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Conical, helically wound, graphite whiskers: a limiting member of the “fullerenes”?
Journal of Crystal Growth 121 (1992) 543—558 North-Holland
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CRYSTAL GROWTH
Conical, helically wound, graphite whiskers: a limiting member of the “fullerenes”? S. Amelinckx, W. Luyten, T. Krekels, G. Van Tendeloo and J. Van Landuyt University of Antwerp (RUCA), Groenenborgerlaan 171, B-2020 Antwerpen, Belgium Received 15 April 1992
Conically structured columnar graphite crystals have been studied by electron microscopy and electron diffraction and a formation mechanism is proposed explaining their shape and unusual diffraction effects consisting in periodically interrupted circular ring patterns. A growth mechanism is proposed whereby the initial graphite layer adopts a slitted dome shaped configuration by introducing a fivefold carbon ring in the sixfold carbon network. Successive sheets are rotated with respect to the previous one over a constant angle, thus realizing a helical cone around a “disclination” with a fivefold carbon ring core. This model explains the morphological features and the particular diffraction effects observed on these reproducibly prepared columnar graphite crystals. The growth mechanism has a direct relationship to the formation of the fullerene “bucky ball” molecules.
1. Introduction More than twenty years ago, Haanstra et al. [11 described needle-shaped graphite columns which were grown at temperatures above 1800°C on f3-silicon carbide substrates in a carbon-monoxide atmosphere. The columns apparently nucleated on multiply twinned crystals of a-SiC. The carbon columns were described by the authors as “being stacked from parallel layers of carbon atoms in a hexagonal network bent into a cone with top angle of about 141°”.This angle was supposed to have been inherited from the f3-SiC substrate. The successive layers were assumed to be “randomly rotated with respect to each other over angles being multiples of 60° or 30° divided by a prime number” [1]. Already much earlier, related observations had been made by Tsuzuku [2] who investigated the structure of conical crystals with a “lotus-leaflike” form grown on the surface of soot heated to 2500°C. In both cases interrupted or segmented diffraction rings were observed in the c * zone of electron diffraction patterns of thin sections of the columns. 0022-0248/92/$05.OO © 1992
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A detailed analysis of the formation mechanism and a consistent interpretation of the diffraction effects are still lacking. From the descriptions and reproduction of the diffraction patterns given in the two references [1] and [21we can conclude that the crystals must have had similar microtextures. Since in ref. [21 the growth conditions were not unlike those under which “fullerenes” could be formed, we considered it worthwhile to re-examine some of the crystals obtained more than twenty years ago from Dr. Knippenberg, one of the authors of ref. [11. The columnar crystals used in our experiments were prepared on SiC substrates, using the method described in detail therein [1]. Recently, microtubules of graphite have been described by lijima [31, but these have a very different microstructure, the sheets forming a scroll-type arrangement.
2. Morphological observations 2.1. Scanning electron microscopy The “crystals” were examined “as-received”, i.e. without an~’treatment, in a scanning electron
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S. Amelinckx ci al.
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Conical, helically wound, graphite whiskers: limiting member of ‘fullerenes”?
microscope which revealed well-formed rounded needle-shaped “crystals” with cross sections of the order of 3 ~m, sometimes with flat end surfaces, sometimes ending in a “sharp” point (fig. la). When broken, one end of the cylindrical crystal fragments always exhibited a shallow corncal depression and the other end a rather flat conical hill, the semi-apex angle of the two features always being complementary (fig. ib). It is
clear that the cylindrically shaped crystal fragments had been formed by cleavage along the graphite sheets and that the cleavage faces were not flat but had a conical shape (fig. ic). .
.
2.2. Transmission electron microscopic observa
-
Thin cleavage flakes of such crystals obtained by crushing the needles were mounted on a car-
I—
2,5 ~.tm
Fig. 1. Scanning electron microscope images of conically wound carbon columns. (a) Crystal ending in a sharp tip. (b) Crystals cleaved in cylindrical segments. (c) Cylindrical part of a column: note the shape of the two end faces.
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bon coated holey film and examined in transmission electron microscopy. Fig. 2a shows such a quasi-circular, pancake-shaped flake. It clearly reveals the shape of the cross section of the column. Such thin flakes usually exhibit some narrow strips of material lying on the crystal, and which are highly deformed, as judged from the density of inclination extinction contours, and
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which always end abruptly at the periphery of the circular flake. Such features are presumably “wrinkles” in the foil resulting from flattening a conical thin sheet as a result of crushing. They may also be folded parts of the pancake. In ref. [2], double-folded crystals were shown to adopt the shape of a fan, confirming their conical shape. A similar observation is reproduced in fig. 2b;
I —I.
C
I
~
]_,.
JP~TUIIII~
~
1~.im
2~ixn
~
Fig. 2. Quasi-circular pancake-shaped thin cleavage section ot a columnar crystal. (a) Bright field image of a c-cleavage disk. (b) Dark field image of a cleavage disk exhibiting folds and wrinkles. (c) Bright field image of a column fragment revealing the conical geometry.
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Conical, helically wound, graphite whiskers: limiting member of ‘fullerenes”?
part of the disc was folded back on itself, revealing a fan-like periphery. In some cases also shadows of column fragments were formed which confirm the conical geometry revealed by the scanning electron microscopic observations (fig. 2c) and allow one to measure the semi-apex angle ~
3. Description of the electron diffraction patterns 3.1. The c * zone pattern The c * zone diffraction patterns of such circular crystal fragments are quite remarkable. A number of examples are shown in fig. 3. The
100
-
fbi
100
1-ig. 3. Ditterent clectron diffraction patterns along the — zone of cleavage flakes ot conically s~oundgraphite columns. The rings correspond with hk0 reflections of graphite. The circles are divided into (a) 18 segments, (b) 30 segments, (c) 42 segments and (d) 126 segments.
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patterns mostly consist of “segmented” concentric circles with radii in accordance with the hk0 lattice spacings of graphite. The segments are almost uniformly spaced along the circles. Each “segment” or spot cluster is often further composed of a number of equidistant spots, the same number in almost all segments. Apparently the dotted fine structure of the segments remained unresolved or unnoticed in previous work [1,21.
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The angular range of the segments is constant along the different circles, their length thus increases proportionally to the radii. Occasionally the dots are so closely spaced that the segments look continuous (fig. 3b). Moreover, sometimes the segments are joined, head to tail, leading to continuous circles, often with periodic reinforcements in brightness (fig. 3c). In the latter case the pattern is identical to a textured powder pattern
004
:;~
002
S Fig. 4. Diffraction patterns along zones different from c*. (a) Diffraction pattern along a zone which is practically perpendicular to the c* direction; note the elongated shape of the 001 reflections. (b) Fine structure of high order 00! spots; the spot splitting is indicated by the arrows.
548
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Conical, helically wound, graphite whiskers: limiting member of “fullerenes”?
consisting of cleavage flakes randomly rotated around the c axis normal to the plane of the flakes, so as to exhibit only hkO reflections. The positions of the segments in one circle are roughly staggered in radial directions with respect to those in the second circle of the pattern. The length of the segments depends on the sample, and their angular separation is constant for seetions made from the same column, but it is usu ally different for different crystals. It is worth noting that in ref. [21the segmented nature of the diffraction rings was attributed to the “polygonization” of a conical sheet, by the formation of radially oriented edge dislocation walls. The number of segments or spots on a circle is extremely variable; observed numbers are 18, 30, 42, 66, 126, 222.
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000 /1
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Fig. 5. Schematic representation of the conical surfaces representing the loci of diffuse intensity in reciprocal space.
Fig. 6. Oblique sections ot the diffuse scattering distributions at various tilt angles. The tilt axis is represented by the straight line in (c) . Note the ellipse-shaped loci which are the intersections of Ewald’s sphere with the surfaces represented in fig. 5.
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3.2. Patterns along zones normal to c * A typical diffraction pattern along a zone normal to the c* axis is shown in fig. 4a. Only one row of 001 spots is present; they are elongated along a direction normal to the c* axis. The elongation of the spots increases with increasing 1, although this is not immediately obvious from a photograph because of the simultaneously decreasing overall intensity of the spots with increasing I. The high order spots are sometimes visibly split into two components of which the separation increases with increasing 1, as shown in fig. 4b. The other reflections have all degenerated into continuous streaks, which broaden with increasing 1, the width being a minimum for l = 0. This type of diffraction pattern is consistent with a reciprocal lattice in which the lattice points of the 00/ row have been transformed into ring shaped regions situated at the levels of 001 reflections, on a double cone with its apex in the origin of reciprocal space and with a semi-apex angle of about 20°.The other regions of diffuse scattering form complicated 3D distributions in reciprocal space. They result from the fact that each diffraction spot along the circles is the apex of a double cone with the same semi-apex angle as for the 001 reflections. These cones overlap and the final result is a region of diffuse intensity limited by the conical surfaces represented in fig. 5. This shape of the diffuse intensity distribution is consistent with diffraction patterns such as fig. 4a and figs. 6a, 6b and 6c. For small tilt angles of the specimen the spot pattern along the circles does not change. For larger tilt angles ellips shaped loci of diffuse intensity are formed (figs. 6a—6c).
4. Diffraction contrast imaging One or a few segments of the diffraction rings can be selected to produce a dark field image. It turns out that the whole circular sample is imaged, showing that the whole flake contributes to all diffraction segments. The image consists of a bright—dark line pattern, which is roughly perpendicular to the direction of the curved clusters
549
of spots used to make the DF image (fig. 2b). Diffraction patterns made from different areas of the circular flake produce the same diffraction pattern, but not necessarily in the same orientation, showing that the complete set of differently oriented sheets is present in each part of the sample.
5. Model for the growth of conically wound columnar crystals 5.1. Growth mechanism All observations appear to be consistent with a model in which the needle-shaped “crystal” consists of a helical surface wound on a rather “flat” cone. We can form the “crystal” by the following imaginary operations. In a sheet of graphite we make a cut, for instance by following the sides of the hexagons and ending somewhere in the middle of the sheets. Along this cut we lift one part of the sheet with respect to the other part over one sheet thickness, and slide this part over the other one in such a way as to achieve in the sector of overlap the normal stacking of graphite, at least for a certain fraction of the hexagons. We construct in fact in the overlap part a “coincidence site” lattice [41of two graphite sheets rotated over a small angle. This can of course only be achieved by deforming the planar sheet into a cone with its apex in the endpoint of the cut. The angle of the sector of overlap depends on the density of the coincidence site lattice (i.e. on the .~-index)which in turn is related to the semi-apex angle of the cone as will be shown below. Graphite sheets are known to grow almost exclusively along their edges; if subsequent growth takes place, a helical surface will thus be formed by growth along steps, somewhat like in spiral growth [5].However, growth now takes place on a helical cone and homologous crystallographic directions in successive sheets are no longer parallel but enclose a small constant angle /3, determined by the original wedge angle of the overlap area. Between two successive sheets a low-angle pure twist boundary is thus present. The defect along the core of the cone is now a “disclination”,
550
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/ Conical,
helically wound, graphite whiskers: limiting member of “fullerenes”?
R I
mal graphite, is a consequence of the presence of I
the dislocation grid forming the twist boundary. 5.2. Geometrical features of the model.
R0
Quantitative confirmation of principle possible by verifying between the semi-apex angle cP 7) and the angle 13 over which I
I
I
this model is in the relationship of the cone (fig. the two parts of
the sheet overlap. The angle cP can be measured either in scanning electron microscopic images or in shadow images in TEM, whereas the angle /3
a
Fig. 7. Schematic representation of the geometry of the conical crystal, illustrating the derivation of the relation between Pand/3.
rather than a “dislocation”. In the literature, disclinations characterized by a screw rotation operation are sometimes called “dispirations” (e.g. ref. [7]). The very easy cleavage along the helical surface, easier than the c-cleavage in nor-
can be derived from the diffraction pattern along the c * zone, as will be shown below. In practice it is unfortunately difficult to perform the two measurements on the same crystal. In fig. 8 we represent the flat circular sheet of graphite with radius R before introducing the “disclination”. We now remove an angular sector /3, and close the gap by making the point P coincide with 0 where a=Rf3.
(1)
On doing this we form the cone of fig. 7, having a circular base with radius R0. Expressing the circumference of the base of the cone in two different ways we obtain the relation: 2~rRo 2~R a. (2) =
—
b~ Fig. 8. Continuous model illustrating the formation of a conical helix. (a) A sector /3 is removed from a disc. (b) The angular gap is closed and the cone is formed.
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/ Conical,
helically wound, graphite whiskers: limiting member of “fullerenes
‘?
551
Overlap angle: /3 (deg)
Semi-apex angle: cP (deg)
Multiplicity: N = 360°//3
27.7 21.7 15.1
67.37 70.00 73.34
13 16.59 23.84
closing this angular gap, accompanied by sliding the two parts along a direction normal to the plane of the disk (fig. 8). Such a defect is called a “disclination” or a “dispiration” line [6,7]. Summarizing: Instead of growth around a screw dislocation normal to the c-planes, the model assumes that growth takes place around a “dis-
3458
clination”, angles of successive forming a sheets helical with cone.respect The rotation to an
Table 1 Coincidence site lattices angles
~
initial reference orientation are cumulative: each sheet is rotated over the same angle /3 with
Taking into account that sin j
=
R /R ~
(3) ‘
and using (1), we obtain the desired relation from (2) and (3): /3
=
2’ir(l
—
sin cP).
(4)
A theoretical angle /3 can be obtained from the rotation angles corresponding to possible coincidence site lattices, as listed in table 1. In a continuum approximation this operation consists in deforming a pancake-shaped disk, by removing a wedge shaped sector of material, followed by
-
respect to the previous one. It is clear that rightand left-handed conical helices are equally possible. The angle /3 is directly visualized in the double-folded fan-shaped cleavage foil of fig. 9; it is clear that the lacking sector is /3/2. In normal spiral growth around a dislocation line, the successive lattice planes have all the same orientation, exhibiting perhaps an elastic twist in sufficiently thin filaments [8]. In conical spiral growth, as proposed here, successive layers are rotated over a constant angle. Our model accounts in a natural way for the fact that the .
“~
j —s05 ____
Inn
Fig. 9. Bright field image of a double folded fan shaped cleavage foil. The indicated lacking sector corresponds with /3/2.
552
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Conical, helically wound, graphite whiskers: limiting member of “fullerenes”?
rotations between successive sheets are not random but systematic, a fact which remained unexplained in previous work.
(pQ+
5.3. Structural features of the model
Qo
For a detailed discussion of the diffraction effects, a detailed model of the texture is required. In our description of the model, we shall make use of a reference system of “helical polar coordinates”. These coordinates are the polar distance r and the azimuth angle ~P,varying from ~ to + ~, counted from the same reference direction SO. The coordinates are helical in the sense that a rotation over 2~raround S introduces a displacement over c/2 (c lattice parameter of graphite) parallel to the axis of the cone. The points P0(r0, 1P0) and P1(r~,1P1~ + 2ir) are thus located in successive sheets, separated by c/2, but they project along c in the same point. We now consider the structure in the vicinity of a column centred on the point P0(r0, 11’~), situated in the sheet n. We specify the orientation of this sheet by the angle 4~ enclosed by SP0 and some reference direction associated with the graphite structure in that sheet. A convenient reference direction is for instance one of the base directions of the hexagonal coincidence site lattice corresponding with the rotation angle /3, as defined above and shown in fig. 10. Along the line SP0, the angle 4~is constant for varying r. Along the same direction SP0, but in the next sheet n + 1, this reference direction will havereference rotated direction over an angle /3, 4~ i.e. +for 2ir the becomes /3. ~I’~ + In general, along the line SP,, situated vertically above SP 0 with azimuth ~P= + n2~,the reference direction will have become 4~ + n/3, and a vertical displacement over n(c/2) has taken place. Consider now first the simple case mj3 = 60° (m = small integer). The graphite texture in the column centred on P0 then becomes periodic along the normal to the sheet, with period m(c/2), since in the sheets n and n + m the coincidence site lattices become parallel. One can then define a structural unit, something like a “local” unit cell defined in the c-direction by the
(~
(p 0
s
o
—
=
Fig. 10. Reference system of helical polar coordinates.
parallel meshes of the coincidence site lattices in the sheets n and n + m and separated by m(c/2). The carbon atoms in the interleaving sheets form the filling of this structural unit. In a strip along the direction SP0, the texture contains a number of such “unit cells” in parallel orientation. Also along the local normal to the sheets there are a number of such parallel structural units, depending on the sample thickness. Instead of considering this structural unit as being located between the sheets n and n + m, we can consider just as well this structural unit as being located between the sheets n + 1 and n + m These twoover structural units arelocal then related by +a 1.rotation /3 about the c axis, accompanied by a displacement c/2 parallel to this axis, i.e. this structural unit has a non-crystallographic 6m-fold screw axis as a symmetry element. As viewed along the c-direction, the graphite texture has a 6m-fold rotation axis. If the angle /3 is not a simple fraction of 60° but still commensurate with 60°, we can always write /3N = k > 360° (5 ‘
where N is the multiplicity of the rotation axis as observed in the diffraction pattern (in the previ-
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ous case, N = 6m), and where k is a small integer. For instance for N = 30 and /3 in the range 20°—25°,one would have k = 2, leading to /3 = 24°. For N = 126 and the same range of /3 values, one would have k = 7 (/3 = 20°)or k = 8 (/3 = 22.85°). We now note that the rotation of the graphite sheet over an angle /3, when proceeding to the next sheet, must occur progressively with ‘P. If we consider the line SQ0 at ‘I’d + ~l’~ enclosing an angle ~W’0 with SP0, the graphite sheet in the point Q0, or its associated coincidence site latlice, will have rotated over an angle =
(/3/360°)z.t’P0
(6)
and will have been displaced vertically over =
(/3/360°)(c/2).
(7)
The same applies to the sheets n, n + 1,. . . , n + m,... As a result, the column of local unit cells, centred on QO, i.e. the whole texture will differ L14. in orientation. Again along the line SQ0 all structural units will be parallel. The whole conical texture can thus be considered as consisting of sectors limited by radial lines within which the “structures” are approximately periodic along a radial direction and along columns parallel to the local c direction, the translation symmetry being that of the coincidence site lattice and the rotation symmetry being N-fold. The limits of these sectors are not strictly defined since the orientation of the “unit cells” changes in fact continuously with the azimuth ‘I’. With a change of the azimuth LW’ corresponds a change in the orientation of the texture of L~4 given by (6). In case the angle /3 is incommensurate with 360°,there is strictly speaking no periodicity along the c axis and hence no rotation symmetry if the number of sheets remains finite. If the number of sheets becomes very large the rotation symmetry becomes D00.
6. Diffraction effects The electron diffraction effects and the related observations will be reported in detail in a separate paper. In the present paper we shall there-
553
fore limit ourselves to a brief description of those diffraction effects which are specific of the model and thus allow one to establish the latter unambiguously. A spot position on one of the circles can be characterized by its azimuth angle counted from an arbitrary origin. These azimuth angles can be obtained by applying a simple algorithm as we shall see. It is well known that the kinematical diffraction theory is adequate to describe the geometry of diffraction patterns, but it is not particularly successful in predicting the intensity of spots in electron diffraction patterns. Since in any given volume element successive graphite sheets are rotated one relative to the previous one about the local c-axis over a constant angle /3, the kinematical theory predicts that in the diffraction pattern of that volume element the spot positions can be described by the following relations. The azimuths of the spots in the first circle are given by ict~ =
(I—i) X 60°+n~f3,
(8)
where n3 is an integer. The index j refers to the six different starting points for sequences j = 1, 2, . .. , 6. The spots on the second circle are similarly represented by their azimuths
~
= (i 1) x 60°+ 30°+ n1/3. (9) Although these algorithms are very simple, the resulting patterns may become rather complex since they depend very sensitively on two main parameters: the angle /3 and the total number n of sheets in the sample. The pattern may become complex if /3 is not commensurate with 60°.None of these parameters follows directly and unambiguously from the observed diffraction pattern. Whereas it is simple to generate a pattern for given values of /3 and n, the reverse operation, i.e. deducing /3 and n from an observed pattern, is not trivial. Moreover, since the samples considered are not single crystals but well-defined regular textures, at least when undeformed by the preparation method of the sample, the orientation of the diffraction pattern is found to depend on the position of the selected area and even on the size of the selective aperture. Specimen de—
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/ Conical,
helically wound, graphite whiskers: limiting member of “fullerenes”?
formation may change the shape of spots and segments, but will change neither the pseudosymmetry nor the multiplicity. Two diffraction patterns taken from two Selected areas both along the rim of the circular sample, but differing in azimuth by z.1’P, will be rotated one with respect to the other over an angle ~D given by ~cP = (/3/360°)zl’P. A statically detectable rotation of the diffraction pattern will thus result only if ~D differs from an integer multiple of the angular separation of the spots or
simple, ideal case where m/3 = 60°, the rotation over /3 is a 6rn-fold symmetry rotation in projection. This also implies that the angular range of the texture over the sample area of which the diffraction pattern is recorded depends on the aperture size. If the selector aperture collects an angular range of textures z~tO(~,the diffraction spots or segments will have an angular width
of the segments. The latter is given by 360°/N, where N is the multiplicity or quasi-rotation symmetry of the pattern, A dynamical experiment, consisting in displacing the selective aperture (or the sample), whilst observing the diffraction pattern, would result in a visible rotation of the diffraction pattern, often over several interspot distances, depending on the change in azimuth accompanying the displacement of the aperture. The latter effect can always be detected since it does not depend on the precise magnitude of /3 or of LW’, as does the static observation. As shown above in section 5.3 and as also implied in the relation (6), the local texture rotates continuously with the azimuth of the considered volume element, the rotation angle being /3 when the azimuth has changed by 360°. In the
The spot or segment widths will thus increase with increasing aperture size. However, this is a small effect since /3/360°is of the order of 0.06.
1i’P0
=
(/3/360°) ~i@O.
(10)
It will therefore only be observable for patterns with a small multiplicity, i.e. N < 30, and for which sharp spots are formed when using the smaller aperture. For segmented circles the effect remains unobservable in practice; in deformed samples it is wiped out completely.
7. Diffraction experiments and computer simulations The predicted features of the diffraction patterns have been verified and found to be in agreement with the experiments; they will be
Fig. 11. Magnified part of one of the segmented rings of the diffraction pattern represented in fig. 3d . The spacing between the spots in one segment differs from segment to segment, sometimes leading to (nearly) overlapping spots (arrowed).
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helically wound, graphite whiskers: limiting member of “fullerenes”?
555
discussed in detail in a forthcoming paper. We limit ourselves here to showing the excellent agreement between the two observed diffraction patterns, and the corresponding computer simulated patterns, based on the algorithm (9). The observed pattern of fig. 3d has a multiplicity of N 126. A magnified part of this pattern is shown in fig. 11. Although it is clear that every segment or spot cluster contains four spots, the positions of the spots in each segment differ from segment to segment. This is most probably due to the fact that the crystal was slightly deformed during crushing. The main features of this diffraction pattern, however, are correctly reproduced in the simulated pattern, using as parameters /3 22.842°and a number of sheets n 84 (fig. 12a). Clearly the spacing between the spots
and the number of sheets was n 76. Again the agreement is quite good. In these simulation experiments the finite aperture size effect was neglected; in addition we have to stress that the resulting patterns are only representative for ideal, undeformed textures, present in a small volume element, i.e. for the smallest aperture. The “dynamical” experiment described above was performed in the electron microscope on a number of patterns. Displacing the aperture along the rim of a specimen always resulted in a rotation of the diffraction pattern. A static experiment performed on the diffraction pattern of a specimen with N 30, i.e. with an interspot angular separation of 12°,leads to a rotation over 6° for a change in the azimuth M’1~of the aperture
in a cluster would remain constant in the case of an undeformed crystal, as can be judged from the magnified part of the simulated pattern of fig. 12b. The pattern of fig. 3b, on the other hand, consists of unresolved segments, the multiplicity being N 30. For the simulated pattern represented in fig. 12b, the value /3 23.9°was used
by 90°in agreement with the prediction for this particular case for which /3 24°and thus /3/4 6°.
=
=
=
=
=
a
~
.----..-
=
=
=
=
8. Discussion One important question has not yet been answered. Why does such a helical cone form at all?
b /
/ /
I
\
\
1~Th
~
~
/
/
I
Fig. 12. Simulated diffraction patterns for: (a) /3 = 22.842 and n = 84 (N = 126). The magnified part reveals the fine strucure of the pattern; in an undeformed crystal, all spots in a cluster are equally spaced. (b) /3 = 23.9 and n = 76 (N = 30); the magnified part clearly shows that the spots in a cluster are so closely spaced that the segments will look continuous.
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The authors in ref. [1] assume that the cone shape is inherited from the twinned SiC substrate. The formation mechanism proposed in ref. [2], on the other hand, is summarized in the following quote: “If a thin monocrystal layer is subjected to the rotational slip immediately after the mechanical bucklingproducing the central screw dislocation it may partly overlap by an angle 0, forming a cone”. However, a different answer is suggested by a recent proposal for the nucleation mechanism of fullerenes [7]. The non-planar character of the graphite nucleus finds its origin in the occurrence during the nucleation process of a five-ring of carbon atoms. If such a pentagon is surrounded symmetrically by edge sharing hexagons, a dome is formed which on the subsequent addition of the correct ratio of pentagons and hexagons grows into a closed cage such as C60 or C71~[7]. In the present case a single pentagon may be sufficient to form a twisted nucleus (fig. 13). On further lateral growth by the addition of hexagons exclusively, a symmetrical dome cannot grow further without leaving gaps. Instead a “slitted” cone forms, consisting of one helical
/ i-
.~-
~
7~
‘~
~.
~
~-
~--~
-~
I
/
-
-
1
Fig. 14. Coincidence site lattice of two superposed graphite sheets rotated over an angle of 2 arctan (1/31i). The unit of misfit is a hexagonal grid with a lattice parameter a = where a is the C—C distance in a six-ring.
sheet. The angular gap left along the pentagon in fig. 13 is theoretically 2 x 12° 24°. The two “lids” of the helix being at different levels, the sheet can now expand laterally by growing on top of itself with a small orientation difference in the vicinity of 24°,but somewhat smaller. Hereby the overlapping graphite sheets will realize the best fit compatible with the presence of one or perhaps a small number of pentagons. Hereby one or a small number of preferred rotation angles will occur, whereby the density of coincidence sites is a maximum. The coincidence site lattice which best fits the twist angle introduced by the presence of the pentagons is presented in fig. 14; it has a hexagonal unit mesh with a lattice parameter a~/~f(a C—C distance) and corresponds with a /3-value of 2 arctan(1/3V~) 21.8°. Because of the weak bonding between the successive graphite sheets, the tendency to realize this particular coincidence site lattice may not be very pronounced and in practice the value of the /3 angle may be somewhat variable. However all observations of semi-apex angles and of the fanlike double-folded sheets are consistent with a /3 =
/
/ .
/
/
--
/
/
Fig. 13. 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,
=
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Conical, helically wound, graphite whiskers: limiting member of “fullerenes”?
angle which does not differ very much from this ideal value. The “core” of the helical surface finally consists of vertically stacked pentagons. As pointed out above, the growth conditions of the helical cones as described in ref. [2] are not unlike those of the “fullerenes”, which makes the formation of a small concentration of pentagons a probable event, According to our model, the angle /3 is the rotation angle of a low energy low angle twist boundary, i.e. a boundary with the maximum density of coincidence sites taken into account the fact that the presence of the pentagons determines to a large extent the angular range. In fig. 14 we have represented an example for which the angle /3 is approximately 21° with .~ 7. The corresponding semi-apex angle of the cone is then 70°±2°,which is the only observed angle, as e.g. in fig. 8. =
9. Conclusions
A surprising feature of the model is the fact that the wide variety of multiplicities or pseudorotation symmetries of the diffraction pattern turns out to be compatible with a narrow range of /3 angles as derived from the observations and from theoretical considerations. The /3 angle was derived: (i) from the semi-apex angle of the conical cleavage faces such as fig. 2c; (ii) from the fan-like double folded cleavage sheets, such as fig. 9. Two independent theoretical considerations suggest f3 angles in the same narrow range: (i) the lacking angular segment in the fit between the singular pentagon and the adjacent hexagons as represented in fig. 13; (ii) the value of the rotation angle of the best coincidence site lattice, as represented in fig. 14. Even though the preparation methods in refs. [1] and [21 are rather different, the similarity between the diffraction patterns reproduced in refs. [1] and [2] strongly suggests that the same microstructure was observed in both cases. This also means that the specific form of /3-SiC which
557
was considered to be required as a substrate in ref. [1] is actually not essential, since a quite different procedure, approaching more the conditions under which “fullerenes” are formed, allows one to obtain the same type of conical crystals. It seems, however, that the use of a /3-SiC substrate provides a reproducible method to prepare this particular microstructure. The fact that heating soot at temperatures above 2500°C in a closed graphite crucible [2] allows one to produce conical graphite crystals as well makes our proposed microscopic mechanism plausible. Our macroscopic model allows a consistent and natural explanation for all the observed morphological characteristics and the geometrical features of the diffraction patterns and allows to link these patterns to the morphology of the “crystals”, which are in fact “textures”. The microscopic model is more speculative in the sense that we have no direct evidence for the occurrence of five-rings. However a consistent geometrical structure in accordance with the observations can be built on this assumption.
Acknowledgements We would like to thank Dr. W.F. Knippenberg for providing us with samples and references. Professor D. Van Dyck (University of Antwerp) and Dr. U. Dahmen (L.B.L. Berkeley, California) are gratefully acknowledged for useful discus. sions. The . Fund for Scientific Research (11KW) . . and the Ministry of Science Policy of the Belgian Government (IUAP Project 48) are gratefully acknowledged for financial support. .
.
Appendix. Diffraction patterns We call basic multiplicity N the number of sharp spots or of spot clusters (segments) along a circle. The angular separation of the spots (or segments) is then 360°/N.Note that N is always a 6-fold, which is a direct consequence of the 6-fold symmetry of the graphite sheets.
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The angular range of /3 being restricted to the interval 20°—24°, as follows from the observations of the geometrical features, we have 3/3 60°+ c. (A.1) If e = 0, we have the simplest possible case, i.e. =
Table 2 Observed basic pseudo-periods (N = number of spots, p N/6, /3=(l+1/p)X20° and e=360°/N N p /3 (deg) e (deg) 30 42
5 7
66 126
11 21
24 226 7
N= 18, /3 with
=
20°. If
E’’O
( <<60°), we note that
21~ 20~ 21
=
360°/N,
=
60° (p
12 8~ 7
26 7
(A.2)
we obtain a strictly periodic N-fold pattern of isolated spots provided pe
=
=
integer).
(A.3)
The number of spots is then finite (i.e. N) even for an infinite number of sheets. The angle /3 follows from (A.1) and (A.3), /3 (1 + l/P) x 200, and the value of p follows from (A.2) and (A.3), p N/6. All observed basic pseudo-periods are obtained in this way, as shown in table 2. The equidistant distribution of spots in clusters, exhibiting the same pseudo-periodic arrangements with the same multiplicities, is due to small deviations from the /3-values indicated here. The multiplicities observed in ref. [1], N 102, 132 and 222, correspond to p 17, 22 and 37, respectively. =
=
=
=
References [1] H.B. Haanstra, W.F. Knippenberg and G. Verspui, J. Crystal Growth 16 (1972) 71. [2] T. Tsuzuku, in: Proc. 3rd Conf. on Carbon, Buffalo (Pergamon, New York, 1957) p. 433. [31S. lijima, Nature 56 (1991) 354. [4] W. Bollmann, Phil. Mag. 16 (1967) 363, 383. [5] F.C. Frank, Discussions Faraday Soc. 5 (1949) 48. [6] J. Friedel, in: Dislocations in Solids, Vol. 1, Ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1979) p. 13. [7] R.C. Bond in: Dislocations in Solids, Vol. 8, Ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1979) p. 8. [81R.F. Curl and R.E. Smalley, Sci. Am. 4 (1991) 32. [9] J.D. Eshelby, in: Dislocations in Solids, Vol. 1, Ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1979) p. 169.
~
CRYSTAL GROWTH
Conical, helically wound, graphite whiskers: a limiting member of the “fullerenes”? S. Amelinckx, W. Luyten, T. Krekels, G. Van Tendeloo and J. Van Landuyt University of Antwerp (RUCA), Groenenborgerlaan 171, B-2020 Antwerpen, Belgium Received 15 April 1992
Conically structured columnar graphite crystals have been studied by electron microscopy and electron diffraction and a formation mechanism is proposed explaining their shape and unusual diffraction effects consisting in periodically interrupted circular ring patterns. A growth mechanism is proposed whereby the initial graphite layer adopts a slitted dome shaped configuration by introducing a fivefold carbon ring in the sixfold carbon network. Successive sheets are rotated with respect to the previous one over a constant angle, thus realizing a helical cone around a “disclination” with a fivefold carbon ring core. This model explains the morphological features and the particular diffraction effects observed on these reproducibly prepared columnar graphite crystals. The growth mechanism has a direct relationship to the formation of the fullerene “bucky ball” molecules.
1. Introduction More than twenty years ago, Haanstra et al. [11 described needle-shaped graphite columns which were grown at temperatures above 1800°C on f3-silicon carbide substrates in a carbon-monoxide atmosphere. The columns apparently nucleated on multiply twinned crystals of a-SiC. The carbon columns were described by the authors as “being stacked from parallel layers of carbon atoms in a hexagonal network bent into a cone with top angle of about 141°”.This angle was supposed to have been inherited from the f3-SiC substrate. The successive layers were assumed to be “randomly rotated with respect to each other over angles being multiples of 60° or 30° divided by a prime number” [1]. Already much earlier, related observations had been made by Tsuzuku [2] who investigated the structure of conical crystals with a “lotus-leaflike” form grown on the surface of soot heated to 2500°C. In both cases interrupted or segmented diffraction rings were observed in the c * zone of electron diffraction patterns of thin sections of the columns. 0022-0248/92/$05.OO © 1992
—
A detailed analysis of the formation mechanism and a consistent interpretation of the diffraction effects are still lacking. From the descriptions and reproduction of the diffraction patterns given in the two references [1] and [21we can conclude that the crystals must have had similar microtextures. Since in ref. [21 the growth conditions were not unlike those under which “fullerenes” could be formed, we considered it worthwhile to re-examine some of the crystals obtained more than twenty years ago from Dr. Knippenberg, one of the authors of ref. [11. The columnar crystals used in our experiments were prepared on SiC substrates, using the method described in detail therein [1]. Recently, microtubules of graphite have been described by lijima [31, but these have a very different microstructure, the sheets forming a scroll-type arrangement.
2. Morphological observations 2.1. Scanning electron microscopy The “crystals” were examined “as-received”, i.e. without an~’treatment, in a scanning electron
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Conical, helically wound, graphite whiskers: limiting member of ‘fullerenes”?
microscope which revealed well-formed rounded needle-shaped “crystals” with cross sections of the order of 3 ~m, sometimes with flat end surfaces, sometimes ending in a “sharp” point (fig. la). When broken, one end of the cylindrical crystal fragments always exhibited a shallow corncal depression and the other end a rather flat conical hill, the semi-apex angle of the two features always being complementary (fig. ib). It is
clear that the cylindrically shaped crystal fragments had been formed by cleavage along the graphite sheets and that the cleavage faces were not flat but had a conical shape (fig. ic). .
.
2.2. Transmission electron microscopic observa
-
Thin cleavage flakes of such crystals obtained by crushing the needles were mounted on a car-
I—
2,5 ~.tm
Fig. 1. Scanning electron microscope images of conically wound carbon columns. (a) Crystal ending in a sharp tip. (b) Crystals cleaved in cylindrical segments. (c) Cylindrical part of a column: note the shape of the two end faces.
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bon coated holey film and examined in transmission electron microscopy. Fig. 2a shows such a quasi-circular, pancake-shaped flake. It clearly reveals the shape of the cross section of the column. Such thin flakes usually exhibit some narrow strips of material lying on the crystal, and which are highly deformed, as judged from the density of inclination extinction contours, and
545
which always end abruptly at the periphery of the circular flake. Such features are presumably “wrinkles” in the foil resulting from flattening a conical thin sheet as a result of crushing. They may also be folded parts of the pancake. In ref. [2], double-folded crystals were shown to adopt the shape of a fan, confirming their conical shape. A similar observation is reproduced in fig. 2b;
I —I.
C
I
~
]_,.
JP~TUIIII~
~
1~.im
2~ixn
~
Fig. 2. Quasi-circular pancake-shaped thin cleavage section ot a columnar crystal. (a) Bright field image of a c-cleavage disk. (b) Dark field image of a cleavage disk exhibiting folds and wrinkles. (c) Bright field image of a column fragment revealing the conical geometry.
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Conical, helically wound, graphite whiskers: limiting member of ‘fullerenes”?
part of the disc was folded back on itself, revealing a fan-like periphery. In some cases also shadows of column fragments were formed which confirm the conical geometry revealed by the scanning electron microscopic observations (fig. 2c) and allow one to measure the semi-apex angle ~
3. Description of the electron diffraction patterns 3.1. The c * zone pattern The c * zone diffraction patterns of such circular crystal fragments are quite remarkable. A number of examples are shown in fig. 3. The
100
-
fbi
100
1-ig. 3. Ditterent clectron diffraction patterns along the — zone of cleavage flakes ot conically s~oundgraphite columns. The rings correspond with hk0 reflections of graphite. The circles are divided into (a) 18 segments, (b) 30 segments, (c) 42 segments and (d) 126 segments.
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patterns mostly consist of “segmented” concentric circles with radii in accordance with the hk0 lattice spacings of graphite. The segments are almost uniformly spaced along the circles. Each “segment” or spot cluster is often further composed of a number of equidistant spots, the same number in almost all segments. Apparently the dotted fine structure of the segments remained unresolved or unnoticed in previous work [1,21.
547
The angular range of the segments is constant along the different circles, their length thus increases proportionally to the radii. Occasionally the dots are so closely spaced that the segments look continuous (fig. 3b). Moreover, sometimes the segments are joined, head to tail, leading to continuous circles, often with periodic reinforcements in brightness (fig. 3c). In the latter case the pattern is identical to a textured powder pattern
004
:;~
002
S Fig. 4. Diffraction patterns along zones different from c*. (a) Diffraction pattern along a zone which is practically perpendicular to the c* direction; note the elongated shape of the 001 reflections. (b) Fine structure of high order 00! spots; the spot splitting is indicated by the arrows.
548
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Conical, helically wound, graphite whiskers: limiting member of “fullerenes”?
consisting of cleavage flakes randomly rotated around the c axis normal to the plane of the flakes, so as to exhibit only hkO reflections. The positions of the segments in one circle are roughly staggered in radial directions with respect to those in the second circle of the pattern. The length of the segments depends on the sample, and their angular separation is constant for seetions made from the same column, but it is usu ally different for different crystals. It is worth noting that in ref. [21the segmented nature of the diffraction rings was attributed to the “polygonization” of a conical sheet, by the formation of radially oriented edge dislocation walls. The number of segments or spots on a circle is extremely variable; observed numbers are 18, 30, 42, 66, 126, 222.
/
_______________________________
-_____________
--
//
~‘\
000 /1
~o~u ,‘~\
I
Fig. 5. Schematic representation of the conical surfaces representing the loci of diffuse intensity in reciprocal space.
Fig. 6. Oblique sections ot the diffuse scattering distributions at various tilt angles. The tilt axis is represented by the straight line in (c) . Note the ellipse-shaped loci which are the intersections of Ewald’s sphere with the surfaces represented in fig. 5.
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3.2. Patterns along zones normal to c * A typical diffraction pattern along a zone normal to the c* axis is shown in fig. 4a. Only one row of 001 spots is present; they are elongated along a direction normal to the c* axis. The elongation of the spots increases with increasing 1, although this is not immediately obvious from a photograph because of the simultaneously decreasing overall intensity of the spots with increasing I. The high order spots are sometimes visibly split into two components of which the separation increases with increasing 1, as shown in fig. 4b. The other reflections have all degenerated into continuous streaks, which broaden with increasing 1, the width being a minimum for l = 0. This type of diffraction pattern is consistent with a reciprocal lattice in which the lattice points of the 00/ row have been transformed into ring shaped regions situated at the levels of 001 reflections, on a double cone with its apex in the origin of reciprocal space and with a semi-apex angle of about 20°.The other regions of diffuse scattering form complicated 3D distributions in reciprocal space. They result from the fact that each diffraction spot along the circles is the apex of a double cone with the same semi-apex angle as for the 001 reflections. These cones overlap and the final result is a region of diffuse intensity limited by the conical surfaces represented in fig. 5. This shape of the diffuse intensity distribution is consistent with diffraction patterns such as fig. 4a and figs. 6a, 6b and 6c. For small tilt angles of the specimen the spot pattern along the circles does not change. For larger tilt angles ellips shaped loci of diffuse intensity are formed (figs. 6a—6c).
4. Diffraction contrast imaging One or a few segments of the diffraction rings can be selected to produce a dark field image. It turns out that the whole circular sample is imaged, showing that the whole flake contributes to all diffraction segments. The image consists of a bright—dark line pattern, which is roughly perpendicular to the direction of the curved clusters
549
of spots used to make the DF image (fig. 2b). Diffraction patterns made from different areas of the circular flake produce the same diffraction pattern, but not necessarily in the same orientation, showing that the complete set of differently oriented sheets is present in each part of the sample.
5. Model for the growth of conically wound columnar crystals 5.1. Growth mechanism All observations appear to be consistent with a model in which the needle-shaped “crystal” consists of a helical surface wound on a rather “flat” cone. We can form the “crystal” by the following imaginary operations. In a sheet of graphite we make a cut, for instance by following the sides of the hexagons and ending somewhere in the middle of the sheets. Along this cut we lift one part of the sheet with respect to the other part over one sheet thickness, and slide this part over the other one in such a way as to achieve in the sector of overlap the normal stacking of graphite, at least for a certain fraction of the hexagons. We construct in fact in the overlap part a “coincidence site” lattice [41of two graphite sheets rotated over a small angle. This can of course only be achieved by deforming the planar sheet into a cone with its apex in the endpoint of the cut. The angle of the sector of overlap depends on the density of the coincidence site lattice (i.e. on the .~-index)which in turn is related to the semi-apex angle of the cone as will be shown below. Graphite sheets are known to grow almost exclusively along their edges; if subsequent growth takes place, a helical surface will thus be formed by growth along steps, somewhat like in spiral growth [5].However, growth now takes place on a helical cone and homologous crystallographic directions in successive sheets are no longer parallel but enclose a small constant angle /3, determined by the original wedge angle of the overlap area. Between two successive sheets a low-angle pure twist boundary is thus present. The defect along the core of the cone is now a “disclination”,
550
S. Amelinckx et a!.
/ Conical,
helically wound, graphite whiskers: limiting member of “fullerenes”?
R I
mal graphite, is a consequence of the presence of I
the dislocation grid forming the twist boundary. 5.2. Geometrical features of the model.
R0
Quantitative confirmation of principle possible by verifying between the semi-apex angle cP 7) and the angle 13 over which I
I
I
this model is in the relationship of the cone (fig. the two parts of
the sheet overlap. The angle cP can be measured either in scanning electron microscopic images or in shadow images in TEM, whereas the angle /3
a
Fig. 7. Schematic representation of the geometry of the conical crystal, illustrating the derivation of the relation between Pand/3.
rather than a “dislocation”. In the literature, disclinations characterized by a screw rotation operation are sometimes called “dispirations” (e.g. ref. [7]). The very easy cleavage along the helical surface, easier than the c-cleavage in nor-
can be derived from the diffraction pattern along the c * zone, as will be shown below. In practice it is unfortunately difficult to perform the two measurements on the same crystal. In fig. 8 we represent the flat circular sheet of graphite with radius R before introducing the “disclination”. We now remove an angular sector /3, and close the gap by making the point P coincide with 0 where a=Rf3.
(1)
On doing this we form the cone of fig. 7, having a circular base with radius R0. Expressing the circumference of the base of the cone in two different ways we obtain the relation: 2~rRo 2~R a. (2) =
—
b~ Fig. 8. Continuous model illustrating the formation of a conical helix. (a) A sector /3 is removed from a disc. (b) The angular gap is closed and the cone is formed.
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‘?
551
Overlap angle: /3 (deg)
Semi-apex angle: cP (deg)
Multiplicity: N = 360°//3
27.7 21.7 15.1
67.37 70.00 73.34
13 16.59 23.84
closing this angular gap, accompanied by sliding the two parts along a direction normal to the plane of the disk (fig. 8). Such a defect is called a “disclination” or a “dispiration” line [6,7]. Summarizing: Instead of growth around a screw dislocation normal to the c-planes, the model assumes that growth takes place around a “dis-
3458
clination”, angles of successive forming a sheets helical with cone.respect The rotation to an
Table 1 Coincidence site lattices angles
~
initial reference orientation are cumulative: each sheet is rotated over the same angle /3 with
Taking into account that sin j
=
R /R ~
(3) ‘
and using (1), we obtain the desired relation from (2) and (3): /3
=
2’ir(l
—
sin cP).
(4)
A theoretical angle /3 can be obtained from the rotation angles corresponding to possible coincidence site lattices, as listed in table 1. In a continuum approximation this operation consists in deforming a pancake-shaped disk, by removing a wedge shaped sector of material, followed by
-
respect to the previous one. It is clear that rightand left-handed conical helices are equally possible. The angle /3 is directly visualized in the double-folded fan-shaped cleavage foil of fig. 9; it is clear that the lacking sector is /3/2. In normal spiral growth around a dislocation line, the successive lattice planes have all the same orientation, exhibiting perhaps an elastic twist in sufficiently thin filaments [8]. In conical spiral growth, as proposed here, successive layers are rotated over a constant angle. Our model accounts in a natural way for the fact that the .
“~
j —s05 ____
Inn
Fig. 9. Bright field image of a double folded fan shaped cleavage foil. The indicated lacking sector corresponds with /3/2.
552
S. Amelinckx ci a!.
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Conical, helically wound, graphite whiskers: limiting member of “fullerenes”?
rotations between successive sheets are not random but systematic, a fact which remained unexplained in previous work.
(pQ+
5.3. Structural features of the model
Qo
For a detailed discussion of the diffraction effects, a detailed model of the texture is required. In our description of the model, we shall make use of a reference system of “helical polar coordinates”. These coordinates are the polar distance r and the azimuth angle ~P,varying from ~ to + ~, counted from the same reference direction SO. The coordinates are helical in the sense that a rotation over 2~raround S introduces a displacement over c/2 (c lattice parameter of graphite) parallel to the axis of the cone. The points P0(r0, 1P0) and P1(r~,1P1~ + 2ir) are thus located in successive sheets, separated by c/2, but they project along c in the same point. We now consider the structure in the vicinity of a column centred on the point P0(r0, 11’~), situated in the sheet n. We specify the orientation of this sheet by the angle 4~ enclosed by SP0 and some reference direction associated with the graphite structure in that sheet. A convenient reference direction is for instance one of the base directions of the hexagonal coincidence site lattice corresponding with the rotation angle /3, as defined above and shown in fig. 10. Along the line SP0, the angle 4~is constant for varying r. Along the same direction SP0, but in the next sheet n + 1, this reference direction will havereference rotated direction over an angle /3, 4~ i.e. +for 2ir the becomes /3. ~I’~ + In general, along the line SP,, situated vertically above SP 0 with azimuth ~P= + n2~,the reference direction will have become 4~ + n/3, and a vertical displacement over n(c/2) has taken place. Consider now first the simple case mj3 = 60° (m = small integer). The graphite texture in the column centred on P0 then becomes periodic along the normal to the sheet, with period m(c/2), since in the sheets n and n + m the coincidence site lattices become parallel. One can then define a structural unit, something like a “local” unit cell defined in the c-direction by the
(~
(p 0
s
o
—
=
Fig. 10. Reference system of helical polar coordinates.
parallel meshes of the coincidence site lattices in the sheets n and n + m and separated by m(c/2). The carbon atoms in the interleaving sheets form the filling of this structural unit. In a strip along the direction SP0, the texture contains a number of such “unit cells” in parallel orientation. Also along the local normal to the sheets there are a number of such parallel structural units, depending on the sample thickness. Instead of considering this structural unit as being located between the sheets n and n + m, we can consider just as well this structural unit as being located between the sheets n + 1 and n + m These twoover structural units arelocal then related by +a 1.rotation /3 about the c axis, accompanied by a displacement c/2 parallel to this axis, i.e. this structural unit has a non-crystallographic 6m-fold screw axis as a symmetry element. As viewed along the c-direction, the graphite texture has a 6m-fold rotation axis. If the angle /3 is not a simple fraction of 60° but still commensurate with 60°, we can always write /3N = k > 360° (5 ‘
where N is the multiplicity of the rotation axis as observed in the diffraction pattern (in the previ-
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ous case, N = 6m), and where k is a small integer. For instance for N = 30 and /3 in the range 20°—25°,one would have k = 2, leading to /3 = 24°. For N = 126 and the same range of /3 values, one would have k = 7 (/3 = 20°)or k = 8 (/3 = 22.85°). We now note that the rotation of the graphite sheet over an angle /3, when proceeding to the next sheet, must occur progressively with ‘P. If we consider the line SQ0 at ‘I’d + ~l’~ enclosing an angle ~W’0 with SP0, the graphite sheet in the point Q0, or its associated coincidence site latlice, will have rotated over an angle =
(/3/360°)z.t’P0
(6)
and will have been displaced vertically over =
(/3/360°)(c/2).
(7)
The same applies to the sheets n, n + 1,. . . , n + m,... As a result, the column of local unit cells, centred on QO, i.e. the whole texture will differ L14. in orientation. Again along the line SQ0 all structural units will be parallel. The whole conical texture can thus be considered as consisting of sectors limited by radial lines within which the “structures” are approximately periodic along a radial direction and along columns parallel to the local c direction, the translation symmetry being that of the coincidence site lattice and the rotation symmetry being N-fold. The limits of these sectors are not strictly defined since the orientation of the “unit cells” changes in fact continuously with the azimuth ‘I’. With a change of the azimuth LW’ corresponds a change in the orientation of the texture of L~4 given by (6). In case the angle /3 is incommensurate with 360°,there is strictly speaking no periodicity along the c axis and hence no rotation symmetry if the number of sheets remains finite. If the number of sheets becomes very large the rotation symmetry becomes D00.
6. Diffraction effects The electron diffraction effects and the related observations will be reported in detail in a separate paper. In the present paper we shall there-
553
fore limit ourselves to a brief description of those diffraction effects which are specific of the model and thus allow one to establish the latter unambiguously. A spot position on one of the circles can be characterized by its azimuth angle counted from an arbitrary origin. These azimuth angles can be obtained by applying a simple algorithm as we shall see. It is well known that the kinematical diffraction theory is adequate to describe the geometry of diffraction patterns, but it is not particularly successful in predicting the intensity of spots in electron diffraction patterns. Since in any given volume element successive graphite sheets are rotated one relative to the previous one about the local c-axis over a constant angle /3, the kinematical theory predicts that in the diffraction pattern of that volume element the spot positions can be described by the following relations. The azimuths of the spots in the first circle are given by ict~ =
(I—i) X 60°+n~f3,
(8)
where n3 is an integer. The index j refers to the six different starting points for sequences j = 1, 2, . .. , 6. The spots on the second circle are similarly represented by their azimuths
~
= (i 1) x 60°+ 30°+ n1/3. (9) Although these algorithms are very simple, the resulting patterns may become rather complex since they depend very sensitively on two main parameters: the angle /3 and the total number n of sheets in the sample. The pattern may become complex if /3 is not commensurate with 60°.None of these parameters follows directly and unambiguously from the observed diffraction pattern. Whereas it is simple to generate a pattern for given values of /3 and n, the reverse operation, i.e. deducing /3 and n from an observed pattern, is not trivial. Moreover, since the samples considered are not single crystals but well-defined regular textures, at least when undeformed by the preparation method of the sample, the orientation of the diffraction pattern is found to depend on the position of the selected area and even on the size of the selective aperture. Specimen de—
554
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helically wound, graphite whiskers: limiting member of “fullerenes”?
formation may change the shape of spots and segments, but will change neither the pseudosymmetry nor the multiplicity. Two diffraction patterns taken from two Selected areas both along the rim of the circular sample, but differing in azimuth by z.1’P, will be rotated one with respect to the other over an angle ~D given by ~cP = (/3/360°)zl’P. A statically detectable rotation of the diffraction pattern will thus result only if ~D differs from an integer multiple of the angular separation of the spots or
simple, ideal case where m/3 = 60°, the rotation over /3 is a 6rn-fold symmetry rotation in projection. This also implies that the angular range of the texture over the sample area of which the diffraction pattern is recorded depends on the aperture size. If the selector aperture collects an angular range of textures z~tO(~,the diffraction spots or segments will have an angular width
of the segments. The latter is given by 360°/N, where N is the multiplicity or quasi-rotation symmetry of the pattern, A dynamical experiment, consisting in displacing the selective aperture (or the sample), whilst observing the diffraction pattern, would result in a visible rotation of the diffraction pattern, often over several interspot distances, depending on the change in azimuth accompanying the displacement of the aperture. The latter effect can always be detected since it does not depend on the precise magnitude of /3 or of LW’, as does the static observation. As shown above in section 5.3 and as also implied in the relation (6), the local texture rotates continuously with the azimuth of the considered volume element, the rotation angle being /3 when the azimuth has changed by 360°. In the
The spot or segment widths will thus increase with increasing aperture size. However, this is a small effect since /3/360°is of the order of 0.06.
1i’P0
=
(/3/360°) ~i@O.
(10)
It will therefore only be observable for patterns with a small multiplicity, i.e. N < 30, and for which sharp spots are formed when using the smaller aperture. For segmented circles the effect remains unobservable in practice; in deformed samples it is wiped out completely.
7. Diffraction experiments and computer simulations The predicted features of the diffraction patterns have been verified and found to be in agreement with the experiments; they will be
Fig. 11. Magnified part of one of the segmented rings of the diffraction pattern represented in fig. 3d . The spacing between the spots in one segment differs from segment to segment, sometimes leading to (nearly) overlapping spots (arrowed).
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helically wound, graphite whiskers: limiting member of “fullerenes”?
555
discussed in detail in a forthcoming paper. We limit ourselves here to showing the excellent agreement between the two observed diffraction patterns, and the corresponding computer simulated patterns, based on the algorithm (9). The observed pattern of fig. 3d has a multiplicity of N 126. A magnified part of this pattern is shown in fig. 11. Although it is clear that every segment or spot cluster contains four spots, the positions of the spots in each segment differ from segment to segment. This is most probably due to the fact that the crystal was slightly deformed during crushing. The main features of this diffraction pattern, however, are correctly reproduced in the simulated pattern, using as parameters /3 22.842°and a number of sheets n 84 (fig. 12a). Clearly the spacing between the spots
and the number of sheets was n 76. Again the agreement is quite good. In these simulation experiments the finite aperture size effect was neglected; in addition we have to stress that the resulting patterns are only representative for ideal, undeformed textures, present in a small volume element, i.e. for the smallest aperture. The “dynamical” experiment described above was performed in the electron microscope on a number of patterns. Displacing the aperture along the rim of a specimen always resulted in a rotation of the diffraction pattern. A static experiment performed on the diffraction pattern of a specimen with N 30, i.e. with an interspot angular separation of 12°,leads to a rotation over 6° for a change in the azimuth M’1~of the aperture
in a cluster would remain constant in the case of an undeformed crystal, as can be judged from the magnified part of the simulated pattern of fig. 12b. The pattern of fig. 3b, on the other hand, consists of unresolved segments, the multiplicity being N 30. For the simulated pattern represented in fig. 12b, the value /3 23.9°was used
by 90°in agreement with the prediction for this particular case for which /3 24°and thus /3/4 6°.
=
=
=
=
=
a
~
.----..-
=
=
=
=
8. Discussion One important question has not yet been answered. Why does such a helical cone form at all?
b /
/ /
I
\
\
1~Th
~
~
/
/
I
Fig. 12. Simulated diffraction patterns for: (a) /3 = 22.842 and n = 84 (N = 126). The magnified part reveals the fine strucure of the pattern; in an undeformed crystal, all spots in a cluster are equally spaced. (b) /3 = 23.9 and n = 76 (N = 30); the magnified part clearly shows that the spots in a cluster are so closely spaced that the segments will look continuous.
556
5. Ame!inckx ci a!.
/ Conical,
helically wound, graphite whiskers: limiting member of “fullerenes”?
The authors in ref. [1] assume that the cone shape is inherited from the twinned SiC substrate. The formation mechanism proposed in ref. [2], on the other hand, is summarized in the following quote: “If a thin monocrystal layer is subjected to the rotational slip immediately after the mechanical bucklingproducing the central screw dislocation it may partly overlap by an angle 0, forming a cone”. However, a different answer is suggested by a recent proposal for the nucleation mechanism of fullerenes [7]. The non-planar character of the graphite nucleus finds its origin in the occurrence during the nucleation process of a five-ring of carbon atoms. If such a pentagon is surrounded symmetrically by edge sharing hexagons, a dome is formed which on the subsequent addition of the correct ratio of pentagons and hexagons grows into a closed cage such as C60 or C71~[7]. In the present case a single pentagon may be sufficient to form a twisted nucleus (fig. 13). On further lateral growth by the addition of hexagons exclusively, a symmetrical dome cannot grow further without leaving gaps. Instead a “slitted” cone forms, consisting of one helical
/ i-
.~-
~
7~
‘~
~.
~
~-
~--~
-~
I
/
-
-
1
Fig. 14. Coincidence site lattice of two superposed graphite sheets rotated over an angle of 2 arctan (1/31i). The unit of misfit is a hexagonal grid with a lattice parameter a = where a is the C—C distance in a six-ring.
sheet. The angular gap left along the pentagon in fig. 13 is theoretically 2 x 12° 24°. The two “lids” of the helix being at different levels, the sheet can now expand laterally by growing on top of itself with a small orientation difference in the vicinity of 24°,but somewhat smaller. Hereby the overlapping graphite sheets will realize the best fit compatible with the presence of one or perhaps a small number of pentagons. Hereby one or a small number of preferred rotation angles will occur, whereby the density of coincidence sites is a maximum. The coincidence site lattice which best fits the twist angle introduced by the presence of the pentagons is presented in fig. 14; it has a hexagonal unit mesh with a lattice parameter a~/~f(a C—C distance) and corresponds with a /3-value of 2 arctan(1/3V~) 21.8°. Because of the weak bonding between the successive graphite sheets, the tendency to realize this particular coincidence site lattice may not be very pronounced and in practice the value of the /3 angle may be somewhat variable. However all observations of semi-apex angles and of the fanlike double-folded sheets are consistent with a /3 =
/
/ .
/
/
--
/
/
Fig. 13. 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,
=
S. .4me!inckx ci a!.
/
Conical, helically wound, graphite whiskers: limiting member of “fullerenes”?
angle which does not differ very much from this ideal value. The “core” of the helical surface finally consists of vertically stacked pentagons. As pointed out above, the growth conditions of the helical cones as described in ref. [2] are not unlike those of the “fullerenes”, which makes the formation of a small concentration of pentagons a probable event, According to our model, the angle /3 is the rotation angle of a low energy low angle twist boundary, i.e. a boundary with the maximum density of coincidence sites taken into account the fact that the presence of the pentagons determines to a large extent the angular range. In fig. 14 we have represented an example for which the angle /3 is approximately 21° with .~ 7. The corresponding semi-apex angle of the cone is then 70°±2°,which is the only observed angle, as e.g. in fig. 8. =
9. Conclusions
A surprising feature of the model is the fact that the wide variety of multiplicities or pseudorotation symmetries of the diffraction pattern turns out to be compatible with a narrow range of /3 angles as derived from the observations and from theoretical considerations. The /3 angle was derived: (i) from the semi-apex angle of the conical cleavage faces such as fig. 2c; (ii) from the fan-like double folded cleavage sheets, such as fig. 9. Two independent theoretical considerations suggest f3 angles in the same narrow range: (i) the lacking angular segment in the fit between the singular pentagon and the adjacent hexagons as represented in fig. 13; (ii) the value of the rotation angle of the best coincidence site lattice, as represented in fig. 14. Even though the preparation methods in refs. [1] and [21 are rather different, the similarity between the diffraction patterns reproduced in refs. [1] and [2] strongly suggests that the same microstructure was observed in both cases. This also means that the specific form of /3-SiC which
557
was considered to be required as a substrate in ref. [1] is actually not essential, since a quite different procedure, approaching more the conditions under which “fullerenes” are formed, allows one to obtain the same type of conical crystals. It seems, however, that the use of a /3-SiC substrate provides a reproducible method to prepare this particular microstructure. The fact that heating soot at temperatures above 2500°C in a closed graphite crucible [2] allows one to produce conical graphite crystals as well makes our proposed microscopic mechanism plausible. Our macroscopic model allows a consistent and natural explanation for all the observed morphological characteristics and the geometrical features of the diffraction patterns and allows to link these patterns to the morphology of the “crystals”, which are in fact “textures”. The microscopic model is more speculative in the sense that we have no direct evidence for the occurrence of five-rings. However a consistent geometrical structure in accordance with the observations can be built on this assumption.
Acknowledgements We would like to thank Dr. W.F. Knippenberg for providing us with samples and references. Professor D. Van Dyck (University of Antwerp) and Dr. U. Dahmen (L.B.L. Berkeley, California) are gratefully acknowledged for useful discus. sions. The . Fund for Scientific Research (11KW) . . and the Ministry of Science Policy of the Belgian Government (IUAP Project 48) are gratefully acknowledged for financial support. .
.
Appendix. Diffraction patterns We call basic multiplicity N the number of sharp spots or of spot clusters (segments) along a circle. The angular separation of the spots (or segments) is then 360°/N.Note that N is always a 6-fold, which is a direct consequence of the 6-fold symmetry of the graphite sheets.
558
5. Amelinckx ci a!.
/ Conical,
helically wound, graphite whiskers: limiting member of “fullerenes”?
The angular range of /3 being restricted to the interval 20°—24°, as follows from the observations of the geometrical features, we have 3/3 60°+ c. (A.1) If e = 0, we have the simplest possible case, i.e. =
Table 2 Observed basic pseudo-periods (N = number of spots, p N/6, /3=(l+1/p)X20° and e=360°/N N p /3 (deg) e (deg) 30 42
5 7
66 126
11 21
24 226 7
N= 18, /3 with
=
20°. If
E’’O
( <<60°), we note that
21~ 20~ 21
=
360°/N,
=
60° (p
12 8~ 7
26 7
(A.2)
we obtain a strictly periodic N-fold pattern of isolated spots provided pe
=
=
integer).
(A.3)
The number of spots is then finite (i.e. N) even for an infinite number of sheets. The angle /3 follows from (A.1) and (A.3), /3 (1 + l/P) x 200, and the value of p follows from (A.2) and (A.3), p N/6. All observed basic pseudo-periods are obtained in this way, as shown in table 2. The equidistant distribution of spots in clusters, exhibiting the same pseudo-periodic arrangements with the same multiplicities, is due to small deviations from the /3-values indicated here. The multiplicities observed in ref. [1], N 102, 132 and 222, correspond to p 17, 22 and 37, respectively. =
=
=
=
References [1] H.B. Haanstra, W.F. Knippenberg and G. Verspui, J. Crystal Growth 16 (1972) 71. [2] T. Tsuzuku, in: Proc. 3rd Conf. on Carbon, Buffalo (Pergamon, New York, 1957) p. 433. [31S. lijima, Nature 56 (1991) 354. [4] W. Bollmann, Phil. Mag. 16 (1967) 363, 383. [5] F.C. Frank, Discussions Faraday Soc. 5 (1949) 48. [6] J. Friedel, in: Dislocations in Solids, Vol. 1, Ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1979) p. 13. [7] R.C. Bond in: Dislocations in Solids, Vol. 8, Ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1979) p. 8. [81R.F. Curl and R.E. Smalley, Sci. Am. 4 (1991) 32. [9] J.D. Eshelby, in: Dislocations in Solids, Vol. 1, Ed. F.R.N. Nabarro (North-Holland, Amsterdam, 1979) p. 169.