The temperature effect on
buckyball bond length.
Kupás Levente
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Abstract.
In this paper we want to get an answer for the question: how to operate the
temperature on the carbon atoms in soccer-ball-molecule, and how to alternate of this
bond length by influence of atoms flickering. We use the computational modelling to
obtain the answer for the previous question. We demand in computational modelling
the helps of molecular dynamics laws.
In this computational model we will build a global soccer-ball-molecule from
sixty pieces of carbon atoms. The carbon atoms are placed in a fixed order in the three
dimensional space, as follows we creating a soccer-ball-molecule. We let alone the
system and wait to see what processe are jog along between the carbon atoms. To
change the temperature we are able to draw a conclusion about the bigness of bond
lenght what to incline the atoms to stay together.
1. Introduction.
In recent years, properties and structures of nanosize materials have attracted a great
deal of attention; because of their unique properties and small dimensionality they
have a promising future for applications. Since the discovery of carbon fullerenes, a
large body of theoretical and experimental work has been focused on the
characterization of these materials. Significant progress has been made toward
understanding the properties and structures of these nanoscale materials. A typical
nanoscale system has from tens to millions of atoms. From a computational point of
view, the first principle calculations are feasible only for small-size systems; it is
prohibitively expensive to carry out calculations using quantum mechanics for large
systems (>100 atoms). In this case, classical molecular dynamics simulation is the
widely used theoretical method to investigate various mechanical, thermodynamic,
and even chemical properties of these systems. The reliability of molecular dynamics
simulation results is mainly determined by the interaction potentials (force fields)
used in these calculations. It is crucial to know the validation regime of a certain
potential function. Fullerene molecules and carbon clusters (and their crystals) are of
particular interest because of their unique structure and properties. If we want to
understand something about the formation of fullerenes, in the sense of understanding
their constitution out of elements, then an understanding of the chemical bond is
necessary. Fullerenes are complexes of atoms and ions (electrically charged atoms or
groups of atoms), these are connected to each other in the molecules by chemical
bonds. Some of those bonds are covalent bonds.
2. The structure of Buckminsterfullerene.
On the basis of the information provided by mass spectrometry, Kroto, Smalley, and
Coworkers were faced with the problem of rationalizing the unusual stability of 60
carbon atoms bound together to make a molecule of buckyball.
The fundamental issue concerned how to construct a molecular structure that satisfied
normal bonding (four bonds per carbon atom) and comprised exactly 60 atoms of
carbon. With a great leap of both insight and faith, as well as considerable influence
from the geodesic dome structures designed by Buckminster Fuller, these
collaborators proposed that C60 adopts an arrangement of carbon atoms that is similar
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to the stitching on a soccer ball. A soccer ball has
both 6-membered (hexagonal) and 5-membered
(pentagonal) patches. These are sewn together to
make a round ball. Taken in by the beautiful
symmetry of such structures, Kroto and Smalley
conjectured that if 5-membered and 6-membered
rings of carbon were placed together in the same
pattern then a round molecule containing 60
carbon atoms would result. This structure can be
reconciled with the tendency of carbon to form
four bonds if one assumes that each C engaged in
one double bond and two single bonds with the
neighbouring carbon atoms. Overall, this results
in a structure that has patterns of alternating single and double bonds as one traces the
C-C bonding framework. Each carbon lies at the vertex of fused 5- and 6-membered
rings.
3. The stability of Buckminsterfullerene.
The alternation of single and double bonds in a molecule has been found to correlate
with unexpected stability in molecules closely related to C60. By stability we mean
that the forces, or bonds, holding the carbon atoms close together are stronger than
one might have expected from comparison with simpler, related materials. Consider
the example of C6H6. Various experimental measurements indicate that benzene
(C6H6) contains six carbon atoms arranged at the vertices of a hexagon. Attached to
each carbon atom by a single bond is a hydrogen atom.
One can satisfy the bonding rules for carbon and hydrogen by (1) joining each
hydrogen to a carbon atom with a single bond and (2) arranging bonds between the
carbon atoms in an alternating single-double-single pattern as shown in the figure.
Two arrangements of C-C double bonds in benzene are possible; the two structures
differ only in the positioning of the double and single bonds. Although each of the
two structures implies alternation of C-C bond distances, experimentally it is found
that all of the C-C distances are equal. Furthermore, benzene is found to be more
stable than is expected for a molecules that has C-C single bonds and C=C double
bonds. The American chemist, Linus Pauling, suggested a way to resolve this
dilemma: benzene is best considered to be both structures simultaneously. In the
language originated by Pauling, modern chemists state that the best description of
benzene is a superposition of the two resonance structures shown above.
2
By resonance we mean that benzene is at all times a mixture of the two limiting
structures. This is subtly, but importantly, different from saying that benzene is one
structure half of the time and the other structure the rest of the time. The best
description of benzene is that is always half of each structure. Molecules that can
engage in this type of bonding resonance are found to have more stability than
molecules that do not. Thus, we say that the effect of resonance is to (1) even out the
alternation of bond lengths and (2) add stability to the molecule. A general postulate
of Pauling's resonance theory is that the greater the number of resonance forms, the
greater the stability of the material. The special stabilization of C-C bonds associated
with resonance in benzene is manifested in the C-C bond lengths.
An average C-C single bond is 1.54 Å in length, where as an average C-C double
bond length is 1.34 Å. Based on the two resonance structures for benzene one might
approximate each bond to be equivalent to 3/2 of a bond (the average of 1 and 2). This
would lead to a predicted bond length of about 1.44 Å. The experimental C-C bond
length is observed to be 1.39 Å, illustrating that the forces of attraction between
carbon atoms in benzene are unexpectedly strong.
We can extend the concept of resonance to understand the unusual stability of
C60. Just as different patterns of alternating single and double bonds can be drawn for
benzene, different resonance structures can be drawn for C60. Thus, by arranging the
carbon atoms in hexagonal and pentagonal rings, we find that a nearly spherical
molecule with substantial resonance stabilization can be formed from 60 carbon
atoms. These features are consistent with the properties observed for buckyball. For
example, by analogy to benzene we would expect the average C-C bond distances in
buckyball that is less than 1.44 Å.
4. Theory for the simulation.
In Molecular Dynamics Simulations, Newton’s equations of motion (Euler’s
equations for rigid molecules) are solved in order to generate new configurations.
Sampling occurs in the whole phase space, along a continuous path. Consequently,
MD simulations allow for time-dependent properties to be investigated. Relativistic
effects are not considered and quantum effects, if necessary, can be incorporated as
semi classical corrections.
4.1. Equations of motion for MD simulations
The classical MD simulations boil down to numerically integrating Newton’s
equations of motion for the particles (atoms, in the simplest case) which build up the
investigated system:
d2 i
m
 i( 1 2
N ), i=1,2,…,N
2
dt
r F r , r ,...r
Here ri are the position vectors and Fi are the forces acting upon the N particles in the
system.
In our case, in the absence of external forces, the potential can be represented in the
simplest case as a sum of pair wise interactions:
N N
U    u ( rij ),
i 1 j  i
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where rij = ri − rj , rij ≡ |rij |, and the condition j > i prevents the double counting
of the particle pairs. The forces acting on the particles are composed in such a case of
the individual interactions with the rest of the particles:
N
du( rij ) ij
Fi   fij ,
j i
f ij  
.
drij
r
rij
According to Newton’s third law, fji = −fij. The computational effort to solve the set
of equations of motion is proportional to N 2 and is mainly associated with the
evaluation of forces. Therefore, for tractable computations the forces should be
expressed analytically. To further reduce the computational effort the potential can be
cut off at some limiting separation (for rij > rcut) beyond which the potential
becomes negligible.
4.2. The Lennard-Jones potential.
One of the most famous pair potentials for van der Waals systems is the
Lennard-Jones potential:
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6


u LJ ( rij )  4 
 rij







 rij








The parameter ε governs the strength of the interaction and σ defines a length scale.
This potential is strongly repulsive at short distances and it passes through 0 at
rij   , u LJ ( )  0 . The parameters ε and σ are chosen to fit certain physical
properties of the system.
12
The term proportional to rij , dominating at short distances, models the repulsion
due to the non-bonded overlap of the electronic orbitals. The attractive term
6
proportional to rij , dominates at large distances and it models the van der Waals
dispersion forces caused by the dipole-dipole interactions due to fluctuating dipoles.
These weak forces provide the bonding character of closed-shell systems.
The Lennard-Jones potential is totally inadequate for open shell systems, in
which strong localized bonds are formed.
The inter-particle forces arising from the Lennard-Jones potential have the form:
f ij 
48
rij2
  12 1   6 
 rij   2  rij   rij


4.3. Periodic boundary conditions.
The behaviour of finite systems is very different from that of infinite systems. Unless
we want to simulate clusters of atoms or molecules, the number of particles used to
simulate bulk properties of macroscopic systems plays an essential role. In
macroscopic systems, only a small fraction of the atoms are located in the vicinity of
the boundaries.
Using periodic boundary conditions implies that particles are enclosed in a
box, which is mentally replicated to infinity by rigid translation in all the three
Cartesian directions, completely filling the space. All the “image” particles move
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solidary with their ”original” particle from the simulated box and, in fact, only one of
them is effectively simulated. When a particle enters or leaves the simulation region,
an image particle leaves or enters this region, such that the number of particles from
the simulation region is always conserved. One can easily see that the surface effects
are thus virtually eliminated and the position of the box boundaries plays no role.
4.4. Calculating the interactions.
As a result of using periodic boundary conditions, each particle i in the simulation
box appears to be interacting not only with the other particles in the box, but also with
their images. Apparently, the number of interacting pairs increases enormously.
Nevertheless, this inconvenience can be overcome if one uses a potential with a finite
range, since the interaction of two particles separated by a distance exceeding a cut
off distance Rcut can be ignored. Supposing that the box size is larger than 2Rcut
along each Cartesian direction, it is obvious that at most one among the pairs formed
by a particle i in the box and all the periodic images of another particle j will lie
within the cut off distance Rcut and, thus, will interact. This is the essence of the
minimum image criterion: among all images of a particle, consider only the closest
and neglect the rest.
4.5 Vibrational spectra of clusters.
In order to characterize the overall vibration of a cluster, the velocity autocorrelation
function may be employed:
N
(t ). i (0)
i
1 i
Cv (t ) 
,
N
2
 ( 0) 
i 1 i
where vi(t) is the velocity of atom i at time t. This is basically a normalized
projection of the atomic velocity pattern at a particular time t onto the initial pattern.
An alternative to Cv(t) is the displacement autocorrelation function:
N
 i (t ). i (0)
i

1
Cr (t ) 
,
N
2
 (0)
i 1 i
with δri(t) representing the displacement of atom i at time t. If the initial preparation
of the vibrational state is to obey some predefined pattern, the latter autocorrelation
function may be sometimes more convenient. Aside from this aspect, the information
contained in the two autocorrelation functions is absolutely equivalent.
The vibration of molecules with an inversion point can be additionally characterized
by the parity, defined as:
N
 I
sym( I )
I 1
 v v
 v
 r r
 r
P

r .r
N
 I 1rI 2
,
where sym(I) designates the symmetric atom for atom I with respect to the inversion
point.
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6. The Simulation.
The simulation program by the helps of methods presented above, are able to render
answer for the bigness of bond length between the carbon atoms.
The carbon atoms are placed in a fixed order in the three dimensional space.
The starting location of carbon atoms are same in any temperature.
To send up the temperature, each atoms increasingly starting to shake fervently
around the starting location. On account of this theoritically the difference between
the maximum and the minimum bond length more in the higher temperature.
Starting from the 0.1K the temperature was extend every time with 100K in four steps.
With analyse the system in every 0.05 ps, the next data’s come to out:
0.1 K
Bmin
1.4016
1.4033
1.4030
1.4031
1.4031
1.4031
1.4031
Bmax
1.4027
1.4035
1.4030
1.4031
1.4031
1.4031
1.4031
100 K
Bmin
1.3981
1.4022
1.4027
1.4030
1.4030
1.4031
1.4031
Bmax
1.4057
1.4042
1.4033
1.4032
1.4031
1.4031
1.4031
200 K
Bmin
1.3981
1.4017
1.4027
1.4030
1.4030
1.4031
1.4031
Bmax
1.4076
1.4049
1.4034
1.4032
1.4031
1.4031
1.4031
300 K
Bmin
1.3966
1.4009
1.4026
1.4029
1.4030
1.4031
1.4031
Bmax
1.4092
1.4049
1.4035
1.4032
1.4031
1.4031
1.4031
400 K
Bmin
1.3958
1.4012
1.4026
1.4029
1.4030
1.4031
1.4031
Bmax
1.4088
1.4056
1.4037
1.4032
1.4031
1.4031
1.4031
Time
[ps]
0.05
0.1
0.15
0.2
0.25
0.3
0.35
Where: Bmin , Bmax - minimum and maximum bond length.
To heat away the start state of a buckyball molecule the bond length difference
even more to come out. Now we analyzing from the 800K to 1000.05K.
800 K
Bmin
1.3926
1.4001
1.4025
1.4028
1.4030
1.4031
1.4031
Bmax
1.4152
1.4068
1.4040
1.4033
1.4032
1.4031
1.4031
1000 K
Bmin
1.3868
1.3996
1.4018
1.4028
1.4030
1.4031
1.4031
1000.05 K
Bmax
1.4168
1.4066
1.4039
1.4033
1.4032
1.4031
1.4031
Bmin
1.3878
1.3368
1.3376
1.3130
1.2856
1.2946
1.3206
Bmax
1.9280
1.9274
1.9761
1.9793
1.9973
1.9955
1.9732
Time
[ps]
0.05
0.1
0.15
0.2
0.25
0.3
0.7
7. Conclusions.
From the data’s presented above we are able to conclude as the theory applied in this
simulation was true.
Truly, the rise of temperature was eventuated bigger distance between the maximum
and minimum bond length.
In the first table the flicker of carbon molecule not to result considerable
aberration between the maximum and minimum bond length. Then again in the
second table we can observe the cataclysm of soccer-ball-molecule. From the big
temperature the deflexion of carbon atoms to snatch off the coupling force in the
soccer-ball-molecule.
By analogy to benzene we must assumed that the average C-C bond distances
in buckyball is less than 1.44 Å, and the simulation issue bond length is observed to
be 1.4031 Å.
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References.
[1] Titus Beu Courses: A3, C2, C3.
[2] M.S. Dresselhaus: Science of Fullerenes (1996).
[3] B.I. Yakobson: For a review of the field of fullerenes.
[4] William A Goddard: bond order potential.
[5] G. Wildöer, C. Venema: The chemical bond.
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