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THE GEOMETRY OF HYPOTHETICAL CURVED GRAPHITE STRUCTURES
THE GEOMETRY OF HYPOTHETICAL CURVED GRAPHITE STRUCTURES H . T E R R O N E S a n d A. L . M A C K A Y Department of Crystallography, Birkbeck College, (University of London), Malet Street, London W C 1 E 7 H X , U.K. (Accepted
2S April
1992)
Abstract—Assuming that rings of 5 , 7 , and 8 carbon atoms are allowable (as well as hexagons) in graphitic sheets, a great variety of finite and infinite structures can be postulated. Besides the shells of the fullerenes, which have positive Gaussian curvature, and the cylinders which have zero Gaussian curvature, infinite periodic networks with negative Gaussian curvature are possible. The latter promise to have lower energies than the convex fullerenes. Key Words—Fullerenes, ring structures, cylindrical structures. "The human mind has first to construct forms, indepen dently, before we can find them in things." Albert Einstein. "Take Carbon for example then / What shapely towers it constructs / T o house the hopes of men!" A. M. Sullivan (1896-).
given by a(h + hk + / c ) , where h and k are the steps along the two hexagonal axes (which are here taken to be 60° apart). Starting from one lattice point a hexagonal super-lattice of side a(h + hk + / c ) can be marked out where all its points lie on points of the original lattice. Each cell of the larger lattice will contain (h + hk + / c ) = Γ of the smaller cells. This sequence (h + hk + k ) = Γ runs 1, 3, 4, 7, 9, 12, 1 3 , . . . This kind of tessellation has long been known in mineralogy where a fraction of the atoms in a hex agonal lattice may be vacant or substituted by other types of atom. The vacancies or substituting atoms are arranged symmetrically as far apart from each other as possible. If h > k and k Φ 0 then the super-lattice is unsymmetrically disposed with respect to the original lat tice. If the lattice is turned over and superimposed on itself so that the super-lattice points coincide, then we have a coincidence site lattice (in which a fraction \/(h + hk + k ) of the original lattice points coin cide). Coincidence site lattices can also be found in three dimensions, particularly for cubic lattices. Networks of the graphite type can be twinned when the super-lattice of one part coincides in direc tion with the super-lattice of the other component. The boundary involves the appearance of rings of 5 and 7 in equal numbers as is shown in Fig. 1. It is not certain whether, in fact, such twin boundaries are ob served in graphite, but they appear in the correspond ing grain boundaries in metals[3] and they may be in volved in the formation of the cones observed by Iijima[4]. 2
2
The characteristics of the process of X-ray crystal structure analysis have led to an undue emphasis on classically crystalline materials to the neglect of or ganised structures which do not give diffraction pat terns with sharp spots. Gradually, even in the inorganic field, curved lay ers have become recognised as essential structural components. These were first recognised in asbestos and halloysite[l,2], where concentric cylinders and spiral windings of silicate sheets were disclosed. We can now begin to assemble the basic geometry of such curved structures under the rubric of "flexi-crysialo graphy." We will assume here that we are discussing graph ite layers, but most of the geometry applies to other layers, such as silicate sheets, with hexagonal or square or lower symmetry. 2. P L A N E S H E E T S
The structural components which we will chiefly consider here are the hexagonal sheets of three-con nected carbon atoms found in graphite. In graphite itself these sheets are stacked in hexagonal sequences (repeating every two sheets) or rhombohedrally (re peating every three sheets) or in disordered combi nations. The dimensions of the hexagonal graphite structure are: a = 2.47 À, c = 6.79 À, so that the C C distance is about 1.42 À. 3. TESSELLATIONS
In a planar hexagonal lattice of lattice constant a, the distances from one lattice point to another are
7
2
2
2
2
2
1
7
2
2
2
1. I N T R O D U C T I O N
1
2
2
4 . REGULAR A N D SEMI-REGULAR POLYHEDRA
A regular polygon is a planar polygon with all its sides of equal length, all its inter-edge angles equal and all its vertices symmetrically equivalent. If stel lated, edges may intersect each other. For example the pentagram is the stellation of the pentagon in which, tracing the edges round the centre, more than one circuit is necessary to return to the starting point. 113
H. T E R R O N E S and A . L . M A C K A Y
114
Fig. 2. The truncated icosahedron, the shape of the C o mol ecule. 6
semiregular polyhedra with which we will not be con cerned here. 5. E U L E R ' S L A W
Fig. 1. The h = 2, k = 1 twin boundary in a planar graph ite-type network.
The regular polygons of order 5, 8, 10, and 12 have each only one stellation, namely {5/2}, {8/3}, {10/3}, and {12/5}. Other orders, such as 7, have more than one stellation. It is probably no coincidence that quasi-crystals may have symmetry axes of orders 5, 8, 10, and 12 as compared with the axes of order 2, 3, 4, and 6 (for which the corresponding polygons have no stellations) allowable in real crystals. Stellation might be considered as a first step in the generalisation of the concept of axis of symmetry. The 5 regular polyhedra (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron) each have faces which are all the same regular polygon and all their vertices symmetrically equivalent. The 13 semi-regular or Archimedean solids are convex poly hedra which have all their faces regular polygons of two or more kinds and all their vertices symmetri cally equivalent. They can be designated by the num bers of faces meeting at a vertex. For example the truncated icosahedron of C is 5.6 (Fig. 2). This fig ure is also obtained by deep truncation of the dodec ahedron. If we start with a truncated icosahedron then if the hexagonal faces are extended, we obtain an icosahedron and if the pentagonal faces are extended, then we obtain a dodecahedron. It is conventional to exclude the prisms N.4 from this definition. There are also a large number (53) of stellated regular and 60
2
For a convex polyhedron, topologically like a sphere, with F faces, V vertices, and Ε edges, Euler's k sphere has genus law states that V—E+F=2. zero and if another more complex polyhedron can be deformed to take the shape of a sphere with Ν han dles, then it is said to have the genus N. This is a useful but not a complete characterisation. Knot theory, which is still developing, is needed for a better classi fication. A torus has the genus 1 and the P-surface in side a cubic unit cell (see below) has the genus 3. The imposition of periodic boundary conditions is equiv alent to putting three handles across outside the cube connecting opposite faces. The more general expres sion of Euler's Law in three dimensions is V — Ε + F = χ, where χ is the Euler characteristic and χ = 2 — 2g, where g is the genus. Thus, for a torus, V — Ε + F = 0, and for a cell of the ^-surface, V — Ε + F = -4. For a network of the graphite type, assuming that there are only pentagons, hexagons, heptagons, or oc tagons, each edge is shared by two polygonal faces and each vertex is shared by three polygons. If N is the number of polygons with η sides and η vertices, we have: n
F= N + N + N + A , 2E = 3V = 5N -F 6A + 7A + 8A . 5
6
5
2
7
8
6
V
8
Putting this into Euler's expression we have N - N - 2N = 6χ = 12(1 - g) 5
7
S
so that for a torus N — N-, — 2N = 0. For a sphere N — Ν — 2N% = 12 and thus in such a figure, if 5
5
Ί
8
Geometry of hypothetical curved graphite structures
there are no heptagons or octagons, there can only be hexagons, in an indefinite number and 12 pentagons. Since V = 20 + 2jV , the number of vertices must be even. For the unit cell of the P-surface N — N — 2N = —24. Thus, if there are no pentagons, 24 Ν or 12 7V are necessary. 6
5
7
Ί
8
6. I N F I N I T E P O L Y H E D R A
The concept of polyhedra can be extended to in clude infinitely periodic regular and semiregular pol yhedra by allowing nonconvex arrangements. Many of these have been discussed by Wells[5]. One ex ample is shown in Fig. 3, which is made up of hexa gons and squares and where all vertices are equiva lent. Each vertex can be given the symbol 6.4 meaning that in going round a vertex we meet a hex agon and three squares. It might be noted that this polyhedron divides all space into two congruent regions. It will be seen below that this semiregular polyhedron is of importance in its relationship to pe riodic minimal surfaces serving as a conceptual ref erent. It is a polygonal version of the .P-surface. It should be noted that for the convex semiregular polyhedra, which have genus zero, the sum of the face angles meeting at vertex adds up to less than 360° and that, if there are Ν vertices the Ν deficits add to 720°. The vertex sum divided into 720° is an integer or, for a stellation, a fraction. For the infinite semiregular polyhedra the vertex sums are greater than 360°, depending on the genus g, the total excess being 4n(g — 1 ). If the polyhedron is stellated the genus may be fractional, but we are not concerned with this case. We may ask what combinations of three faces (pentagons, hexagons, heptagons, etc.) can meet at a vertex under the condition that all vertices should be symmetrically equivalent. It can readily be seen that, 3
S
115
if any of the faces has an odd number of sides, then the other two polygons must be the same (as two dif ferent polygons cannot alternate around an odd axis). It is more difficult to enumerate vertices at which four or more faces meet but, since we are considering graphite sheets, this is not necessary. We may consider the tessellations in two ways, ei ther as packings of pentagons, hexagons, etc. meeting three at a vertex, or as the repetitions of asymmetric triangular units. For example, the regular tessella tions by heptagons with the symbol 7 (for three hep tagons meeting at a vertex) is equivalent to the tessel lation by triangles 14.6.4 of angles π / 7 , π / 3 , π / 2 . By enumerating all the possibilities we find that the com binations 8.6.6, 8.8.5, 7 . 7 . 7 , 7.6.6 have vertex sums a little above 360°. To make a surface of genus 3 (as for the P, D, and G surfaces) the excess vertex sum of the vertices within the primitive unit cell should be 8 π . This is given by 56 vertices of the type 7 . 7 . 7 . This is the sur face from which Vanderbilt and TersoffI6] began to develop their model. If this tessellation is now trun cated, that is, a hexagon is placed at each vertex, the tessellation becomes 7.6.6 (with all the vertices equiv alent to each other). To make up 8 π , 168 vertice are needed. This number gives a more relaxed structure than using heptagons alone. The process is exactly parallel to the truncation of the regular dodecahe dron, with vertices 5.5.5, to give the truncated icosa hedron 5.6.6 of C . Lenosky et al.[7] have somewhat different com binations where 216 vertices of the type 7.6.6 and 48 of the type 6.6.6 add to give the vertex excess of 8 π . In our case[8] 144 points of the type 8.6.6 and 4 8 of the type 6.6.6 make up the 192 vertices per cell. In each case any number of 6.6.6 points may occur with out changing the vertex excess. We might consider points of type 8.8.5 of which 80 per cell of genus 3 would be needed plus any number of 6.6.6 points. The polyhedron corresponding to the Neovius surface (C(P)) has the same arrangement of points as that of the surface 6.4 discussed already, but the spaces between the points are differently filled with polygons so that each of the 48 points per cubic cell has the configuration 8.4.8.6 and this leads to a sur face of genus 9. 3
6 0
3
7. D E L T A H E D R A
Fig. 3. The infinite semiregular polyhedron 6 . 4 which is a polygonal version of the Ρ periodic minimal surface. 2
A deltahedron is a polyhedron, each of the faces of which is an equilateral triangle. Deltahedra include the tetrahedron, the octahedron, the icosahedron and many less regular figures, such as the pentagonal bipyramid, the 4m dodecahedron, etc. Given a two-di mensional hexagonal lattice, by cutting away one, two or three sectors around a hexagonal point, it can be folded to cover a deltahedron. Thus, by using the points of a larger super-lattice, a pattern of hexagonal symmetry can be mapped on to the surface of a del tahedron, so that Τ units lie on every face. The most important case is that of the icosahedron. Models of
116
H . T E R R O N E S and A . L . M A C K A Y
angle is 6° and that 120 X 6° is 720° which is 4ττ. If there are no mirror planes the asymmetric unit can be taken as a triangle with five-, three- and twofold axes at its vertices of which there will be 60 to make up the whole surface.
9.
CURVATURE
The curvature of a surface at a point can be ex pressed by fitting two circles, with their planes includ ing the normal to the surface and at right angles to each other. The curvatures are the reciprocals of the radii of curvature. When both curvatures are either maxima or minima then they are principal curva tures K and K . If the surface is convex at the point in question then the curvatures are both positive. If the point on the surface is a saddle point (is anti-clas tic), then the curvatures have oppposite sign (Fig. 5c). The mean curvature is defined as Η = (K + K )/2 and the Gaussian curvature as Κ = K K . Thus, for a convex surface Κ is positive and for a saddle-surface negative. For a spherical shell Η = 1/r and Κ = l/r . Η has the dimensions L~ and AT the dimensions L . On a surface with Κ > 0, such as a closed convex shell, the perimeter of a small circle of radius r is a little less than 2wr and the area of this circle is a little less than πι . The perimeter of the circle is 2irr{\ — Kr /! + Oir )). x
2
{
X
2
2
2
1
2
2
2
Fig. 4. Albrecht Durer's drawing of the construction of the icosahedron truncum by folding up a sheet of cardboard.
the convex fullerenes can be constructed in this way. Albrecht Durer seems to have invented the construc tion of polyhedra by folding up cardboard and Fig. 4 shows his drawing of the icosahedron truncum. D'Arcy Thompson[9] generalised the folding method and showed that all the convex semiregular solids could be constructed by folding plane tilings.
4
10. F U L L E R E N E S
Fullerenes are symmetrical closed convex graph ite shells for which Κ > 0 consisting of 12 pentagons and various numbers of hexagons. They have been il lustrated recently by Smalley and Curl [10] and by many others so that it is not necessary to discuss them further here. Being topologically equivalent to the sphere they have genus 0 and faces + vertices = edges + 2. Further, N - N - 2N = 12. 5
7
S
8. ICOSAHEDRAL TESSELLATIONS
The regular icosahedron has the symmetry 5m3m (I ) or 532 (/) according to whether or not there are mirror planes of symmetry. If there are mir ror planes the order of the group is 120 and if not, then 60. This means that 120 asymmetric units can be identically situated, if mirror images of the units are considered to be identical with the originals, or if not then 60 asymmetric units can have identical sur roundings. Asymmetry is usual case for the units of viruses which frequently form icosahedral shells. The asymmetric region of a spherical shell with icosahe dral symmetry is a spherical triangle with axes of order 5, 3, and 2 at its vertices, with mirror planes forming its sides and thus with angles of 36°, 60°, and 90°, respectively. The whole surface comprises 120 such triangles. We might note that the angular excess over 180° for the sum of the interior angles of the tri h
Fig. 5. Different types of curvature, (a) Positive Gaussian curvature, (b) Negative Gaussian curvature, (c) Zero Gaus sian curvature.
Geometry of hypothetical curved graphite structures
117
11. C Y L I N D R I C A L L A T T I C E S
For a circular cylinder of radius r, K = 1/r, K = 0 so that H = Vir and Κ = 0. Other cross sec tions may be considered as distortions of circles. A lattice point is marked at one identifiable point in the pattern and all identical points are similarly marked (identical meaning also identical in orientation). Cy lindrical lattices can readily be handled by taking a cylindrical projection (Fig. 6) where the surface is un rolled to give a plane sheet of width lirr. There are two kinds of cylindrical pattern, rational and irratio nal. In rational lattices further lattice points lie ex actly above others with a displacement parallel to the axis of the cylinder. With irrational lattices a second lattice point never occurs exactly above the first. Dif fraction from helices of both types has been devel oped by Cochran et al.[\\]. The dense packings of equal spheres around a cyl inder have been examined by Erickson[12] who de rived formulae for their generation. Some are shown in Fig. 7. These can be used to produce the corre{
2
Fig. 6. Cylindrical projection. This shows the 3 - 8 irrational helix which Iijima observed by electron diffraction from very small cylinders.
Fig. 7. Tubular arrangements of close-packed equal spheres (Erickson). A hexagonal mesh can be ob tained simply by omitting one third of the spheres. (Reproduced from Science by kind permission.)
H. T E R R O N E S and A . L. M A C K A Y
Fig. 8. The P-surface.
sponding graphite nets by omitting the sphere at the centre of rings of six. Iijima[4] has observed hollow cylinders of graph ite by high resolution electron microscopy and by electron diffraction has demonstrated the orientation of the lattice with respect to the axes of the cylinders. Tubes consist of 5 or more layers, probably separate tubes but possibly spirals. The minimum internal di ameter found was about 2 nm. Only hexagons are needed for cylinders but in order to make a cone a rolled sheet must be joined by a seam which contains a row of dislocations as in Fig. 1 and to join a cone to larger and smaller cylinders requires ring of 5 and 7 (or 8), respectively. 12.
PERIODIC MINIMAL
SURFACES
Minimal surfaces are surfaces with H = 0 so that K = —K and Κ < 0. They are thus saddle-shaped (anti-clastic) or flat everywhere. Surfaces may be minimal either because, as for a soap-film spanning a {
2
nonplanar loop of wire, it minimises its energy by having a minimum of area or, for a membrane sur face made of lipid molecules, because it minimises the splay energy. The mathematical condition for a surface to have zero mean curvature is that the diver gence of its unit normal should be zero. The splay en ergy is the integral of H over the area. It is not possible to construct an infinite surface with constant negative Gaussian curvature. Such a surface with a constant, imaginary radius of curva ture defines the hyperbolic plane. However, it was found by H. A. Schwarz[13] that patches of negative Gaussian curvature and Η - 0 could be smoothly joined to give an infinite triply pe riodic surface of zero mean curvature. Typically a patch might be a triangle everywhere saddle-shaped and thus with angles which add to less than 180°. These triangles are repeated by diad axes. The Fig. 8 shows the P-surface. Exact procedures for determining the shapes of the periodic minimal surfaces are available and have 2
Table 1. Images of the normals of the atomic positions onto the complex plane for the D, G and F T P M S (u, and υ!) i
u,
0 1 2 3 4 5 6
0 0.04 0.149 0.41854 0.49365 0.1 0.245
0 0.493656 0.41854 0.149 0.04 0.245 0.11
Ki
η
x
y
-0.0783 -0.0109 -0.0395 -0.0395 -0.0109 -0.0598 -0.0598
±3.573 ±9.573 ±5.031 ±5.031 ±9.573 ±4.089 ±4.089
0 0.7088 1.4163 2.8419 3.5494 0.7673 1.7288
0 -3.5494 -2.8419 -1.4163 -0.7088 - 1.7288 -0.7673
0 -2.0 -1.0 1.0 2.0 -0.3533 0.3533
Gaussian Curvature Kj (in  ) . Radii of curvature r, (in A). Coordinates in real space for the D patch (in A). 2
Geometry of hypothetical curved graphite structures
119
Table 2. Properties of D, G, and Ρ T P M S made with graphite, a is the lattice parameter in A, for a C - C distance of 1.42 Â Surface a N
c
Sc Symmetry
D surface
G surface
Ρ surface
24.09 192 768 3 9 _ Prihm — Fdhm
18.98 192 384 3 5 7a3
15.41 192 192 3 3 Im3m — Pm3m
Ν ρ is the number of atoms per primitive rhombohedral cell. N is the number of atoms per cubic unit cell. g is the genus of the primitive cell. g the genus of the cubic unit cell. c
p
c
been described elsewhere[14]. They involve repre senting each point on the surface by a point in the complex plane. The coordinates x , y, ζ of a point / in the surface are related to this complex number w,- + Wj by the Weierstrass integrals which can be com puted. The Z>, P, and G surfaces are closely related to each other and the coordinates of one are obtained from those of another by multiplying the complex number by exp(/0), where θ is the Bonnet angle; θ = 0, θ = 38.0147°, θ = 90° for the D, G, and Ρ patches, respectively. In Table 1, atomic coordinates in complex and real spaces for the D patch are given. From these po sitions the whole D surface decorated with graphite can be obtained. The Gaussian curvature and the radii of curvature are also given. It is interesting to note that the magnitude of the maximum value of the Gaussian curvature (0.0783 A~ ) is less than the Gaussian curvature of C which is 0.08076 A . Therefore, C is more curved than the D, G, and Ρ minimal surfaces decorated with graphite; this can 2
60
60
2
have important implications when considering the stability of curved graphite since the elastic energy in volves the Gaussian and mean curvatures. Table 2 shows some properties of the exact TPMS decorated with graphite. Surfaces can also be found by finite element anal ysis methods. These methods look quite promising since they are suitable for generating surfaces which minimize energy functionals. 1 3 . GRAPHITE STRUCTURES WITH NEGATIVE GAUSSIAN CURVATURE
Many of the periodic minimal surfaces can be dec orated with graphite-type networks. We have suc cessfully built models of the P, D, (7, and Η surfaces (using three-way joints and connector tubes)[8] (Figs. 9 and 10.). Triangular patches of various sizes of the graphite network with angles 90°, 30°, and 45° (Fig. 9) can be joined so that the 45° vertices combined to give rings of eight carbon atoms. In the actual structures
Fig. 9. Possible graphite patches for triply periodic minimal surfaces decorated with graphite, (a) Ρ surface; (b) D surface; (c) G surface. The angles of the curved triangle are 90°, 30°, and 45° not 60°.
120
H . T E R R O N E S and A . L. M A C K A Y
(a)
(b) Fig. 11. Possible graphite structures based on surfaces par allel to periodic minimal surfaces curved by the introduc tion of heptagons, (a) PI surface; (b) Dl surface. (Coordi nates provided by T h o m a s Lenosky.)
Fig. 10. Possible graphite structures based on periodic min imal surfaces which are curved by the introduction of octa gons, (a) Ρ surface; (b) D surface; (c) G surface.
these sit appropriately on the saddle points of highest curvature (the highest curvature is at the centres of the octagons). Given the basic patch, the asymmetric unit of pattern, this can be twisted to give either the P, the D, or the G surfaces. It is possible to begin by embedding C atoms in the exactly determined surfaces but, since the struc tures are to be refined, approximate positions mea sured from physical models are quite adequate. Moreover, Lenosky's and Vanderbilt and TersofFs structures are parallel surfaces, where the two subspaces are unequal, and not the more symme trical minimal surfaces.
Thomas Lenosky[7] has built surfaces which are parallel to the Ρ and D periodic minimal surfaces and divide space into two unequal regions (and are thus less symmetrical than the periodic minimal surfaces. (See Fig. 11.) He has refined the atomic positions and has calculated the physical properties of the struc tures. These patches contain heptagons rather than octagons to introduce the negative Gaussian curva ture. Such structures are found, by calculation, to be much more energetically favourable than Buckminsterfullerenes. Lenosky, Gonze, Teter, and Elser pro pose the name Schwarzites, in memory of H. A. Schwarz, the mathematician who developed the first few of the periodic minimal surfaces, for this category of graphites with nonpositive curvatures. We are much indebted to Lenosky for calculating energies of our structures, those of Vanderbilt and Tersoff and his own on the same basis (Table 3), so that their stabilities can be compared with that of C . It will be seen that there are several such structures with energies less than that of C . Physical properties of cubic forms of carbon cal culated by Lenosky[7]. Dl and ΡΊ are the surfaces produced by himself, the D- and P-parallel surfaces 60
60
Geometry of hypothetical curved graphite structures
121
Table 3. Properties for different negatively curved graphite structures Structure
Ρ
AE
Β
a
a A
Ν
Schwarzite Dl Schwarzite PI Schwarzite P8 Schwarzite DT fee. C Diamond
1.15 1.02 1.16 1.28 1.71 3.52
0.18 0.20 0.19 0.22 0.42 0.02
9.4 7.5 10.3 11.5 1.4 44.3
17.39 11.4 10.47 15.42 10.0 2.31
24.7 16.2 14.9 21.9 14.12 3.5595
216 216 192 168 60 2
6 0
ρ is the density in g / c m . AE is the total energy relative to graphite in e V / a t o m . Β is the bulk modulus in units of 10' d y n e / c m . a is the cubic unit cell size in units of the bond length. a A is the cubic unit cell size in Angstroms. Ν is the number of C atoms per primitive unit cell (the f e e . cell contains four primitive cells). 3
1
2
incorporating heptagons. DT is the D- parallel sur face found by Vanderbilt and Tersoff. P8, D8, and G8 are the symmetrical P-, and G-surfaces incorpo rating octagons found by Mackay and Terrones. Tables 4-6, give the coordinates of the corre sponding structures. Vanderbilt and TersoffI6] have proposed a graph ite structure based on a surface parallel to the D-surface, so that the two subspaces are not equivalent (Fig. 12). A tetrahedral joint is built out of 84 atoms in hexagons and heptagons so that each point is a member of two hexagons and one heptagon (6 .7) and is topologically (but not metrically) equiv alent to every other. There are thus 2 X 84 atoms per primitive unit cell with space group Fd3 and 8 X 84 = 672 per cubic unit cell with a = 21.8 À (assuming graphite-type bonds). The fractional coordinates of the nonequivalent atoms are given in Table 7. The density is expected to be 1.29 g/cm . The au thors calculate the energy of formation to be 0.11 eV/ atom as compared with 0.67 eV/atom for C , but their calculation is not exactly the same as that of Lenosky.
Table 4. Fractional coordinates of the PS surface with ori gin at the centre of the cubic cell (Im3m) Atom
X
y
ζ
1 2 3
0.286 0.314 0.387
0.173 0.0908 0.0494
0.286 0.314 0.2729
Table 5. Fractional coordinates of the G 8 surface^vith ori gin at one of the corners of the cubic cell (Ia3d)
2
Atom
x
y
ζ
1 2 3 4
0.1903 0.1348 -0.0844 -0.0279
0.2117 0.1703 0.2014 0.1520
-0.2265 -0.1988 -0.1511 -0.1217
3
60
Fig. 12. Structure proposed by Vanderbilt and Tersoff (DT).
Table 6. Fractional coordinates of the D 8 surface with origin at one of the corners of the cubic cell (Fdlm) Atom
X
y
ζ
1 2 3 4 5 6
-0.0416 -0.0831 -0.1668 -0.2083 -0.0967 -0.1532
0.0 0.0 0.0 0.0 0.05172 0.05172
0.54163 0.58315 0.6668 0.7083 0.61032 0.63966
Table
7. Fractional
coordinates structure
of
Vanderbilt's
Dl'
Atom
X
y
ζ
1 2 3 4 5 6 7
0.1134 -0.0109 0.0272 -0.0984 -0.0662 -0.0743 -0.1915
0.0050 0.0303 0.0756 0.0826 0.1382 0.0402 0.0177
0.2415 0.2200 0.2435 0.1704 0.1542 0.2146 0.1649
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H . T E R R O N E S and A. L. M A C K A Y
tion of Penrose tilings and would bring the mean cur vature towards zero. When we attempt to execute the H tilings in real space the curvature causes the sur face to fold like seaweed (as fucus letuca) since more area is produced within a given radius than is appro priate for a planar tiling[16]. After a certain number of units have been added the surface can be closed on itself in various ways, periodically or irregularly. One of these ways is that found by Vanderbilt and Tersoff. 2
REFERENCES
Fig. 13. A tessellation of the hyperbolic plane by regular heptagons (7.3.2). The 7.6 tessellation is obtained by trun cation from this, a hexagon replacing each vertex. 2
14. T H E H Y P E R B O L I C PLANE—H
2
The hyperbolic plane (called T/ , the surface of the sphere being S ), has a constant negative Gaussian curvature. Regular and irregular tessellations of this surface by polygons can represent topologically the local connectivities of real space polygonal and con tinuous surfaces which have everywhere nonpositive curvatures. In particular, the regular tessellation 7.3.2 of H by heptagons (Fig. 13)[ 15] can be truncated to give the tessellation 7.6.6 which corresponds to Van derbilt and Tersoffs surface and is the least curved of the regular tessellations. Further truncations would increase the ratio of hexagons to heptagons, but all vertices would then be no longer equivalent to each other. This process corresponds to the recursive infla 2
2
2
1. E. J. W. Whittaker, Acta Cryst. 10, 149 (1957), and ear lier papers. 2. K. Yada, Acta Cryst. Ml, 6 5 9 - 6 6 4 (1971). 2. D . L. D . Caspar and A. Klug, Cold Spring Harbor Symp. Quant. Biol. 27, 1-24 (1962). 3. M. F. Ashby, F. Spaepen, and S. Williams, Acta Metallurgica 26, 1 6 4 7 - 1 6 6 3 ( 1978). 4. S. Iijima, Nature 3 5 4 , 5 6 - 5 8 (1991). 5. A. F. Wells, Three-dimensional Nets and Polyhedra. Wiley, N e w York (1977). 6. D . Vanderbilt and I. Tersoff, Phys. Rev. Lett. 68 (4), 51 1-513 (1992). 7. T. Lenosky, X. Gonze, M. Teter, and V. Elser, Nature 3 5 5 , 3 3 3 - 3 3 5 (1992). 8. A. L. Mackay and H . Terrones, Nature 3 5 2 , 7 6 2 ( 1991 ). 9. D'A. W. T h o m p s o n , Proc. Roy. Soc. A107, 181-188 (1925). 10. R. E. Smalley and R. F. Curl, Sci. Amer. 3 2 - 3 7 (Oct. 1991). 11. W. Cochran, F. H . C. Crick, and V. Vand, Acta Cryst. 5,581-586(1952). 12. R. O. Erickson (1973). Science 181 (1973). 13. H . A. Schwarz, Gesammelte Mathematische Abhandlunger, Vol. 1. Springer, Berlin (1890). 14. H . Terrones, Ph.D. Thesis, University of London (1992). 14. G. G. Tibbetts, M. G. Devour, and E. J. Rodda, Carbon 2 5 ( 3 ) , 3 6 7 - 3 7 5 (1987). 15. A. L. Mackay, Acta Cryst. A 4 2 , 5 5 - 5 6 (1986). 16. W. P. Thurston and J. R. Weeks, Sci. Amer. 251 (1), 94-106(1984).
2S April
1992)
Abstract—Assuming that rings of 5 , 7 , and 8 carbon atoms are allowable (as well as hexagons) in graphitic sheets, a great variety of finite and infinite structures can be postulated. Besides the shells of the fullerenes, which have positive Gaussian curvature, and the cylinders which have zero Gaussian curvature, infinite periodic networks with negative Gaussian curvature are possible. The latter promise to have lower energies than the convex fullerenes. Key Words—Fullerenes, ring structures, cylindrical structures. "The human mind has first to construct forms, indepen dently, before we can find them in things." Albert Einstein. "Take Carbon for example then / What shapely towers it constructs / T o house the hopes of men!" A. M. Sullivan (1896-).
given by a(h + hk + / c ) , where h and k are the steps along the two hexagonal axes (which are here taken to be 60° apart). Starting from one lattice point a hexagonal super-lattice of side a(h + hk + / c ) can be marked out where all its points lie on points of the original lattice. Each cell of the larger lattice will contain (h + hk + / c ) = Γ of the smaller cells. This sequence (h + hk + k ) = Γ runs 1, 3, 4, 7, 9, 12, 1 3 , . . . This kind of tessellation has long been known in mineralogy where a fraction of the atoms in a hex agonal lattice may be vacant or substituted by other types of atom. The vacancies or substituting atoms are arranged symmetrically as far apart from each other as possible. If h > k and k Φ 0 then the super-lattice is unsymmetrically disposed with respect to the original lat tice. If the lattice is turned over and superimposed on itself so that the super-lattice points coincide, then we have a coincidence site lattice (in which a fraction \/(h + hk + k ) of the original lattice points coin cide). Coincidence site lattices can also be found in three dimensions, particularly for cubic lattices. Networks of the graphite type can be twinned when the super-lattice of one part coincides in direc tion with the super-lattice of the other component. The boundary involves the appearance of rings of 5 and 7 in equal numbers as is shown in Fig. 1. It is not certain whether, in fact, such twin boundaries are ob served in graphite, but they appear in the correspond ing grain boundaries in metals[3] and they may be in volved in the formation of the cones observed by Iijima[4]. 2
2
The characteristics of the process of X-ray crystal structure analysis have led to an undue emphasis on classically crystalline materials to the neglect of or ganised structures which do not give diffraction pat terns with sharp spots. Gradually, even in the inorganic field, curved lay ers have become recognised as essential structural components. These were first recognised in asbestos and halloysite[l,2], where concentric cylinders and spiral windings of silicate sheets were disclosed. We can now begin to assemble the basic geometry of such curved structures under the rubric of "flexi-crysialo graphy." We will assume here that we are discussing graph ite layers, but most of the geometry applies to other layers, such as silicate sheets, with hexagonal or square or lower symmetry. 2. P L A N E S H E E T S
The structural components which we will chiefly consider here are the hexagonal sheets of three-con nected carbon atoms found in graphite. In graphite itself these sheets are stacked in hexagonal sequences (repeating every two sheets) or rhombohedrally (re peating every three sheets) or in disordered combi nations. The dimensions of the hexagonal graphite structure are: a = 2.47 À, c = 6.79 À, so that the C C distance is about 1.42 À. 3. TESSELLATIONS
In a planar hexagonal lattice of lattice constant a, the distances from one lattice point to another are
7
2
2
2
2
2
1
7
2
2
2
1. I N T R O D U C T I O N
1
2
2
4 . REGULAR A N D SEMI-REGULAR POLYHEDRA
A regular polygon is a planar polygon with all its sides of equal length, all its inter-edge angles equal and all its vertices symmetrically equivalent. If stel lated, edges may intersect each other. For example the pentagram is the stellation of the pentagon in which, tracing the edges round the centre, more than one circuit is necessary to return to the starting point. 113
H. T E R R O N E S and A . L . M A C K A Y
114
Fig. 2. The truncated icosahedron, the shape of the C o mol ecule. 6
semiregular polyhedra with which we will not be con cerned here. 5. E U L E R ' S L A W
Fig. 1. The h = 2, k = 1 twin boundary in a planar graph ite-type network.
The regular polygons of order 5, 8, 10, and 12 have each only one stellation, namely {5/2}, {8/3}, {10/3}, and {12/5}. Other orders, such as 7, have more than one stellation. It is probably no coincidence that quasi-crystals may have symmetry axes of orders 5, 8, 10, and 12 as compared with the axes of order 2, 3, 4, and 6 (for which the corresponding polygons have no stellations) allowable in real crystals. Stellation might be considered as a first step in the generalisation of the concept of axis of symmetry. The 5 regular polyhedra (the tetrahedron, cube, octahedron, dodecahedron, and icosahedron) each have faces which are all the same regular polygon and all their vertices symmetrically equivalent. The 13 semi-regular or Archimedean solids are convex poly hedra which have all their faces regular polygons of two or more kinds and all their vertices symmetri cally equivalent. They can be designated by the num bers of faces meeting at a vertex. For example the truncated icosahedron of C is 5.6 (Fig. 2). This fig ure is also obtained by deep truncation of the dodec ahedron. If we start with a truncated icosahedron then if the hexagonal faces are extended, we obtain an icosahedron and if the pentagonal faces are extended, then we obtain a dodecahedron. It is conventional to exclude the prisms N.4 from this definition. There are also a large number (53) of stellated regular and 60
2
For a convex polyhedron, topologically like a sphere, with F faces, V vertices, and Ε edges, Euler's k sphere has genus law states that V—E+F=2. zero and if another more complex polyhedron can be deformed to take the shape of a sphere with Ν han dles, then it is said to have the genus N. This is a useful but not a complete characterisation. Knot theory, which is still developing, is needed for a better classi fication. A torus has the genus 1 and the P-surface in side a cubic unit cell (see below) has the genus 3. The imposition of periodic boundary conditions is equiv alent to putting three handles across outside the cube connecting opposite faces. The more general expres sion of Euler's Law in three dimensions is V — Ε + F = χ, where χ is the Euler characteristic and χ = 2 — 2g, where g is the genus. Thus, for a torus, V — Ε + F = 0, and for a cell of the ^-surface, V — Ε + F = -4. For a network of the graphite type, assuming that there are only pentagons, hexagons, heptagons, or oc tagons, each edge is shared by two polygonal faces and each vertex is shared by three polygons. If N is the number of polygons with η sides and η vertices, we have: n
F= N + N + N + A , 2E = 3V = 5N -F 6A + 7A + 8A . 5
6
5
2
7
8
6
V
8
Putting this into Euler's expression we have N - N - 2N = 6χ = 12(1 - g) 5
7
S
so that for a torus N — N-, — 2N = 0. For a sphere N — Ν — 2N% = 12 and thus in such a figure, if 5
5
Ί
8
Geometry of hypothetical curved graphite structures
there are no heptagons or octagons, there can only be hexagons, in an indefinite number and 12 pentagons. Since V = 20 + 2jV , the number of vertices must be even. For the unit cell of the P-surface N — N — 2N = —24. Thus, if there are no pentagons, 24 Ν or 12 7V are necessary. 6
5
7
Ί
8
6. I N F I N I T E P O L Y H E D R A
The concept of polyhedra can be extended to in clude infinitely periodic regular and semiregular pol yhedra by allowing nonconvex arrangements. Many of these have been discussed by Wells[5]. One ex ample is shown in Fig. 3, which is made up of hexa gons and squares and where all vertices are equiva lent. Each vertex can be given the symbol 6.4 meaning that in going round a vertex we meet a hex agon and three squares. It might be noted that this polyhedron divides all space into two congruent regions. It will be seen below that this semiregular polyhedron is of importance in its relationship to pe riodic minimal surfaces serving as a conceptual ref erent. It is a polygonal version of the .P-surface. It should be noted that for the convex semiregular polyhedra, which have genus zero, the sum of the face angles meeting at vertex adds up to less than 360° and that, if there are Ν vertices the Ν deficits add to 720°. The vertex sum divided into 720° is an integer or, for a stellation, a fraction. For the infinite semiregular polyhedra the vertex sums are greater than 360°, depending on the genus g, the total excess being 4n(g — 1 ). If the polyhedron is stellated the genus may be fractional, but we are not concerned with this case. We may ask what combinations of three faces (pentagons, hexagons, heptagons, etc.) can meet at a vertex under the condition that all vertices should be symmetrically equivalent. It can readily be seen that, 3
S
115
if any of the faces has an odd number of sides, then the other two polygons must be the same (as two dif ferent polygons cannot alternate around an odd axis). It is more difficult to enumerate vertices at which four or more faces meet but, since we are considering graphite sheets, this is not necessary. We may consider the tessellations in two ways, ei ther as packings of pentagons, hexagons, etc. meeting three at a vertex, or as the repetitions of asymmetric triangular units. For example, the regular tessella tions by heptagons with the symbol 7 (for three hep tagons meeting at a vertex) is equivalent to the tessel lation by triangles 14.6.4 of angles π / 7 , π / 3 , π / 2 . By enumerating all the possibilities we find that the com binations 8.6.6, 8.8.5, 7 . 7 . 7 , 7.6.6 have vertex sums a little above 360°. To make a surface of genus 3 (as for the P, D, and G surfaces) the excess vertex sum of the vertices within the primitive unit cell should be 8 π . This is given by 56 vertices of the type 7 . 7 . 7 . This is the sur face from which Vanderbilt and TersoffI6] began to develop their model. If this tessellation is now trun cated, that is, a hexagon is placed at each vertex, the tessellation becomes 7.6.6 (with all the vertices equiv alent to each other). To make up 8 π , 168 vertice are needed. This number gives a more relaxed structure than using heptagons alone. The process is exactly parallel to the truncation of the regular dodecahe dron, with vertices 5.5.5, to give the truncated icosa hedron 5.6.6 of C . Lenosky et al.[7] have somewhat different com binations where 216 vertices of the type 7.6.6 and 48 of the type 6.6.6 add to give the vertex excess of 8 π . In our case[8] 144 points of the type 8.6.6 and 4 8 of the type 6.6.6 make up the 192 vertices per cell. In each case any number of 6.6.6 points may occur with out changing the vertex excess. We might consider points of type 8.8.5 of which 80 per cell of genus 3 would be needed plus any number of 6.6.6 points. The polyhedron corresponding to the Neovius surface (C(P)) has the same arrangement of points as that of the surface 6.4 discussed already, but the spaces between the points are differently filled with polygons so that each of the 48 points per cubic cell has the configuration 8.4.8.6 and this leads to a sur face of genus 9. 3
6 0
3
7. D E L T A H E D R A
Fig. 3. The infinite semiregular polyhedron 6 . 4 which is a polygonal version of the Ρ periodic minimal surface. 2
A deltahedron is a polyhedron, each of the faces of which is an equilateral triangle. Deltahedra include the tetrahedron, the octahedron, the icosahedron and many less regular figures, such as the pentagonal bipyramid, the 4m dodecahedron, etc. Given a two-di mensional hexagonal lattice, by cutting away one, two or three sectors around a hexagonal point, it can be folded to cover a deltahedron. Thus, by using the points of a larger super-lattice, a pattern of hexagonal symmetry can be mapped on to the surface of a del tahedron, so that Τ units lie on every face. The most important case is that of the icosahedron. Models of
116
H . T E R R O N E S and A . L . M A C K A Y
angle is 6° and that 120 X 6° is 720° which is 4ττ. If there are no mirror planes the asymmetric unit can be taken as a triangle with five-, three- and twofold axes at its vertices of which there will be 60 to make up the whole surface.
9.
CURVATURE
The curvature of a surface at a point can be ex pressed by fitting two circles, with their planes includ ing the normal to the surface and at right angles to each other. The curvatures are the reciprocals of the radii of curvature. When both curvatures are either maxima or minima then they are principal curva tures K and K . If the surface is convex at the point in question then the curvatures are both positive. If the point on the surface is a saddle point (is anti-clas tic), then the curvatures have oppposite sign (Fig. 5c). The mean curvature is defined as Η = (K + K )/2 and the Gaussian curvature as Κ = K K . Thus, for a convex surface Κ is positive and for a saddle-surface negative. For a spherical shell Η = 1/r and Κ = l/r . Η has the dimensions L~ and AT the dimensions L . On a surface with Κ > 0, such as a closed convex shell, the perimeter of a small circle of radius r is a little less than 2wr and the area of this circle is a little less than πι . The perimeter of the circle is 2irr{\ — Kr /! + Oir )). x
2
{
X
2
2
2
1
2
2
2
Fig. 4. Albrecht Durer's drawing of the construction of the icosahedron truncum by folding up a sheet of cardboard.
the convex fullerenes can be constructed in this way. Albrecht Durer seems to have invented the construc tion of polyhedra by folding up cardboard and Fig. 4 shows his drawing of the icosahedron truncum. D'Arcy Thompson[9] generalised the folding method and showed that all the convex semiregular solids could be constructed by folding plane tilings.
4
10. F U L L E R E N E S
Fullerenes are symmetrical closed convex graph ite shells for which Κ > 0 consisting of 12 pentagons and various numbers of hexagons. They have been il lustrated recently by Smalley and Curl [10] and by many others so that it is not necessary to discuss them further here. Being topologically equivalent to the sphere they have genus 0 and faces + vertices = edges + 2. Further, N - N - 2N = 12. 5
7
S
8. ICOSAHEDRAL TESSELLATIONS
The regular icosahedron has the symmetry 5m3m (I ) or 532 (/) according to whether or not there are mirror planes of symmetry. If there are mir ror planes the order of the group is 120 and if not, then 60. This means that 120 asymmetric units can be identically situated, if mirror images of the units are considered to be identical with the originals, or if not then 60 asymmetric units can have identical sur roundings. Asymmetry is usual case for the units of viruses which frequently form icosahedral shells. The asymmetric region of a spherical shell with icosahe dral symmetry is a spherical triangle with axes of order 5, 3, and 2 at its vertices, with mirror planes forming its sides and thus with angles of 36°, 60°, and 90°, respectively. The whole surface comprises 120 such triangles. We might note that the angular excess over 180° for the sum of the interior angles of the tri h
Fig. 5. Different types of curvature, (a) Positive Gaussian curvature, (b) Negative Gaussian curvature, (c) Zero Gaus sian curvature.
Geometry of hypothetical curved graphite structures
117
11. C Y L I N D R I C A L L A T T I C E S
For a circular cylinder of radius r, K = 1/r, K = 0 so that H = Vir and Κ = 0. Other cross sec tions may be considered as distortions of circles. A lattice point is marked at one identifiable point in the pattern and all identical points are similarly marked (identical meaning also identical in orientation). Cy lindrical lattices can readily be handled by taking a cylindrical projection (Fig. 6) where the surface is un rolled to give a plane sheet of width lirr. There are two kinds of cylindrical pattern, rational and irratio nal. In rational lattices further lattice points lie ex actly above others with a displacement parallel to the axis of the cylinder. With irrational lattices a second lattice point never occurs exactly above the first. Dif fraction from helices of both types has been devel oped by Cochran et al.[\\]. The dense packings of equal spheres around a cyl inder have been examined by Erickson[12] who de rived formulae for their generation. Some are shown in Fig. 7. These can be used to produce the corre{
2
Fig. 6. Cylindrical projection. This shows the 3 - 8 irrational helix which Iijima observed by electron diffraction from very small cylinders.
Fig. 7. Tubular arrangements of close-packed equal spheres (Erickson). A hexagonal mesh can be ob tained simply by omitting one third of the spheres. (Reproduced from Science by kind permission.)
H. T E R R O N E S and A . L. M A C K A Y
Fig. 8. The P-surface.
sponding graphite nets by omitting the sphere at the centre of rings of six. Iijima[4] has observed hollow cylinders of graph ite by high resolution electron microscopy and by electron diffraction has demonstrated the orientation of the lattice with respect to the axes of the cylinders. Tubes consist of 5 or more layers, probably separate tubes but possibly spirals. The minimum internal di ameter found was about 2 nm. Only hexagons are needed for cylinders but in order to make a cone a rolled sheet must be joined by a seam which contains a row of dislocations as in Fig. 1 and to join a cone to larger and smaller cylinders requires ring of 5 and 7 (or 8), respectively. 12.
PERIODIC MINIMAL
SURFACES
Minimal surfaces are surfaces with H = 0 so that K = —K and Κ < 0. They are thus saddle-shaped (anti-clastic) or flat everywhere. Surfaces may be minimal either because, as for a soap-film spanning a {
2
nonplanar loop of wire, it minimises its energy by having a minimum of area or, for a membrane sur face made of lipid molecules, because it minimises the splay energy. The mathematical condition for a surface to have zero mean curvature is that the diver gence of its unit normal should be zero. The splay en ergy is the integral of H over the area. It is not possible to construct an infinite surface with constant negative Gaussian curvature. Such a surface with a constant, imaginary radius of curva ture defines the hyperbolic plane. However, it was found by H. A. Schwarz[13] that patches of negative Gaussian curvature and Η - 0 could be smoothly joined to give an infinite triply pe riodic surface of zero mean curvature. Typically a patch might be a triangle everywhere saddle-shaped and thus with angles which add to less than 180°. These triangles are repeated by diad axes. The Fig. 8 shows the P-surface. Exact procedures for determining the shapes of the periodic minimal surfaces are available and have 2
Table 1. Images of the normals of the atomic positions onto the complex plane for the D, G and F T P M S (u, and υ!) i
u,
0 1 2 3 4 5 6
0 0.04 0.149 0.41854 0.49365 0.1 0.245
0 0.493656 0.41854 0.149 0.04 0.245 0.11
Ki
η
x
y
-0.0783 -0.0109 -0.0395 -0.0395 -0.0109 -0.0598 -0.0598
±3.573 ±9.573 ±5.031 ±5.031 ±9.573 ±4.089 ±4.089
0 0.7088 1.4163 2.8419 3.5494 0.7673 1.7288
0 -3.5494 -2.8419 -1.4163 -0.7088 - 1.7288 -0.7673
0 -2.0 -1.0 1.0 2.0 -0.3533 0.3533
Gaussian Curvature Kj (in  ) . Radii of curvature r, (in A). Coordinates in real space for the D patch (in A). 2
Geometry of hypothetical curved graphite structures
119
Table 2. Properties of D, G, and Ρ T P M S made with graphite, a is the lattice parameter in A, for a C - C distance of 1.42 Â Surface a N
c
Sc Symmetry
D surface
G surface
Ρ surface
24.09 192 768 3 9 _ Prihm — Fdhm
18.98 192 384 3 5 7a3
15.41 192 192 3 3 Im3m — Pm3m
Ν ρ is the number of atoms per primitive rhombohedral cell. N is the number of atoms per cubic unit cell. g is the genus of the primitive cell. g the genus of the cubic unit cell. c
p
c
been described elsewhere[14]. They involve repre senting each point on the surface by a point in the complex plane. The coordinates x , y, ζ of a point / in the surface are related to this complex number w,- + Wj by the Weierstrass integrals which can be com puted. The Z>, P, and G surfaces are closely related to each other and the coordinates of one are obtained from those of another by multiplying the complex number by exp(/0), where θ is the Bonnet angle; θ = 0, θ = 38.0147°, θ = 90° for the D, G, and Ρ patches, respectively. In Table 1, atomic coordinates in complex and real spaces for the D patch are given. From these po sitions the whole D surface decorated with graphite can be obtained. The Gaussian curvature and the radii of curvature are also given. It is interesting to note that the magnitude of the maximum value of the Gaussian curvature (0.0783 A~ ) is less than the Gaussian curvature of C which is 0.08076 A . Therefore, C is more curved than the D, G, and Ρ minimal surfaces decorated with graphite; this can 2
60
60
2
have important implications when considering the stability of curved graphite since the elastic energy in volves the Gaussian and mean curvatures. Table 2 shows some properties of the exact TPMS decorated with graphite. Surfaces can also be found by finite element anal ysis methods. These methods look quite promising since they are suitable for generating surfaces which minimize energy functionals. 1 3 . GRAPHITE STRUCTURES WITH NEGATIVE GAUSSIAN CURVATURE
Many of the periodic minimal surfaces can be dec orated with graphite-type networks. We have suc cessfully built models of the P, D, (7, and Η surfaces (using three-way joints and connector tubes)[8] (Figs. 9 and 10.). Triangular patches of various sizes of the graphite network with angles 90°, 30°, and 45° (Fig. 9) can be joined so that the 45° vertices combined to give rings of eight carbon atoms. In the actual structures
Fig. 9. Possible graphite patches for triply periodic minimal surfaces decorated with graphite, (a) Ρ surface; (b) D surface; (c) G surface. The angles of the curved triangle are 90°, 30°, and 45° not 60°.
120
H . T E R R O N E S and A . L. M A C K A Y
(a)
(b) Fig. 11. Possible graphite structures based on surfaces par allel to periodic minimal surfaces curved by the introduc tion of heptagons, (a) PI surface; (b) Dl surface. (Coordi nates provided by T h o m a s Lenosky.)
Fig. 10. Possible graphite structures based on periodic min imal surfaces which are curved by the introduction of octa gons, (a) Ρ surface; (b) D surface; (c) G surface.
these sit appropriately on the saddle points of highest curvature (the highest curvature is at the centres of the octagons). Given the basic patch, the asymmetric unit of pattern, this can be twisted to give either the P, the D, or the G surfaces. It is possible to begin by embedding C atoms in the exactly determined surfaces but, since the struc tures are to be refined, approximate positions mea sured from physical models are quite adequate. Moreover, Lenosky's and Vanderbilt and TersofFs structures are parallel surfaces, where the two subspaces are unequal, and not the more symme trical minimal surfaces.
Thomas Lenosky[7] has built surfaces which are parallel to the Ρ and D periodic minimal surfaces and divide space into two unequal regions (and are thus less symmetrical than the periodic minimal surfaces. (See Fig. 11.) He has refined the atomic positions and has calculated the physical properties of the struc tures. These patches contain heptagons rather than octagons to introduce the negative Gaussian curva ture. Such structures are found, by calculation, to be much more energetically favourable than Buckminsterfullerenes. Lenosky, Gonze, Teter, and Elser pro pose the name Schwarzites, in memory of H. A. Schwarz, the mathematician who developed the first few of the periodic minimal surfaces, for this category of graphites with nonpositive curvatures. We are much indebted to Lenosky for calculating energies of our structures, those of Vanderbilt and Tersoff and his own on the same basis (Table 3), so that their stabilities can be compared with that of C . It will be seen that there are several such structures with energies less than that of C . Physical properties of cubic forms of carbon cal culated by Lenosky[7]. Dl and ΡΊ are the surfaces produced by himself, the D- and P-parallel surfaces 60
60
Geometry of hypothetical curved graphite structures
121
Table 3. Properties for different negatively curved graphite structures Structure
Ρ
AE
Β
a
a A
Ν
Schwarzite Dl Schwarzite PI Schwarzite P8 Schwarzite DT fee. C Diamond
1.15 1.02 1.16 1.28 1.71 3.52
0.18 0.20 0.19 0.22 0.42 0.02
9.4 7.5 10.3 11.5 1.4 44.3
17.39 11.4 10.47 15.42 10.0 2.31
24.7 16.2 14.9 21.9 14.12 3.5595
216 216 192 168 60 2
6 0
ρ is the density in g / c m . AE is the total energy relative to graphite in e V / a t o m . Β is the bulk modulus in units of 10' d y n e / c m . a is the cubic unit cell size in units of the bond length. a A is the cubic unit cell size in Angstroms. Ν is the number of C atoms per primitive unit cell (the f e e . cell contains four primitive cells). 3
1
2
incorporating heptagons. DT is the D- parallel sur face found by Vanderbilt and Tersoff. P8, D8, and G8 are the symmetrical P-, and G-surfaces incorpo rating octagons found by Mackay and Terrones. Tables 4-6, give the coordinates of the corre sponding structures. Vanderbilt and TersoffI6] have proposed a graph ite structure based on a surface parallel to the D-surface, so that the two subspaces are not equivalent (Fig. 12). A tetrahedral joint is built out of 84 atoms in hexagons and heptagons so that each point is a member of two hexagons and one heptagon (6 .7) and is topologically (but not metrically) equiv alent to every other. There are thus 2 X 84 atoms per primitive unit cell with space group Fd3 and 8 X 84 = 672 per cubic unit cell with a = 21.8 À (assuming graphite-type bonds). The fractional coordinates of the nonequivalent atoms are given in Table 7. The density is expected to be 1.29 g/cm . The au thors calculate the energy of formation to be 0.11 eV/ atom as compared with 0.67 eV/atom for C , but their calculation is not exactly the same as that of Lenosky.
Table 4. Fractional coordinates of the PS surface with ori gin at the centre of the cubic cell (Im3m) Atom
X
y
ζ
1 2 3
0.286 0.314 0.387
0.173 0.0908 0.0494
0.286 0.314 0.2729
Table 5. Fractional coordinates of the G 8 surface^vith ori gin at one of the corners of the cubic cell (Ia3d)
2
Atom
x
y
ζ
1 2 3 4
0.1903 0.1348 -0.0844 -0.0279
0.2117 0.1703 0.2014 0.1520
-0.2265 -0.1988 -0.1511 -0.1217
3
60
Fig. 12. Structure proposed by Vanderbilt and Tersoff (DT).
Table 6. Fractional coordinates of the D 8 surface with origin at one of the corners of the cubic cell (Fdlm) Atom
X
y
ζ
1 2 3 4 5 6
-0.0416 -0.0831 -0.1668 -0.2083 -0.0967 -0.1532
0.0 0.0 0.0 0.0 0.05172 0.05172
0.54163 0.58315 0.6668 0.7083 0.61032 0.63966
Table
7. Fractional
coordinates structure
of
Vanderbilt's
Dl'
Atom
X
y
ζ
1 2 3 4 5 6 7
0.1134 -0.0109 0.0272 -0.0984 -0.0662 -0.0743 -0.1915
0.0050 0.0303 0.0756 0.0826 0.1382 0.0402 0.0177
0.2415 0.2200 0.2435 0.1704 0.1542 0.2146 0.1649
122
H . T E R R O N E S and A. L. M A C K A Y
tion of Penrose tilings and would bring the mean cur vature towards zero. When we attempt to execute the H tilings in real space the curvature causes the sur face to fold like seaweed (as fucus letuca) since more area is produced within a given radius than is appro priate for a planar tiling[16]. After a certain number of units have been added the surface can be closed on itself in various ways, periodically or irregularly. One of these ways is that found by Vanderbilt and Tersoff. 2
REFERENCES
Fig. 13. A tessellation of the hyperbolic plane by regular heptagons (7.3.2). The 7.6 tessellation is obtained by trun cation from this, a hexagon replacing each vertex. 2
14. T H E H Y P E R B O L I C PLANE—H
2
The hyperbolic plane (called T/ , the surface of the sphere being S ), has a constant negative Gaussian curvature. Regular and irregular tessellations of this surface by polygons can represent topologically the local connectivities of real space polygonal and con tinuous surfaces which have everywhere nonpositive curvatures. In particular, the regular tessellation 7.3.2 of H by heptagons (Fig. 13)[ 15] can be truncated to give the tessellation 7.6.6 which corresponds to Van derbilt and Tersoffs surface and is the least curved of the regular tessellations. Further truncations would increase the ratio of hexagons to heptagons, but all vertices would then be no longer equivalent to each other. This process corresponds to the recursive infla 2
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