E will result. The algorithm (2) was used to generate a number of spot sequences for different values of E. The results are shown in fig. 7 and can be compared with the observed patterns of fig. 2.
7. Diffraction experiments
A~
Fig. 6. Schematic representation of a cleavage fragment seen face-on, illustrating the rotation of the texture in a graphite sheet. A displacement of the selector aperture (3') along the rim of the flake will therefore give rise to a rotation of the segments in the selected-area diffraction pattern (fig. 8). When a smaller selector aperture (3") is used, the segments in the selected-area diffraction pattern will be less elongated (fig. 10).
The following experiment was performed on a nearly undeformed circular cleavage foil exhibiting a well defined 30-fold rotation axis. For the latter the angular separation of the spots is 12°. The orientation of the diffraction pattern could be determined unambiguously by using the shadow of the built-in pointer of the microscope as a marker. In fig. 8 this shadow has been indicated by arrows as a visual aid to the reader. The selective aperture was put along the specimen rim in a number of positions with different azimuths. In fig. 8a the pointer was positioned on a diffraction spot, and it was left untouched, as was the specimen, whilst recording diffraction patterns for different azimuthal positions of the selector aperture. In (c) the azimuth was 90 °, considering (a) as the origin; the rotation of the
130
Iv.. Luyten et al. / Conical, helically wound, graphite whiskers
p a t t e r n was 6 ° since t h e p o i n t e r was m i d w a y b e t w e e n the two spots. In (d), with a z i m u t h 180 °, the c o r r e s p o n d i n g r o t a t i o n was 12 ° since the p o i n t e r p o s i t i o n now coincides with t h e a d j a c e n t spot. In an i n t e r m e d i a t e p o s i t i o n of t h e a p e r t u r e (fig. 8b) also the r o t a t i o n angle was i n t e r m e d i a t e . A s s u m i n g a l i n e a r r e l a t i o n of the type (1) bet w e e n a z i m u t h c h a n g e A~b a n d r o t a t i o n angle A T we have for the p r e s e n t case: A q t = (1/15)AqS. This shows t h a t the a n g l e / 3 = ( 1 / 1 5 ) 2 7 r = 24 °. In t h e p r e s e n t case, the r e l a t i o n (3) yields 3 × 24 ° = 72 ° = 60 ° + 12 °, i.e. • = 12 °. This e x p e r i m e n t confirms the m o d e l quantitatively a n d allows us to m e a s u r e /3 directly. Fig. 9 illustrates t h e s a m e type of e x p e r i m e n t on a s a m p l e exhibiting a multiplicity of 66, for A 4 ) = 90 °. U n f o r t u n a t e l y , d u e to d e f o r m a t i o n of the s p e c i m e n , as is e v i d e n t from the s t r e a k e d c h a r a c t e r of the diffraction spots, it is not m e a n i n g f u l to i n t e r p r e t this observation quantitatively. It d o e s c o n f i r m the prese n c e of the local lattice r o t a t i o n , however. T h e size d e p e n d e n c e of the spots on the a p e r ture size is much m o r e difficult to d e m o n s t r a t e u n a m b i g u o u s l y since it is a small effect a n d since d e f o r m a t i o n of the foil a l r e a d y l e a d s to spot b r o a d e n i n g i n d e p e n d e n t of the a p e r t u r e size. T h e two diffraction p a t t e r n s of fig. 10 n e v e r t h e l e s s d e m o n s t r a t e the effect on a p a t t e r n with a multiplicity e q u a l to 30. T h e left p a t t e r n was t a k e n with a small a p e r t u r e (0.5 ~ m ) a n d the right one with a large a p e r t u r e (1.5 ~ m ) , b o t h p l a c e d at the s a m e position along t h e rim of t h e s p e c i m e n . It is clear t h a t t h e a n g u l a r extent of spots is l a r g e r in the right p a t t e r n than in the left one.
8. Conclusions T h e o b s e r v a t i o n s of the diffraction effects prod u c e d by cleavage sheets of the type of g r a p h i t e
Fig. 8. Electron diffraction patterns for different azimuthal positions of the selector aperture. The shadow of the built-in pointer (arrowed) is used to measure the rotation of the pattern. In (a) the pointer is positioned on a diffraction spot. In (c) the azimuth was 90° with respect to (a). The rotation of the pattern is 6°. In (d), with azimuth 180°, the corresponding rotation is 12°. In an intermediate position of the aperture (b) also the rotation angle is intermediate.
W. Luyten et al. / Conical, helically wound, graphite whiskers
131
Q
V
a
e
6
q
m
8~
o
Q
Fig. 10. Electron diffraction patterns demonstrating the effect of the size of the selector aperture on the length of the segments. The left pattern was taken with a small aperture (0.5 izm) and the right one with a large aperture (1.5 ~m). Both apertures were placed at the same position along the rim of the specimen.
m O
B
b
Fig. 9. Electron diffraction patterns for different azimuthal positions of the selector aperture. The shadow of the built-in pointer (arrowed) is used to measure the rotation of the pattern. In (a) the pointer is directed on a diffraction spot. In (b) the azimuth is 90 ° with respect to (a). The presence of the local lattice rotation is confirmed. The deformation of the specimen, which does not allow a quantitative interpretation of the experiment, is evident from the streaked character of the diffraction spots.
whiskers studied previously [3] confirm unambiguously the assumed microstructure. The whiskers are not single crystals but possess geometrically well defined textures producing diffraction patterns with rotation symmetries of high multiplicity. The wide variety of the observed multiplication can simply be explained by small changes in the conical shape, which is completely defined by a single parameter, the overlap angle/3. A tentative atomistic model suggests the
presence of pentagons of carbon atoms, like in the fullerenes.
Acknowledgements We would like to acknowledge Dr. Knippenberg for providing us with samples and references. This work was possible with the financial help of the National Fund for Scientific Research and the Services for Science Policy (IUAP48) of the Belgian National Government.
References [1] T. Tsuzuku, in: Proc. 3rd Conf. on Carbon, Buffalo (Pergamon, New York, 1957) p. 433. [2] H.B. Haanstra, W.F. Knippenberg and G. Verspui, J. Cryst. Growth 16 (1972) 71. [3] S. Amelinckx, W. Luyten, T. Krekels, G. Van Tendeloo and J. Van Landuyt, J. Cryst. Growth 121 (1992) 543. [4] F.C. Frank, Adv. Phys. I (1952) 91.
Electron diffraction effects of conical, helically wound, graphite whiskers W. L u y t e n , T. Krekels, S. A m e l i n c k x , G. V a n T e n d e l o o , D. V a n D y c k and J. V a n L a n d u y t UniL,ersity of Antwerp (RUCA), Groenenborgerlaan 171, 2020 Antwerpen, Belgium Received 25 June 1992; at Editorial Office 11 September 1992
Recently a new model for the structure of columnar graphite whiskers has been proposed; this model includes the presence of five-rings of carbon atoms, as in the fullerenes. The whiskers are assumed not to be single crystals but have geometrically well defined textures. Their unusual diffraction patterns exhibit rotation symmetry of high multiplicity. The wide variety of the observed multiplicities is explained by assuming small changes in the conical shape of the whiskers. We describe a series of electron diffraction experiments which have been performed on fracture cones of these graphite whiskers. These experiments confirm unambiguously the proposed microstructure.
I. Introduction Columnar growth of carbon was first reported by Tsuzuku et al. [1] and later by Haanstra et al. [2]. In a recent paper [3] the mechanism by which conically wound graphite whiskers grow was described and a possible relationship with the fivefold fullerene-like carbon rings was suggested. The proposed growth mechanism implies that the whisker is not a single crystal but has in fact a geometrically well defined texture. The electron diffraction effects associated with such a texture arc quite striking, giving rise to diffraction patterns with rotation symmetries of a very large multiplicity such as 30, 66, 126 and 222. It is the purpose of this paper to describe specific diffraction experiments which allow one to determine unambiguously the lattice geometry associated with the whiskers. We shall first describe briefly the formation mechanism of the graphite columns and their "lattice" geometry, and subsequently we discuss in detail the characteristic diffraction effects to be expected for such entities. Finally we present some diffraction experiments, which confirm the expected features.
2. Material morphology Under the given experimental conditions [2] needle-shaped carbon whiskers are formed. They are easily resolved under the optical microscope or in the scanning electron microscope (fig. la). They are up to 0.1 mm long and have a cross-section of about 3 mm. Most whiskers are broken and exhibit a shallow conical top on one side and a similar conical depression on the other side (fig. lb). During crushing in liquid nitrogen the needles break perpendicular to their length axis. The obtained thin flakes were deposited on a copper grid, which was covered with a carbon film and examined by transmission electron microscopy.
3. Electron diffraction patterns The diffraction patterns along the normal to thin circular cleavage slices of the whiskers are shown in fig. 2. The patterns exhibit different rotation symmetries with frequently observed multiplicities such as 30, 66, 126 and 222. The radii of the circles, on which the spot clusters are situated, correspond to the length of low-order
0304-3991/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
124
I,~ Luyten et al. / Conical, helically wound, graphite whiskers
Fig. 1. (a) Scanning electron micrograph of needle-shaped carbon whiskers. (b) Scanning electron micrograph of a cleavage fragment. Note the conical surfaces at both sides of the whisker.
W. Luyten et al. / Conical, helically wound, graphite whiskers
125
d Fig. 2. Different electron diffraction patterns along the zone normal to the cleavage flakes of conically wound graphite whiskers. The circles are divided in 18 (a), 30 (b), 42 (c) and 126 (d) segments.
diffraction vectors of graphite with I = 0 i.e. 1010, 1120 . . . . All such patterns have hexagonal symmetry, which simply reflects the hexagonal symmetry of graphite along [0001]. The segments or spot clusters in successive rings are in staggered positions. This is consistent with the assumption that the ring patterns are due to the superposition of many single-crystal diffraction patterns,
differing in orientation by discrete rotation angles about the [0001]* axis. Along zones inclined with respect to the foil normal the diffraction spot clusters are situated on ellipse-shaped curves; only spot clusters along two diametrically opposite arcs remain sharp, the larger part of the ellipses becoming continuously streaked (fig. 3).
126
W. Luyten et aL / Conical, helically wound, graphite whiskers
Along zones roughly perpendicular to the foil normal only rather broad contours of diffuse intensity are formed. In most patterns broadened and split 0001 reflections are visible. The separation of the split spots increases with increasing index I.
4. Whisker formation mechanism
The formation mechanism of such structures can be described by the following succession of imaginary operations. Suppose we start with a flat circular sheet of graphite with radius R (fig. 4a). We remove an angular sector/3 and we close the gap by making the point P coincide with the point Q, introducing simultaneously a relative offset of the two parts of the sheet over a distance equal to the sheet thickness (fig. 4b). We have now formed a helical surface situated on a cone. Along the junction line of the two parts steps are now present on each side of the sheet. Assuming that growth only occurs by the addition of growth units along the exposed ledges, a conically wound helical surface will be generated. Hereby the growth fronts adopt spiral shapes, similar to those associated with spiral growth around a screw dislocation, as proposed by Frank [41. However, whereas the usual spiral growth mechanism generates a succession of layers, all in the same crystallographic orientation, and therefore produces a single crystal, this is no longer the case here. The structures in successive sheets of the helical surface differ in orientation by the constant angles /3 and this orientation difference is cumulative. The semi-apex angle ~b of the cone is directly related to the angle /3; it is given by q5 = arcsin (1 - / 3 / 2 ~ - ) [3]. This apex angle is clearly seen in the scanning pictures of fig. lb.
Fig. 3. Electron diffraction patterns obtained by tilting the specimen off the zone normal to the cleavage flake. The tilt axis is chosen perpendicular to the length axis of the fracture cone and is represented by the straight line in (c). The tilt angle increases from (a) to (c).
W. Luyten et al. / Conical, helically wound, graphite whiskers
(b)
~ -
Fig. 4. Model illustrating the formation of a conical helix. In (a) a sector fl is removed from a disc. In (b) the angular gap is closed and a cone is formed.
The interface between successive sheets can be described as a pure twist boundary having a (0001) contact plane and a twist angle equal to /3. The magnitude of the angle/3 is determined by microscopic features of the model. In ref. [3] we assumed that the conical shape resulted from the initial introduction of a pentagon of carbon in a graphite sheet during the nucleation process. The presence of even a single pentagon may cause the graphite sheet to become conically deformed, as represented schematically in fig. 5. The angle of
Fig. 5. Twisted nucleus of a conical helix. The centre is a carbon five-ring. An orientation difference of approximately 20° is generated between the two sheets of the helix in the region of overlap.
127
overlap of the two parts of the same sheet is the angle /3 introduced in the macroscopic model. Taking into account the geometry of the pentagon and of the adjacent edge-sharing hexagons leads to a angle /3 of the order of 200-24 °, depending on the assumed extent of the deformation of the angles involved. An angle of the same magnitude emerges from considerations of the coincidence site lattice of two successive graphite sheets. Excellent fit of the superposed graphite structures is obtained for a twist angle /3 of this magnitude [3].
5. Texture of the whiskers
A detailed understanding of the diffraction effects requires a model of the diffraction grating. Since the whisker is not a single crystal but has a regular texture one can only define a local, or position-dependent lattice. We shall use polar coordinates r and 4) to refer to a site on the conical helix. The origin of this reference system is on the axis of the helix, a chosen reference direction being the origin of the azimuths 4~. The increase of 4~ by 27r (at constant r) is accompanied by a displacement over one layer thickness (i.e. c / 2 ) of graphite in the direction of the local normal and by a rotation over an angle /3 of an arbitrary reference direction defined with respect to the graphite structure. In successive sheets, n and n + 1, when viewed along the local normal to the graphite sheet, the orientation of the graphite structures differs by/3. Along a radial line, i.e. for constant 4) and varying r, the rotation of the graphite sheet remains the same in a given sheet. In a column parallel to the local normal, and centered on the point P0 with coordinates (r0, ~b0), the texture can be approximated by a stack of graphite sheets. Each sheet, with index n, is rotated over an angle /3 about the local normal with respect to the preceding sheet with index n - 1. When proceeding along a given sheet and describing a circuit around the axis of the helix the orientation of the graphite structure changes progressively, the orientation being /3 for a complete circuit, i.e. for an increase of 4) by 27r. As a result columns centered on points differing by A4,
128
W. Luyten et al. / Conical, helically wound, graphite whiskers
in azimuth contain textures which as a whole differ in orientation by an angle Aq,= ( Ad~/ZTr)/3,
(1)
independent of r.
The spotted ring patterns (of fig. 2) are the most informative ones. According to the model described above a simple algorithm allows one to generate the azimuths of the spot positions on a ring: ~bkj = j × 60 ° + k/3,
6. Interpretation of the diffraction patterns It is well known that the kinematical diffraction theory is sufficient to predict the positions of diffraction spots; the intensities are not correctly reproduced, however. In any case, the kinematical theory is applicable to a single sheet of graphite. The shape transform of a sheet is a line in reciprocal space, perpendicular to the plane of the sheet. The reciprocal space of a flat graphite layer thus consists of a set of parallel lines through the two-dimensional hexagonal array of nodes corresponding to the two-dimensional graphite lattice. In the present case the sheet is in fact a rather flat cone; the local orientation of the sheet changes and thus also the local orientation of these lines. The reciprocal space of a single conically wound graphite sheet consists of a hexagonal array of nodes, each one forming the apex of a narrow double cone with a semi-apex angle of about 20 °, having the local normals as generators. A plane (the Ewald " s p h e r e " ) normal to the axis of the cones intersects this configuration in a hexagonal array of points. Assuming the sheets to diffract independently, which is consistent with the kinematical approximation, the local reciprocal space of a volume element of the texture consists of the superposition of a large number of such double cone arrays, one for each sheet. These configurations of cones differ by/3 in orientation for successive sheets. The sets of cones situated on one diffraction ring overlap and thus produce a volume of diffuse intensity, which is roughly generated by having the apex of one of these narrow cones describe a circle keeping the axis of the cone parallel to the c* axis. A section of this reciprocal space parallel to the foil plane at l = 0 will produce the circular arrays of spot clusters, but oblique sections will lead to diffuse scattering along ellipse-shaped regions (fig. 2).
(2)
where j = 0 . . . . . 5 determines one of the six starting positions; k is an integer varying from 1 to n, where n is the number of sheets in the foil. We assume first the angle /3 to be a simple fraction 1/m of 60 °, i.e. m/3 = 60 °. The local structure centered on r 0, &0 then has a "unit cell" defined by the unit mesh of the coincidence site lattice in the basal plane and by a c-parameter equal to m times the sheet thickness normal to the basal plane. The texture has a 6m-fold screw axis as a symmetry element which will show up in the diffraction patterns as a 6m-fold rotation axis along c*. However, we must stress that this is essentially a local property limited to the volume elements centered on the line r, 4~0. In volume elements centered around r, ~h0+ A4~0 every sheet in the texture will have undergone the same rotation over Aq/= (/3/27r)A~b 0 with respect to its orientation in r, &0. The texture has still 6mfold symmetry, but it is rotated as a whole over k ~ and so will be its diffraction pattern. We thus arrive at the conclusion that the orientation of the diffraction pattern will depend on the azimuth of the selected area. Moreover, since the size of the selector aperture and its location r determine the angular range of O values admitted in the aperture, the length of the diffraction segments will increase with increasing aperture size. When assuming that the circular selector aperture contains a central angle ?, (fig. 6), the angular width A X of the diffraction segments will be given by A x = T(/3/2~-). This is a small effect since /3/2W is of the order of 1 / 1 8 and Y of the order of 20 °, for an aperture placed along the rim of the circular cleavage foil, i.e. the angular spread is of the order of 1°-2 ° . The crucial experiment that unambiguously identifies the described geometry of the texture is to demonstrate the rotation of the local lattice, i.e. of the diffraction pattern, with the azimuth of
IV.. Luyten et al. / Conical, helically wound, graphite whiskers
the aperture position with respect to the centre of a circular cleavage foil. Such an experiment is described below. The geometrical constraints, i.e. the formation of a line of pentagons along the axis of the cone and the formation of a low-energy coincidence site lattice, restrict the /3-values to the range 200-24 ° [3]. The only integer value of m is then 3, leading to 18-fold symmetry, which is the smallest observed multiplicity. Most multiplicities are much larger, however, e.g. 126. How can such a variety of multiplicities be reconciled with a very limited range of /3 angles? The reason is that a small change in the /3-value profoundly affects the geometry of the spot pattern generated by the algorithm. The algorithm (2) will generate a finite number of points if/3 is commensurate with 60 °. If/3 is incommensurate the points on the circle will form a dense set provided the number of sheets n ~ o0. In most actually observed cases one has
m/3 = 60 ° + E.
(3)
If now E is commensurate with 60 °, for instance IE = k × 60 ° with k and l integers, the generation of new spots will terminate again and a number
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Fig. 7. (a) Simulated diffraction patterns for /3 = 22.842 and n = 84 (multiplicity = 126). The magnified part reveals the fine structure of the patterns; in an undeformed crystal all spots in a cluster are equally spaced. (b) Simulated diffraction pattern for /3 = 23.9 and n = 76 (multiplicity = 30). The magnified part clearly shows that the spots in each cluster are so closely spaced that the segments will be observed as being
continuous. of points on the circle given by 360°/E will result. The algorithm (2) was used to generate a number of spot sequences for different values of E. The results are shown in fig. 7 and can be compared with the observed patterns of fig. 2.
7. Diffraction experiments
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Fig. 6. Schematic representation of a cleavage fragment seen face-on, illustrating the rotation of the texture in a graphite sheet. A displacement of the selector aperture (3') along the rim of the flake will therefore give rise to a rotation of the segments in the selected-area diffraction pattern (fig. 8). When a smaller selector aperture (3") is used, the segments in the selected-area diffraction pattern will be less elongated (fig. 10).
The following experiment was performed on a nearly undeformed circular cleavage foil exhibiting a well defined 30-fold rotation axis. For the latter the angular separation of the spots is 12°. The orientation of the diffraction pattern could be determined unambiguously by using the shadow of the built-in pointer of the microscope as a marker. In fig. 8 this shadow has been indicated by arrows as a visual aid to the reader. The selective aperture was put along the specimen rim in a number of positions with different azimuths. In fig. 8a the pointer was positioned on a diffraction spot, and it was left untouched, as was the specimen, whilst recording diffraction patterns for different azimuthal positions of the selector aperture. In (c) the azimuth was 90 °, considering (a) as the origin; the rotation of the
130
Iv.. Luyten et al. / Conical, helically wound, graphite whiskers
p a t t e r n was 6 ° since t h e p o i n t e r was m i d w a y b e t w e e n the two spots. In (d), with a z i m u t h 180 °, the c o r r e s p o n d i n g r o t a t i o n was 12 ° since the p o i n t e r p o s i t i o n now coincides with t h e a d j a c e n t spot. In an i n t e r m e d i a t e p o s i t i o n of t h e a p e r t u r e (fig. 8b) also the r o t a t i o n angle was i n t e r m e d i a t e . A s s u m i n g a l i n e a r r e l a t i o n of the type (1) bet w e e n a z i m u t h c h a n g e A~b a n d r o t a t i o n angle A T we have for the p r e s e n t case: A q t = (1/15)AqS. This shows t h a t the a n g l e / 3 = ( 1 / 1 5 ) 2 7 r = 24 °. In t h e p r e s e n t case, the r e l a t i o n (3) yields 3 × 24 ° = 72 ° = 60 ° + 12 °, i.e. • = 12 °. This e x p e r i m e n t confirms the m o d e l quantitatively a n d allows us to m e a s u r e /3 directly. Fig. 9 illustrates t h e s a m e type of e x p e r i m e n t on a s a m p l e exhibiting a multiplicity of 66, for A 4 ) = 90 °. U n f o r t u n a t e l y , d u e to d e f o r m a t i o n of the s p e c i m e n , as is e v i d e n t from the s t r e a k e d c h a r a c t e r of the diffraction spots, it is not m e a n i n g f u l to i n t e r p r e t this observation quantitatively. It d o e s c o n f i r m the prese n c e of the local lattice r o t a t i o n , however. T h e size d e p e n d e n c e of the spots on the a p e r ture size is much m o r e difficult to d e m o n s t r a t e u n a m b i g u o u s l y since it is a small effect a n d since d e f o r m a t i o n of the foil a l r e a d y l e a d s to spot b r o a d e n i n g i n d e p e n d e n t of the a p e r t u r e size. T h e two diffraction p a t t e r n s of fig. 10 n e v e r t h e l e s s d e m o n s t r a t e the effect on a p a t t e r n with a multiplicity e q u a l to 30. T h e left p a t t e r n was t a k e n with a small a p e r t u r e (0.5 ~ m ) a n d the right one with a large a p e r t u r e (1.5 ~ m ) , b o t h p l a c e d at the s a m e position along t h e rim of t h e s p e c i m e n . It is clear t h a t t h e a n g u l a r extent of spots is l a r g e r in the right p a t t e r n than in the left one.
8. Conclusions T h e o b s e r v a t i o n s of the diffraction effects prod u c e d by cleavage sheets of the type of g r a p h i t e
Fig. 8. Electron diffraction patterns for different azimuthal positions of the selector aperture. The shadow of the built-in pointer (arrowed) is used to measure the rotation of the pattern. In (a) the pointer is positioned on a diffraction spot. In (c) the azimuth was 90° with respect to (a). The rotation of the pattern is 6°. In (d), with azimuth 180°, the corresponding rotation is 12°. In an intermediate position of the aperture (b) also the rotation angle is intermediate.
W. Luyten et al. / Conical, helically wound, graphite whiskers
131
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Fig. 10. Electron diffraction patterns demonstrating the effect of the size of the selector aperture on the length of the segments. The left pattern was taken with a small aperture (0.5 izm) and the right one with a large aperture (1.5 ~m). Both apertures were placed at the same position along the rim of the specimen.
m O
B
b
Fig. 9. Electron diffraction patterns for different azimuthal positions of the selector aperture. The shadow of the built-in pointer (arrowed) is used to measure the rotation of the pattern. In (a) the pointer is directed on a diffraction spot. In (b) the azimuth is 90 ° with respect to (a). The presence of the local lattice rotation is confirmed. The deformation of the specimen, which does not allow a quantitative interpretation of the experiment, is evident from the streaked character of the diffraction spots.
whiskers studied previously [3] confirm unambiguously the assumed microstructure. The whiskers are not single crystals but possess geometrically well defined textures producing diffraction patterns with rotation symmetries of high multiplicity. The wide variety of the observed multiplication can simply be explained by small changes in the conical shape, which is completely defined by a single parameter, the overlap angle/3. A tentative atomistic model suggests the
presence of pentagons of carbon atoms, like in the fullerenes.
Acknowledgements We would like to acknowledge Dr. Knippenberg for providing us with samples and references. This work was possible with the financial help of the National Fund for Scientific Research and the Services for Science Policy (IUAP48) of the Belgian National Government.
References [1] T. Tsuzuku, in: Proc. 3rd Conf. on Carbon, Buffalo (Pergamon, New York, 1957) p. 433. [2] H.B. Haanstra, W.F. Knippenberg and G. Verspui, J. Cryst. Growth 16 (1972) 71. [3] S. Amelinckx, W. Luyten, T. Krekels, G. Van Tendeloo and J. Van Landuyt, J. Cryst. Growth 121 (1992) 543. [4] F.C. Frank, Adv. Phys. I (1952) 91.