Journal of Modern Physics, 2011, 2, 97-108
doi:10.4236/jmp.2011.23016 Published Online March 2011 (
Copyright © 2011 SciRes. JMP
Computer Models of Helical Nanostructures
Victor F. Pleshakov
Scientific-Research Institute of Electrical Carbon Products, Elektrougli, Russia,
Received October 22, 2010; revised December 24, 2010; accepted December 27, 2010
A task of mapping a hexagonal grid to different types of helical surfaces including nanocones, nanotubes and
nanoscrolls by unfolding a given surface to a carbon layer plane has been solved. Basing on these models,
polyhedric models with all atomic bonds being constant and equal to 1.42Ǻ as in a flat carbon layer have
been built, and an algorithm of coloring all faces of such models has been developed. Received models can
be utilized for visual demonstration of the helical growth of nanotubes, nanocones, nanofiber and other
nanoobjects, and also for physical properties calculation.
Keywords: Computer Models of Helical Nanostructures
1. Introduction
It seems obvious that 3D-images of nanoobjects play
significant role in understanding their geometry, struc-
ture and properties. Special literature accumulates reach
experimental materials obtained with electronic micro-
scopes confirming the spiral or helical growth of nano-
cones [1], graphite whiskers [2], nanofiber [3-8], nano-
tubes [8-10] and so on. Different helical or spiral growth
of nanoobjects models are suggested and discussed; these
models, however, are merely sketchy. That’s why a need
to develop computer models and general algorithms of
building them has arisen. Such algorithms should exflat
or at least imitate the growth mechanisms of nanoobjects.
The necessity for such models often appears during de-
coding of a nanomaterial structure with x-ray, electronic
microscope and other methods. The current notion of
nanofibers as objects with cones inserted into each other
doesn’t exflat experimentally observed cone’s opening
angles. As shown in [1,11] there are only five types of
nanocones with opening angles δ = 19.19°, 38.94°, 60°,
83.62° and 112.88°, and most of angles observed with
electronic microscope do not match these five standard
ones; and only in a helicoid the δ angle can vary widely.
For instance, [1] adduces opening angle data for dif-
ferent carbon fibers between 2.7° and 14.5°, and suggests
a helical growth model for nanofibers with open and
closed vertex of a cone. In [2], carbon whiskers were
produced during the treatment of wood over a SiC cata-
lyst. They had the δ angle within 110° - 151°. In [3], car-
bon fiber was produced from oil pitch with opening an-
gle varying from 60° to 180°. Reference [1] examines a
helical growth of nanocones and nonofiber, and [10]
examines that of nanotubes. The catalytic synthesis of
carbon nanotubes and nanofiber is described in [12],
where the question of spiral nanofiber synthesis was also
touched. Similar models were used in [4,5] to describe
the result of a nanofiber growth over catalysts.
Graphite has a flake structure, so it delaminates to
separate layers or packs of several layers when it is being
exposed to laser evaporation or an arc discharge. Upon
loosing stability, layers may roll up to a scroll or conical
and cylindrical helixes. Reference [13] suggests a
mechanism of dislocational forming of single-shell or
multi-shell nanotubes from that scroll, and a correspond-
ing disclination mechanism for nanocones. These and
other multiple experimental data have lead scientists to
the idea of a helical or spiral growth of nanoobjects.
In recent days [14-16] helical nanostructures, nano-
helixes and nanotubes attract the attention of many re-
searchers as a possible perspective material to create
terahertz generators still not mastered for now.
In aforementioned documents, as was stated above,
nanoobject models are sketchy, so it is worthwhile to
obtain computer models of surfaces with a carbon nano-
grid mapped onto them, explaining, at least in a qualita-
tive sense, numerous experimental data. Atomic coordi-
nates and parameters of a model could be used then to
calculate physical characteristics of nanomaterials and to
decode their structure.
The purpose of this document is to obtain a general
method of mapping a hexagonal carbon grid (Figure 1)
to different types of helical surfaces by unfolding the
given surface to a carbon layer plane.
These ideas and differential geometry methods often
used in material cutting [17] and Modeling of Tent Fab-
ric Structures [18] have not been used to build complex
nanoobject models, such as helical, as far as we’re con-
cerned. Explicitly or implicitly they were used for simple
surfaces only, such as cylindrical, conical and scrolls [11]
that can always be unfolded to a plane.
2. Problem Definition
A graphite layer being rolled up to a tube, a roll or a
helicoid along the crystallographic direction v = [v1,v2]
(Figure 1), where v1, v2 are integer coprime numbers,
requires all atomic bonds to be constant. This means the
equation system should be solved to build a model in the
general case:
22 2
()()( )
ijiji jr
 
 
,1,2,,ij n
where r0 is the closest spacing between atoms, n – the
number of atoms in the model.
This system, however, is undefined, and has multiple
solutions, so the problem of determining atomic coordi-
nates ξ, η, and ζ in the model is solved in two steps.
At first, a hexagonal grid is mapped onto the given
surface by unfolding it to a plane. Atomic bonds get de-
formed and turn to spiral curve segments on that surface.
It is known that if the Gauss curvature of a surface is
equal to zero, this surface can be unfolded to a plane
while keeping lengths and angles between a pair of
curves lying on it intact. However such surfaces as heli-
cal with the non-zero Gauss curvature can’t be unfolded
directly, but they can be split into flat segments so that
each one can be unfolded to a plane. This is called a
Figure 1. The flat layer atomic coordinate calculation
conditional or an approximate involute. The smaller
these segments are, the more accurately a surface is ap-
Unfolding the surface to a carbon layer plane (Figure
1) this way, we get some area, or simply put - a pattern.
Atoms captured by this area are reflected on the given
surface after “gluing” the pattern back.
Such surface is covered by a grid built of equilateral
curvilinear hexagons with the side (arc) length equal to
the minimal atomic spacing r0 = 1.42. Å
The actual atomic spacing measured along the line
connecting two adjacent atoms is lesser than r0 and de-
pends on the measurement direction.
If the radius of a surface curvature strives for infinity,
all atomic spacings measured along the line strive to the
extreme value r0, like on a plane (Figure 1). For nano-
tubes with the diameter D > 6 the error of spacing
measurement along the line and along the arc lying on a
surface is less than 1%.
This model is called a curved atomic-bond approxima-
tion model, and is a quite good first approximation to the
ideal model. With it, elastic, mechanical, electronic and
other characteristics of nanotubes are computed, and also
hydrogen adsorption and desorption issues are studied.
On the second step, atomic coordinates of the model
are adjusted so that all atomic bonds would equal to r0 =
1.42 as in a flat graphite layer, and polyhedric models
are created basing on the solution of a system of three
quadratic equations. Such models were examined in
[19,20]. Finally, a not so easy problem of coloring all
faces of a polyhedric model is solved.
Now, let’s take a closer look at these questions.
3. Conical Helicoid
Helicoid is a surface produced by rotating a line or a line
segment around the chosen axis with the constant angle
between the axis and the line equal to δ/2, and by simul-
taneous moving that line or a line segment along the axis
by bφ value. There are two types of helicoids: a right
helicoid (δ = 180º) and an oblique helicoid (δ 180º).
To cover a wider class of surfaces, let’s consider a
more complicated version of a helicoid, when a line
segment spires along an Archimedean spiral (ρ = aφ) and
simultaneously moves along the ζ axis by bφ value. Let’s
call this surface a conical helicoid. In particular, having b
= 0 we get a conical scroll named by analogy with a cy-
lindrical scroll examined in [11].
Figure 2 shows a projection of a conical scroll to the
ηζ plane enclosed within two spirals AB and CD, where
L is a scroll’s generatrix, δ-opening angle of a cone. The
BD segment being rotated around the ζ axis at angle of
δ/2 counterclockwise produces conical scroll. If b 0, a
Copyright © 2011 SciRes. JMP
scroll unrolls along the ζ axis producing a conical heli-
coid described in co-ordinate system having point O as
the origin with the following equation:
cos ,sin,cos(2),bt
 (1)
where 0sin( 2)t
 
 00
- t-
ing radius.
In (1) the σ parameter must not be arbitrary, because
the interlayer distance should be constant in a helicoid,
or equal to zero in a cone. So let’s present σ as a sum of
two items: σ = a + c. The a parameter is responsible for
the Archimedean spiral which the BD segment rotates
along, while the c parameter keeps the interlayer distance
H in a helicoid constant. For t = 0 and t = L the surface
enclosed between two spirals (1) is a ruled surface with
Gauss curvature equal or not equal to zero, depending on
the values of ,,,acb
parameters. Coordinates of point
(atoms) on the surface of a helicoid must be expressed
via coordinates of the flat layer.
To map a carbon grid to the given surface we should
know how it is unfolded on the plane and find the de-
pendence ()
between the angle ψ on the involute
and the angle φ on the surface (1). Let’s study these
questions separately.
4. Unfolding Helical Surfaces to a Plane
To calculate coordinates of atoms in each model, two
coordinate systems are introduced – a moving one with
the origin at the О point on the Figure 2 and a fixed one
with the origin at the О point. Further, we will denote
variables ,,
and ,,
yz in the non-stationary
coordinate system with a p index, and in the fixed system
– without an index.
The Equation (1) is expressed in the non-stationary
coordinate system as:
sin( 2)cos
, sin( 2)sin
cos( 2)
One can see, that the expression (2) is an equation of a
conical surface with one fixed point О, where an Archi-
medean spiral is a directrix, and the vector R is a genera-
trix. Such surface can be unfolded to a plane with both
angles and lengths remaining the same. The involute of a
surface to the plane xz is a set of spirals covering area
0 tL
cos ,sin,
xR zR
0sin( 2)sin( 2)Rt
 
 in (2) and
To express the equation of a surface and the equation
of a spiral on a plane in the stationary coordinate sys-
tem one should subtract the value of OO
0cot 2a
 
Figure 2. The projection of a conical scroll to a plane.
z in (3). So we get:
0cot 2
 
 
0cot 2
zz a
 
 
Now the main problem is to find the dependence
we need to map a carbon grid to a surface.
Since the arc length of an arbitrary curve on a surface
remains invariant while unfolding it to a plane, we can
use (2) to get the following equation:
(dd)dsin2dRRRR 2
 
where the left and the right part of the expression are
respectively square of arc length of a curve on the plane
and on the surface, and denotes derivative on the
angle φ.
After several simple transformations we receive the
ddsin2 1
 
; solving it, we find
1sin 2
 
It seems that such approach can be also used to deter-
mine the dependence
for undevelopable surfaces,
however in this case we will get a more complex equa-
The simple way to find the dependence
undevelopable surfaces is as follows. Since an Archi-
medean spiral
0sin 2at
 
 
is a directrix of
a conical helicoid (1), we can approximate dd
where ds – is a length of an arc element of a helix, and
sin 2R
is an absolute value of a radius-vector
drawn from О (Figure 2) to that arc element of a helix.
Considering the calculation is performed in the stationary
coordinate system, we have:
, (7)
(Figure 2) from
in (2) and where
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  
sin 211fqArsh q
  
, qa
If we assume a=0 in the integrand in (7), we auto-
matically receive an accurate relation between angles ψ
and φ for conical surfaces (6).
So we have used (7) to acquire a method of building a
conditional (approximate) involute of helical surfaces to
a plane, that is, we’ve set a one-one mapping between
points (atoms) of the surface and the plane.
The calculations show that the
dependence (7)
is linear and for conical surfaces is approximated by (6)
quite good. Besides, it doesn’t depend on the t parameter
so it is acceptable to use it for undevelopable surfaces
such as helical surfaces as well. In this document, the
equation (6) was used to calculate the dependence
for all types of the below surfaces, except the
general case (1), which has allowed to significantly re-
duce calculation time.
Figure 3 shows an example involute of a conical
scroll to the plane xz in the non-stationary (at the right)
and stationary (at the left) coordinate systems. The invo-
lute displays an area enclosed between two spirals (3) or
(5), when t = 0 and t = L. It is the area, where atoms of a
flat carbon layer must be. One can see that in the
non-stationary coordinate system all radius-vectors come
from the origin (point О’ at the Figure 3). The conical
scroll itself with a carbon grid already mapped to it is
depicted at the Figure 5.
5. The Method of Mapping a Carbon Grid to
a Surface
Finally, let’s study a problem of generating atomic coor-
dinates of a conical helicoid using the known depend-
ency (6), (7) and (8) between φ and ψ angles.
Coordinates at the flat layer xi and zi (i = 1,2, ,n),
where n is the number of atoms enclosed within the area
between two helixes on the xz plane, are known and can
be found by corresponding formulas [11].
11212 220
sinsin ,zalb albz
 (9)
11212 220
cos cos
al balbx
where 12
- parameters of an elementary cell at the
flat layer, 12
, 00
z- coordi-
nates of a nearest hexagon center. The β and α-β angles
can be expressed through the identity period
by formulas [11]. Coordinates of two basis atoms ex-
pressed in fractions of a carbon layer elementary cell
period are
0, 0,13, 23b, if . 12
Figure 3. The involute of a conical scroll to a plane in the non-stationary (at the right) and stationary (at the left) coordinate
Figure 4. The involute of a conical scroll to a carbon layer plane in two coordinate systems - non-stationary (at the left) and
stationary (at the right).
Figure 5. A conical helicoid with non-overlapped layers: φ1 = 0, φ2 = 6π, δ = 60°, ρ0 = 2.3, H = 3.356, L = 5.64, b = 1.2, v = [3,1],
and a conical scroll with a carbon grid mapped to it: φ1 = 0, φ2 = 4π, δ = 60°, ρ0 = 3, H = 3.356, L = 10.47, b = 0, v = [3,1].
Determining coordinates
,,xl l
,,zl l
of atoms in a flat carbon layer from (9) with pre-defined
integer numbers 12
and ,ll
, we can write the follow-
ing equation in the non-stationary coordinate system for
an absolute value of a radius-vector R drawn from the
point O to an atom on the plane, and the angle ψ:
sin ,,
RRRL zRR x z
 (10)
00 sin 2R
 
 .
The main difficulty here is to find angles φi for atoms
lying on the helical surface using known angles ψi on the
involute by solving equations for all atoms
 
, где
 
 (11)
is found with (8), if . 0a
If , then according to (6),
sin 2f
 
fsin 2
Since the function
arcsin p
expressed from
(10) is ambiguous having its principal values lying
within 2π2
 , it is impossible to find one-one
mapping between angles φ and ψ directly. Besides, the
angle ψ at the involute, obviously, should change within
To solve this issue, let us split the interval between
angles 1
onto segments of
the following way:
 
21pjj f , ,
1pk f 1, 2,,jk
, obviously, 1
, and the function rounds the number to
the nearest integer.
Now, for i-th atom let us find the angle
arcsin ii
and determine which quarter of a flat
circle the calculated angle
hits by analyzing signs of
x and z coordinates. For the first quarter of a circle
0, 0xz
0, 0xz
for the second and
the third
0, 0xz
 and, finally, for the
fourth quarter ,
 .
Next, we calculate the angle
 ,
, where p is found with (12), and by using
the function
1, 2,,jk
zero we find the corresponding angle
in according to (11) if for
t = 0. 0a
,,,,,, /Ffzerofib t
where the
ipsi operator calculates the function
,,jl l,f
, and the search of a root of this
function starts from the point
. The integer
numbers 12
identify atoms, while identifies an
interval number as in (12). If , φ is explicitly de-
termined from (6):
(14) 2
Next, let us calculate the radius
00 , , and for all
atoms analyze if the angle
sin 2R
 
and the radius belong
to intervals:
 and ,
RRRL 
If both conditions are met simultaneously, atomic co-
ordinates at the flat layer x and z are stored in arrays
, while helical atomic coordinates in
the stationary coordinate system are calculated using (4).
Atomic coordinates in two coordinate systems on a plane
are computed using Formulas (3) and (5).
Finally, the number of atoms in a model n is calculated.
The variable n initially equal to zero gets increased by
one every time. Arrays of angles 1
 and
needed to draw a flat helical involute and enclosed atoms,
and also to map a carbon grid to a surface, are created for
each atom using (4).
6. Different Surface Types
As was previously stated, depending on the values of
and b parameters in (1), one can get different
types of helical and conical surfaces. Let’s examine these
1) General case. In this case the interlayer distance H
of a helicoid must be constant and roughly equal to that
Copyright © 2011 SciRes. JMP
of turbostratic carbon. To make H constant, the values
and c in (1), as it follows from Figure 2, must be
equal to:
2πcos 2aH
 
, then layers are partially
overlapped; if
 - there’s no over-
lapping. If , we receive a conical scroll.
At Figure 2, the BD segment being rotated is always
located at the surface of a conical scroll (b = 0). The ends
of this segment (points B and D) on that surface describe
two helixes (4), if t = 0 and t = L. Unfolding a conical
scroll turns two helixes (4) into spirals (5) on a plane. It
is the area cut from a plane by spirals that is the area,
where carbon layer atoms must be.
An example involute presented at the Figure 4 shows
a conical scroll unfolded to a carbon layer plane. The
Figure 5 (at the left) shows a conical helicoid and a coni-
cal scroll (b = 0) with a carbon grid mapped to them.
Calculated parameters are shown on the same pictures.
The obtained surfaces can be a visual demonstration of a
nanofiber growth, to some extent.
2) Conical surface (a spiral growth of nanocones).
This case doesn’t contain a spiral movement, and the BD
segment at the Figure 2 rotates at the constant angle to
the ζ axis while staying on the surface of a cone all the
time. To make this possible, we should put a = 0 in (15).
An arbitrary point at the BD segment will describe a
conical helix with the radius
0tan 2b
 
sin 2t
on the surface of a cone.
To make segments not overlap during rotation, the
following condition must be met:
2πcos 2bL
. If
2πcos 2bL
, then two helixes (Figure 2), de-
scribed by ends of the segment on a cone’s surface, join
together forming a stripe with the width L. Such helical
surface being mapped with a carbon grid may visually
imitate a spiral growth of nanocones. As shown in [1,11],
the angle δ at a conical surface can’t be arbitrary as it is
connected with the angle γ at an involute with the fol-
lowing correlation:
2πsin 2
, where
=60, 120, 180, 240,300.
If the condition is met, carbon layers join seamlessly
where the helixes meet.
Figure 6 shows a surface imitating a spiral growth of
a nanocone with
2πcos 2bL
, and a conical
2πcos 2bL
, having in (1). 0H
3) Nanocon es . If one puts σ = 0 and b = 0 in (1) and
limits the angle φ within 02π
 , the following
equation of cone can be written:
cos ,sin,
 
cos2 ,t
 
 , 0tL
The correlation between φ and ψ in this case is calcu-
lated using (6).
Seamless join of a carbon layer on the nanocone’s
surface is only possible if the angle δ meets (16). Nano-
cones are thoroughly examined in [11] by the author by
means of unfolding them to a plane and formulating
conditions for a seamless join of a carbon layer.
4) Oblique helicoid. Assuming σ = 0 in (1), one can
get an equation:
cos ,sin,
 
cos2 ,bt
 
 (17)
 
 , . 0tL
From (15), assuming σ = a + c =0, one can receive c = a.
Figure 7 illustrates this case. The BD segment being
rotated by the angle 2π takes the position, and all
its points move along the axis ζ by the value 2πb. Points
B and D describe two helical curves that entirely lie on
the surface of cylinders with radiuses 0
0sin 2L
 
 . The surface enclosed between
these two helical curves is an oblique helicoid. To make
interlayer distance H constant, the b parameter must be
2πsin 2bH
. Calculation of the angle φ is
performed using formulas (14).
Since the radius ρ doesn't depend on the angle φ, an in-
volute of a helix to a plane in the non-stationary coordinate
Figure 6. A spiral growth of nanocones (at the left) b = (L/2π)cos(δ/2) = 0.70485, and a conical helix (at the right) b = 1.2 >
0.70485, φ1 = 0, φ2 = 7π, δ = 60°, ρ0 = 2.5, H = 0, L = 5.114, v = [5,1].
Copyright © 2011 SciRes. JMP
Figure 7. The oblique helicoid calculation scheme.
system is a circle with radius
sin 2R
, that is:
cos ,sin,
xR zR
 (18)
and the area, where atoms of a flat carbon layer must be,
is enclosed between two concentric circles with radiuses
sin 2R
Since the angle ψ can take a value within
on a plane, then if 2
, then each consequent circle
repeats the previous one, that is, they overlap.
To acquire the equation of a helix in the stationary co-
ordinate system with the origin in O, unlike in (5), one
must subtract the value of
0cot 2OO b
from the variable
z in (18), finally receiving:
 
tan22πcos 2ab
H .
The atomic coordinate
in the stationary coordinate
system, unlike in (4), is calculated as follows:
0cot 2
 
 , where 22
Figure 8 (at the left) shows the surface (17) with a
carbon grid mapped to it.
5) One more case of an oblique helicoid. In this case,
while the BD is being rotated, the point B at Figure 7
always stands on the plane
and circumscribes a
circle with radius ρ, while the point D moves along the ζ
axis by the value of bφ, describing a helical curve. After
turning by 2π, the segment BD takes the CD’ position.
An arbitrary point E lying on the segment BD and dis-
tant from the point B by t in the coordinate system
describes a helical curve similar to the previous
The difference is that t has a variable bottom limit,
cos 2b
From the Equation (17), assuming
cos 2tb
 ,
one can get the equation of a spiral lying on the plane
(described by the point C), namely:
cos ,sin ,
 
, where,
 
and if tL
, we receive a spatial helix described by the
point D, that is:
cos ,sin ,
 
cos2 ,bL
 (20)
0sin 2L
 
 .
The segment CD describes a helical surface, and its
ends move along two helixes (19) and (20).
Since ρ in (19) takes its values within min 0
where ρ0 and ρmin – maximal (or starting) and minimal
values of the radius ρ, the angle φ should be within
 
 2 or, assuming
02πcos 2H
 
Figure 8 (at the right) illustrates this case.
6) Right helicoid (a spiral growth of carbon layers).
Assuming δ = 180º and c = 0 in (1), one can get the fol-
lowing equation:
cos ,sin,,b
 
 (21)
where 0at
 
, 2πbH
, . 0tL
If b = 0, then the Equation (21) is a set of spirals. If a
= 0 as well, we receive a set of flat circles repeating them
selves many times, depending on the angle φ. If
, the areas between two spirals partially
overlap when t = 0 и t = L. If /2πaL - no overlap-
ping occurs. If 2πaL
, the spirals touch each other
and imitate the spiral growth of a flat layer. Finally, if
b0, all spirals unrolls along the ζ axis producing a spa-
tial model that imitates the growth of carbon layers. As a
result of the dislocation entering the surface, we receive
either a stack (a column) of carbon layers in the form of
a cylinder, or a cone, or a ring, or a circle on a plane,
depending on the values of 01
. The angle φ is
calculated from the known angle ψ at the involute using
(13) if 0a
, or (14) if a = 0.
7. Cylindrical Helicoid
Assuming δ = 0 in (1) we can write the following equa-
cos ,sin,bt
 
 
, . (22) 0tL
In this case a line segment with the length L parallel to
the ζ axis rotates along Archimedean spiral and simulta-
tL. The angle φ is calcu-
lated through the angle ψ using the same Formula (14).
Copyright © 2011 SciRes. JMP
Figure 8. Two types of oblique helicoids imitating a spiral growth of nanofiber. The left picture has the following parameters:
φ1 = 0, φ2 = 7π, δ = 60°, ρ0 = 3, H = 3.356, L = 7.44, v = [5,1]. The right picture has the following parameters: φ1 = 0, φ2 = 4.2π,
δ = 40°, ρ0 = 10, ρmin = 3, H = 3.356, L = 6.15, v = [2,1].
neously moves along the ζ axis by bφ. Segment’s ends
describe two helixes with t = 0 и t = L. To unfold the
surface between these two helixes, it is sufficient to
know the length of Archimedean spiral’s arc s and the
variable ζ. Substituting x = s и z = ζ, the surface will be
unfolded at the xz plane.
Considering that 22
, we get the fol-
lowing expression for the length of spiral’s arc:
 
, where
 
222 2
aa Arshaa
 
 (23)
To keep the interlayer distance H constant the follow-
ing condition should be met: a=H/2π.
If a = 0 in (22), we receive two helixes entirely lying
on the surface of a cylinder. To make areas enclosed
between them no overlapping, the inequality must be
fulfilled: 2bL
. If 2πbL, two helixes touch
each other and create a stripe with the width L on
cylider’s surface. Now, if
, where
1, 2,k
is the identity period [11]
in the crystallographic direction
, then car-
bon layers join seamlessly where the helixes meet. Such
surface with a carbon grid mapped onto it visually dem-
onstrates a spiral growth of nanotubes. Since 0
in that case, the involute to the plane xz is a set of line
segments having their slope ratio about the x axis equal
to 0
, namely
zb x
t, , ,
 0tL
 L.
To map a carbon grid to a surface, one can use the fact
that rolling up a flat layer to a helix turns the coordinate
,,xl l
, ,
,1,2,3,ll1, 2,,b
to an
arc of a helix. So we have:
, (24)
is expressed from (23), or
If we calculate an angle φ for all atoms using (24), that
is, for all legitimate values of l1, l2 and ν, then atomic
coordinates on helicoid’s surface can be found from
cos,sin,, ,zll
 (25)
The angle φ here is a function of 12
ing atoms, that is
. In particular, with b=0
we obtain a common cylindrical scroll already examined
in [11].
Figures 9-10 illustrate these cases. We can take as a
hypothesis that when the dislocation enters a surface
along the z axis, figures pictured at Figures 9-10 (at the
left) turn to a scroll or a nanotube (at the right). Further
radial movement of the dislocation in the scroll leads to a
multi-shell nanotube generation, if the angle φ is divisi-
ble by 2π. Otherwise, there is a part of unfinished layer
left on nanotube’s surface, and the growth process con-
tinues according to [13].
Having determined φ from (24) or from
,,xl l
, if a = 0, and assuming z=ζ, one can get the follow-
ing inequities for variables x and z that must be fulfilled
,0.00xxx 1
 
, and
 , where
12 ma
 x
If a = 0 и b = 0, and 0π
, the resulting surface is
a cylindrical surface (nanotube). The condition for a
Copyright © 2011 SciRes. JMP
Figure 9. A cylindrical helicoid with non-overlapped layers: φ1 = 0, φ2 = 6π, ρ0 = 3, H = 3.356, L = 6.44, b = 1.5, v = [3,1] (at
the left), and a scroll b = 0 (at the right).
Figure 10. A spiral growth of a nanotube: φ1 = 0, φ2 = 6π, ρ0 = 8.2869, H = 0, L = 6.44, b = 0, v = [3,1] (at the right), and a heli-
cal cylindrical spiral b = 1.5 (at the left).
seamless join of a carbon layers on such surface look as
follows: 0
kA [11]. In this case, if the tentative
value of cylinder’s radius 0
is known, its actual value
is found with
2πkA, where
2πk ceilA
The range of l1, l2 и ν integers in (24, 25) is calculated as
 
max maxsin 5
Lceil sbLa
 
, . 1, 2i111222
LlLLlLvm 
8. Polyhedric Models
Since each i-th atom on a surface of any model is sur-
rounded by other three nearest atoms, except ones lying
on the border, and is located in the center of curvilinear
triangle, it is always possible to find a point equidistant
from triangle’s vertexes by r0.
Let’s set up three arrays, each one calculated with a
separate program:
k(i) – coordination number (the number of nearest
atomes) for i-th atom; for a hexagonal grid it always
equals to three, except atoms lying on the border, where
k(i) = 1 or 2.
N(i,j) (j = 1,2,3) – the numbers of atoms nearest to i-th
atom in ξ(i), η(i) and ζ(i) arrays.
L(i,j) (j = 1,2,3) – three integer numbers l1, l2 and ν
identifying each i-th atom in (9); ν = 1 or 2 for the first
and the second oblique sub-grids of a carbon layer re-
spectively (Figure 1).
If ξ(i), η(i) and ζ(i) are atomic coordinates in a curved
atomic-bond approximation model, the problem of i-th
atom’s coordinate correction is reduced to solving a sys-
Copyright © 2011 SciRes. JMP
tem of three equation with three unknowns 11 1
one of sub-grids, for instance, when L(i,3) = 1.
,, 2
ij NijNij
  
r . 1,2, 3j
This system is resolved only for atoms having k(i) = 3,
because atoms on the border, where k(i) = 1 or 2, it is
undefined. Analyzing a coordination number k(i), we can
cast out border atoms and finally receive a polyhedric
model with pyramids and empty triangles interchanging
on its surface. All nearest distances of such model are
now equal to r0, and atoms are not located on one con-
tinuous surface. The system is solved in “matlab” with
fsolve operator, with the starting point set to ξ(i), η(i) и
Finally, the last problem to solve is coloring all faces
of polyhedric models. For i-th pyramid in a sub-grid with
ν = 1 faces with coordination numbers k(i) = 3 or 2 (since
for k(i) = 1 it doesn’t make sense) are colored the fol-
lowing way. From an array of coordinates 111
of a
polyhedric model a column vector
K is
formed (where ׳ is a transposition operation); and from
vertex numbers of triangular faces a vector V character-
izing, if a vertex belongs to the given face, is formed.
 
 
,3 ,;,,3 ,,1,
iN iN iiiN i
NiiiNiNi i
If k(i) = 2, the vector V possesses the value of
,,1,,2,iNiN ii 
So numbers of every pyramid’s vertex are associated
with its coordinates.
Now we can draw and color faces in “matlab” to, say,
red color with the following operator patch(‘Vertices’,
Coloring of empty triangles of a model is performed as
follows. At first, the quantity of nearest atoms distant
from the triangle’s center to r0 or lesser value k(i) is
found as well as their numbers N(i,j) and the array of
indices L(i,j). Now, the vector V for k(i) = 3 or 2 pos-
sesses the values:
  
,1, ,2, ,3Ni NiNi 
,1 ,,2Ni Ni 
Since triangle’s vertexes belong to another sub-grid,
the coloring is performed for L(i,3) = 2.
Figures 11-13 illustrate polyhedric models of a nano-
tube, a scroll and a nanocone.
9. Conclusions
Using an unfolding of a given surface to a plane algo-
rithm has been developed. The algorithm allows map-
Figure 11. The polyhedric model of a nanotube with the
diameter D = 8.47, the chirality angle β = 46.1° and the
roll up direction v = [4,3].
Figure 12. The polyhedric model of a scroll v = [3,1].
Figure 13. The polyhedric model of a nanocone δ = 38.94°.
Copyright © 2011 SciRes. JMP
ping a carbon grid to different types of helical surfaces,
including nanotubes, nanocones and a scroll. With the
models acquired, polyhedric models having their atomic
spacing constant and equal to 1.42Ǻ are built, and a al-
gorithm of coloring all faces of such models is developed.
The models can be used to visually imitate spiral or
helical growth of nanotubes, nanocones and nanofiber,
and also to calculate their physical characteristics and
compare them with experimental data.
Despite this work is dealing with mathematical mod-
eling of carbon nanostructures, it is worth saying a cou-
ple of words about the mechanism of their generation.
For nanotubes and nanocones such mechanism can be
briefly described as follows.
Since graphite has a flake structure, so it splits to
separate layers or packs of several layers when it is being
exposed to laser evaporation or to an arc discharge. Upon
loosing stability, layers may roll up to one of helix types:
conical and cylindrical. Then, an instantaneous join of
helixes along helical curves occurs. The further growth
of an object goes by means of joining atoms to the end of
a helix, just like as it does in the growth of crystals on a
helical dislocation.
The joining of helixes, as shown in this document,
must meet several join conditions. For nanotubes it is:
[11], where
is the identity
period in the roll up vector direction
v, D is
the diameter of a tube. For nanocones their opening an-
gle must be equal to δ = 19.19, 38.94, 60, 83.62 and
112.88°. These conditions for the seamless join are met
automatically, because atoms do not have other way of
joining. As a result, we obtain nanomaterials consisting
of nanotubes of different diameters and roll up direc-
Another way of nanotubes generation is discussed in
[13]. The idea is that carbon layers may roll up to a scroll
that may turn to a multi-shell nanotube after the disloca-
tion has entered a surface in a radial direction. The same
considerations can be applied for nanocones, except that
they roll up to a conical scroll. These considerations
seems reasonable, and not only do not contradict with
existing growth models, for example [1,10,13,21], but
also add to them, because the nature of nanoobject gene-
sis is multiform. Cylindrical and conical helixes can be
considered as intermediate (meta-stable) states a system
uses to reach its minimum.
10. References
[1] B. Ekşioğlu and A. Nadarajan, “Structural Analysis of
Conical Carbon Nanofibers,” Carbon, Vol. 44, No. 2,
2006, pp. 360-373. doi:10.1016/j.carbon.2005.07.007
[2] Y. Saito and T. Arima, “Features of Vapor-Grown
Cone-Shaped Graphitic Whiskers Deposited in the Cavi-
ties of Wood Cells,” Carbon, Vol. 45, No. 2, 2007, pp.
248-255. doi:10.1016/j.carbon.2006.10.002
[3] Ch.-T. Lin, W.-C. Chen, M.-Y. Yen, L.-S. Wang, C.-Y.
Lee, T.-S. Chin and T.-T. Chiu, “Cone-Stacked Carbon
Nanofibers with Cone Angle Increasing along the Longi-
tudinal Axis,” Carbon, Vol. 45, No. 2, 2007, pp. 411-415.
[4] A. D. Lueking, H. R. Gutierrez, D. A. Fonseca and E.
Dickey, “Structural Characterization of Exfoliated Graph-
ite Nanofibers,” Carbon, Vol. 45, No. 4, 2007, pp.
751-759. doi:10.1016/j.carbon.2006.11.023
[5] J. Vera-Agullo, H. Varela-Rizo, J. A. Conesa, A. Cristina,
M. César and M.-G. Ignacio, “Evidence for Growth
Mechanism and Helix-Spiral Cone Structure of
Stacked-Cup Carbon Nanofibers,” Carbon, Vol. 45, No.
14, 2007, pp. 2751-2758.
[6] M. H. Al-Saleh and U. Sundararaj, “A Review of Vapor
Grown Carbon Nanofiber/Polymer Conductive Compos-
ites,” Carbon, Vol. 47, No. 1, 2009, pp. 2-22.
[7] C.-W. Huang, H.-C. Wu, W.-H. Lin and Y.-Y. Li, “Tem-
perature Effect on the Formation of Catalysts for Growth
of Carbon Nanofibers,” Carbon, Vol. 47, No. 3, 2009, pp.
795-803. doi:10.1016/j.carbon.2008.11.033
[8] J. Zhao, L. Liu, Q. Guo, J. Shi, G. Zhai, J. Song and Z.
Liu, “Growth of Carbon Nanotubes on the Surface of
Carbon Fibers,” Carbon, Vol. 46, No. 2, 2008, pp.
380-383. doi:10.1016/j.carbon.2007.11.021
[9] L. Zhu, J. Xu, F. Xiao, H. Jiang, D. W. Hess and C. P.
Wong, “The Growth of Carbon Nanotube Stacks in the
Kinetics-Controlled Regime,” Carbon, Vol. 45, No. 2,
2007, pp. 344-348. doi:10.1016/j.carbon.2006.09.014
[10] I. Sumio, “Helical Microtubules of Graphitic Carbon,”
Nature, Vol. 354, No. 7, 1991, pp. 56-58.
[11] V. F. Pleshakov, “Geometry and X-Ray Diffraction Char-
acteristics of Carbon Nanotubes”, Crystallography Re-
ports, Vol. 54, No. 7, 2009, pp. 1230-1241.
[12] P. V. Fursikov and B. P. Tarasov, “Catalytic Synthesis
and Properties of Carbon Nanofibers and Nanotubes,”
International Scientific Journal for Alternative Energy
and Ecology ISJAEE, Vol. 18, No. 10, 2004, pp. 24-40.
[13] V. V. Pokropivny and A. V. Pokropivny, “Dislocation
Mechanism of Nanotube Formation,” Technical Physics
Letters, Vol. 29, No. 6, 2003, pp. 494-495 (Russian text
published in Pis’ma v Zhurnal Tekhnicheskoĭ Fiziki, Vol.
29, No. 12, 2003, pp. 21-24.
[14] O. V. Kibis and M. E. Portnoi, “Carbon Nanotubes: A
New Type of Emitter in the Terahertz Range,” Technical
Physics Letters, Vol. 31, No. 8, 2005, pp. 671-672 (Rus-
sian Text published in Pis’ma v Zhurnal Tekhnicheskoĭ
Fiziki, Vol. 31, No. 15, 2005, pp. 85-89.
[15] O. V. Kibis and M. E. Portnoi, “Semiconductor Nano-
helix in Electric Field: A Superlattice of the New Type,”
Technical Physics Letters, Vol. 33, No. 10, 2007,
Copyright © 2011 SciRes. JMP
Copyright © 2011 SciRes. JMP
878-880 (Russian Text published in Pisma v Zhurnal
Tekhnicheskoĭ Fiziki, Vol. 33, No. 20, 2007, pp. 57-63.
[16] O. V. Kibis, S. V. Malevannyy, L. Hugget, D. G. W.
Parfitt and M. E. Portnoi, “Superlattice Properties of
Helical Nanostructures in a Transverse Electric Field,”
Electromagnetics, Vol. 25, 2005, pp. 425-435.
[17] P. L. Chebyshev, “On the Cutting of Our Clothes,” Com-
plete works, Moscow, Fizmatgiz, Vol. 5, 1955, pp. 165-
[18] E. V. Popov, “Geometrical Modeling of Tent Fabric
Structures with the Stretched Grid Method,” Proceedings
of the 11 International Conference on Computer Graph-
ics&Vision GRAPHICON 2001, UNN, Nizhny Novgorod,
Russia, 2001, pp. 138-143.
[19] B. J. Cox and J. M. Hill, “Exact and Approximate Geo-
metric Parameters for Carbon Nanotubes Incorporating
Curvature,” Carbon, Vol. 45, No. 7, 2007, pp. 1453-1462.
[20] B. J. Cox and J. M. Hill, “Geometric Structure of Ul-
tra-Small Carbon Nanotubes,” Carbon, Vol. 46, No. 4,
2008, pp. 711-713. doi:10.1016/j.carbon.2007.12.011
[21] Y. E. Lozovik and A. M. Popov, “Formation and Growth
of Carbon Nanostructures: Fullerenes, Nanoparticles,
Nanotubes and Cones,” Physics-Uspekhi (Advances in
Physical Sciences), Vol. 40, 1997, pp. 717-737 (Russian
Text published in Uspekhi Fizicheskikh Nauk, Vol. 167,
No. 7, 1997, pp. 751-774.