FA
CO
ACULTY
OM
A
NA
BABEOF CHE
PUT
ARM
ANO
MIKLÓ
Ş‐BOLEMISTRY
TATI
MCHA
TUB
PhD TH
ÓS (că
CLU
LYAI UY AND C
ONA
AIR C
BE JU
HESIS ABS
ăs. NA
UJ‐NAPO
2011
UNIVECHEMIC
AL S
CAR
UNC
STRACT
AGY) K
Prof. Dr
OCA
RSITYAL ENG
STUD
RBON
CTIO
Katalin
Scien
r. Mirce
Y INEERIN
DY O
N
ONS
n
ntific adv
ea V. DIU
NG
OF
visor:
UDEA
Preside
Reviewe
nt
ers
Conf.
Prof. D
Assoc
Assoc
BABEŞ‐BO
Faculty o
Organic C
Arany J. s
ROMANI
Dr. Cornelia
Dr. Titus BE
. Prof. Dr. Is
. Prof. Dr. M
Pu
OLYAI Unive
of Chemistry
Chemistry D
street, no. 1
A
Scie
Prof. Dr.
Thes
a MAJDIK
U
stván László
Mihai MEDE
ublic def
ersity
and Chemic
epartment
1, Cluj‐Napo
entific adv
MIRCEA V
sis commi
ó
ELEANU
ense: Jul
cal Engineeri
oca, 400084
visor:
V. DIUDEA
ittee:
BABEŞ‐BOL
BABEŞ‐BOL
Budapest UEconomics
POLITEHNI
ly 15, 20
ing
A
LYAI Univer
LYAI Univer
University o, Budapest,
CA Univers
11
sity, Cluj‐Na
sity, Cluj‐Na
of Technolog, Hungary
ity Timisoar
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3
Tableofcontents
Chapter1.Introduction.........................................................................................................3
1.1.TheEulerrelationship.........................................................................................................................6
1.2.Singlewallcarbonnanotubes...........................................................................................................8
1.2.1. Chiral Vector Ch ...................................................................................................... 10
1.2.2. Translational Vector T ............................................................................................ 12
1.3.Computationaldetails.......................................................................................................................14
1.4.Mapoperations....................................................................................................................................15
1.4.1. The Leapfrog Le transformation ............................................................................. 15
1.4.2. The Quadrupling Q transformation ........................................................................ 16
1.4.3. The Septupling S transformation ........................................................................... 17
Chapter2.Multiterminalarmchairnanotubejunctions.......................................19
2.1.Tetrahedralgraphiticjunctions....................................................................................................20
2.1.1. Tetrahedral fullerenes ............................................................................................ 20
2.1.2. Classes of tetrahedral fullerenes ............................................................................ 26
2.1.3. Construction of tetrahedral junctions .................................................................... 29
2.1.4. Structure of tetrahedral junctions ......................................................................... 32
2.1.5. Stability of tetrahedral junctions ........................................................................... 36
2.2.Octahedralgraphiticjunctions......................................................................................................39
2.2.1. Construction of octahedral junctions..................................................................... 39
2.2.2. Structure of octahedral junctions .......................................................................... 41
2.2.3. Stability of octahedral junctions ............................................................................ 44
2.3.Tetrahedraljunctionswithattachednanotubes...................................................................46
2.4.MultiterminaljunctionsfromC60fullerene.............................................................................49
2.5.Conclusions............................................................................................................................................51
4
Chapter3.ArmchaircarbonnanotubeYjunctions................................................52
3.1.StructuralmodelsofYjunctions..................................................................................................53
3.2.GrowthmethodsofYjunctions....................................................................................................58
3.2.1. Electric arc .............................................................................................................. 58
3.2.2. Catalytic growth ..................................................................................................... 58
3.2.3. Growth using templates ......................................................................................... 62
3.2.4. Growth by fullerene decomposition ...................................................................... 63
3.2.5. Nano‐welding of crossing tubes ............................................................................. 64
3.3.ComputationalstudyofarmchairYj(n,n)junctions............................................................66
3.3.1. Opened symmetrical armchair Yj(n,n) junctions ................................................... 67
3.3.2. Symmetric capped Yj(n,n) armchair junctions ....................................................... 75
3.3.3. Closed asymmetric Yj(n,n) armchair junctions ...................................................... 81
3.4.Conclusions............................................................................................................................................83
Chapter4.Characterizationofnanoporouscarbonnetworks..........................84
4.1.TheOmegapolynomial.....................................................................................................................85
4.2.Topologicaldescriptionofnanodendrimers.........................................................................89
4.3.TopologicaldescriptionofnetworksbuiltfromP‐typejunctions.................................93
4.4.Conclusions............................................................................................................................................98
References................................................................................................................................99
KEYWORDS: carbon nanotube junctions, fullerenes, molecular modeling, DFT, PM6,
counting polynomial, IPMS
5
Chapter1.Introduction
1.3.Computationaldetails
Starting geometries of fullerenes were obtained using the CaGe [47, 48] software
package, while the geometries of the junctions were built manually or by the aid of the
Nanos Studio [49] software package (used for attaching nanotubes). Full geometry
optimizations with and without symmetry constraints were performed using the PM3 [50]
and PM6 [51] semiempirical methods.
Fullerenes, tetrahedral and octahedral junctions were re‐optimized at the DFT level
of theory using the popular B3LYP [52] exchange‐correlation density functional (Becke 88
exchange functional [53] and the correlation functional of Lee, Yang and Parr [54]). Although
the structures were also optimized with smaller basis sets (3‐21G* and 4‐31G) in the present
work only the single point calculation results obtained with the 6‐31G(d,p) [55, 56] basis set
are presented.
Y‐junctions only at the semiempirical level were studied because of the size of the
molecule.
Geometry optimizations performed with and without symmetry constraints and
resulted in identical structures. Vibrational frequencies were calculated at the same
theoretical level at which the optimization was done, to ensure that a true minima was
found.
All calculations were performed using Gaussian 09 package [57].
Omega polynomials were computed using the Nano Studio package [49]. Map
transformations of selected structures was performed using the CVNet program [58].
2
2.1.2.
T
appears
Each of
other (p
distance
one hex
howeve
Some o
just one
f
A
While s
accordin
2.1.Tetr
Classeso
Tetrahedra
s in the tetr
f these frag
pentagon t
e, while in p
xagon.
It is obviou
er the small
of the fuller
e class, as is
A
Figure 12 Sfragment is o
All tetrahed
some of th
ng to the st
rahedra
oftetrahe
l fullerenes
rahedral ori
gments con
riple), in pa
pattern C th
us that ful
ler member
enes have
s the case of
tructural paoriented alo
ral fulleren
e fullerene
tructural rel
algraph
edralfull
s can be cl
ientation (a
tain three
attern B th
hey are con
lerenes of
rs of the ot
pentagonal
f fullerenes
atterns that ng the C3 sym
es with the
es show mo
lationship to
hiticjunc
lerenes
lassified ac
along the C3
pentagons:
ey share th
nnected to t
type A do
ther fullere
l arrangeme
C44 (T), C52
B
appear in mmetry axis.
number of
ore than o
o the tetrah
ctions
ccording to
3 symmetry
: in pattern
he central h
the central
o not obey
ne classes a
ents such t
(T) and C76
tetrahedral .
f atoms sma
ne structur
hedral junct
the struct
axis) as de
n A they ar
hexagon an
atom and t
y the isolat
also have a
hat they be
(Td).
fullerenes.
aller than 12
ral pattern,
tion (see be
ural fragme
picted in Fi
e adjacent
nd are at on
they share
ted pentag
djacent pen
elong to mo
C
The center
20 were ge
, they were
elow).
6
ent that
gure 12.
to each
ne bond
pairwise
on rule,
ntagons.
ore than
of each
nerated.
e sorted
2.1.4.
a
Structur
Tj84 (T)
Tj112 (T)
T
Figure 20 Teaxis from the
reoftetra
Tj52 (T)
Tj1484‐ (T)
etrahedral jue top.
ahedralj
nctions Tj(3,
unctions
Tj100 (Td)
Tj124 (T)
3) with open
s
ning chirality
Tj64 (Td)
Tj76 (T)
y (3,3) viewe
)
Tj108 (Td)
Tj144 (Td)
d along the t
7
threefold
8
As it was previously presented, all possible fullerenes with tetrahedral symmetry and
a number of atoms smaller than 120 were generated and classified according to the
structural pattern they present. From the fullerenes belonging to class B tetrahedral
junctions with armchair openings were built by homeomorph transformation of the three
edges which are shared between the core hexagon and the pentagons in the structural
fragment.
Attempts to build tetrahedral junctions from the other tetrahedral fullerenes (class A
and C) resulted in the same structures that were obtained from the class B fullerenes. The
correspondence which was found between the junctions and fullerenes is summarized in
Table 3. It is obvious that the structures from a row of Table 3 can be transformed between
each other by modifying (inclusion/extrusion of carbon atoms) the topology of the structural
fragment.
Notice that in the last column no structure is given in the first two rows. This is
because although there is a corresponding structure, it is not a classical fullerene (the
number of atoms is smaller than 20). It must be mentioned that two structures have
icosahedral symmetry (C60 and C80), however they fit into the series not just by their
structure but also energetically (see below).
9
Table 3 The correspondence between the tetrahedral junctions Tj(3,3) and the tetrahedral fullerenes from each class.
# Tetrahedral junctions*
Tjn‐TU[3,3] / f7 = 12
Tetrahedral fullerenes
Cn‐8 / pattern A Cn‐24 / pattern B Cn‐32 / pattern C
1 Tj524‐ (T) / (D2) C44 (T) C28
4‐ (Td) –
2 Tj644‐ (Td) / (D2d) C56 (Td) C40
4‐ (Td) –
3 Tj764‐ (T) / (D2) C68 (T) C52
4‐ (T) C44 (T)
4 Tj84 (T) C764‐ (T) C60 (Ih) C52
4‐ (T)
5 Tj1004‐ (Td) / (D2d) C92 (Td) C76
4‐ (Td) C684‐ (Td)
6 Tj108 (Td) C1004‐ (Td) C84 (Td) C76
4‐ (Td)
7 Tj1124‐ (T) / (D2) C104 (T) C88
4‐ (T) C806‐ (Ih)
8 Tj1244‐ (T) / (D2) C116 (T) C100
4‐ (T) C92 (T)
9 Tj144 (Td) C1364‐ (Td) C120 (Td) C112
4‐ (Td)
10 Tj1484‐ (T) / (D2) C140 (T) C124
4‐ (T) C116 (T)
*Symmetry of the negatively charged tetrahedral junction is given first, and second the symmetry of the neutral molecule (singlet state is
considered).
2.1.5.
A
The jun
were a
Optimiz
and D2 f
f3
Stability
All structur
nctions wer
dded as re
zation of th
for the stru
Figure 21 Plfunction of t31G(d,p) sin
yoftetrah
es were op
re optimize
equired to
ese structu
ctures havi
ot of the totthe total enegle point ene
hedralju
timized at t
d in hydro
be able t
ures in a ne
ng Td and T
tal energy (aergy (a.u.) of ergy calculat
unctions
the B3LYP/6
genated fo
to optimize
eutral form
T topologica
a.u.) the fullthe tetrahedtions.
6‐31G(d,p)
orm. In case
e with the
lead to a lo
l symmetry
erenes with dral junction
level of the
e of some
e highest p
ower symm
y, respective
structural pns Tj(3,3) obt
eory in singl
structures
possible sym
metry struct
ely.
pattern A, B tained from
10
et state.
charges
mmetry.
ture: D2d
and C as B3LYP/6‐
11
In Figure 21 the total energy (Etot in a.u.) of fullerenes from each class is plotted as
function of the total energy (Etot in a.u.) of the corresponding tetrahedral junction. Statistical
data confirms that there is a perfect linear correlation between the total energies. This
concludes that a relationship exists between the junction and the parent fullerene.
Table 4 Single point energy results (total energy Etot and gap energy Egap) obtained at the B3LYP/6‐31G(d,p) level of theory for the neutral and negatively charged tetrahedral junctions. For a better estimation the total energy difference ΔEtot is given in kcal/mol.
Etot (au) ΔEtot
(kcal/mol)
Egap (eV) ΔEgap
(eV) neutral charged (4‐) neutral charged (4‐)
Tj52 (T) / (D2) ‐1995.456 ‐1994.957 ‐313.249 1.433 2.167 ‐0.734
Tj64 (Td) / (D2d) ‐2452.628 ‐2452.215 ‐259.077 1.291 2.199 ‐0.908
Tj76 (T) / (D2) ‐2910.149 ‐2909.807 ‐214.850 1.260 2.330 ‐1.070
Tj100 (Td) / (D2d) ‐3824.692 ‐3824.371 ‐201.644 0.976 1.104 ‐0.128
Tj112 (T) / (D2) ‐4282.120 ‐4281.851 ‐168.892 0.905 1.091 ‐0.187
The stability of the neutral tetrahedral junctions with lowered symmetry (D2d and D2)
and the negatively charged ones were evaluated at the B3LYP/6‐31G(d,p) level of theory and
are compared in Table 4. The energy difference ΔEtot (presented in kcal/mol) shows that the
neutral structures have a lower energy, the energy gain however decreases with the
increase of the structure’s size. Therefore from a thermodynamic point of view the neutral
junctions show a better stability. The gap energy as a measure of kinetic stability gives a
reversed order of the structures, as the last column shows in Table 4 the charged tetrahedral
junctions have a larger gap. This stability difference is more pronounced in the case of the
first three junctions, where ΔEgap is almost one eV.
The plot in Figure 22 presents the total energy per number of carbon atoms (Etot/N in
au) and the gap energy (Egap in au) of the tetrahedral junctions as function of the number of
carbon atoms. It can be seen that the energy increases with the size of the system, although
after a given size (Tj100) the decrease in stability is slightly smaller.
T
than th
gap. Ho
no cert
tetrahe
junction
a3
The neutra
e charged s
owever the f
ain rule of
dral junctio
n resulted fr
Figure 22 Ploa.u. and gap31G(d,p) sin
l junctions s
structures,
first three c
stability, ju
ons Tj524‐ (T)
rom the C60
ot of the tetrp energy in agle point ene
show a bett
especially t
charged jun
udging from
and Tj64 (Td
0 fullerene s
rahedral junca.u. as functergy calculat
ter kinetic s
the large ch
nctions also
m the result
Td) are good
shows the b
ctions Tj(3,3ion of numbtions.
stability, ha
harged struc
have a rela
ts it can be
candidates
best kinetic
) total energber of carbon
ving a muc
ctures have
atively high
e concluded
s for synthe
stability.
gy per numben atoms obt
h larger ene
e a very low
gap. While
d that the f
esis. Notice
er of carbon tained from
12
ergy gap
w energy
e there is
first two
that the
atoms in B3LYP/6‐
2
2.2.2.
2.2.Octa
Structur
C152 (O)
C216 (Oh)
Figure 24 Th
ahedral
reofocta
C104 (O)
)
C288 (Oh)
e octahedra
lgraphi
ahedralju
l junctions w
iticjunct
unctions
C168 (Oh)
C224 (O)
with (4,4) ope
tions
ening chiralit
C128 (Oh
C296 (O)
ty viewed alo
)
C200 (Oh)
C248 (O)
)
ong the four
13
rfold axis.
2.2.3.
T
the octa
in the
junction
differen
the cas
kinetic s
a3
Stability
The plot in
ahedral jun
variation o
ns, the ener
nt stability o
e of tetrah
stability.
Figure 25 Ploa.u. and gap31G(d,p) sin
yofoctah
Figure 25
ctions as a
of the tota
rgy rises wi
ordering. Th
hedral junc
ot of the octp energy in agle point ene
hedraljun
presents th
function of
al energy a
ith the size
he neutral j
tions, the
ahedral junca.u. as functergy calculat
nctions
he variation
f the numbe
as previous
of the syst
unctions ha
smallest m
ctions Oj(4,4)ion of numbtions.
n of the tot
er of carbon
ly observe
tem. The ga
ave the high
members of
) total energber of carbon
tal energy p
n atoms. A
d in the c
ap energy h
hest gap en
these stru
gy per numben atoms obt
per carbon
similar tren
case of tet
however pr
ergy. In con
uctures hav
er of carbon tained from
14
atom of
nd exists
rahedral
esents a
ntrast to
ve a low
atoms in B3LYP/6‐
t
T
the tota
relation
betwee
2, while
twice th
number
Figure 26 Plthe total eneB3LYP/6‐31G
The plot of
al energy (a
nship betwe
n their ene
e the interc
he energy o
r of structur
ot of the totergy of the cG(d,p) single
f the total e
a.u.) of tet
een the tw
rgies (r valu
cept is almo
of the tetra
ral fragmen
tal energy ocorrespondinpoint energ
energy (a.u
rahedral ju
o set of st
ue equals 1
ost zero, me
ahedral one
nts is also tw
f octahedralng tetrahedray calculation
.) of the oc
nctions Tj(3
ructures (F
), and it mu
eaning that
es. This is in
wice in case
l junctions Oal junctions Tns.
ctahedral ju
3,3) reflects
igure 26). T
ust be noted
t the energy
n good agre
of the octa
Oj(4,4) (Etot inTj(3,3) (Etot in
unctions Oj(
s that there
There is a
d that the s
y of the oct
eement wit
ahedral junc
n a.u.) as fun a.u.) obtain
(4,4) as fun
e is a stron
perfect cor
slope equal
tahedral jun
th the fact
ctions.
15
nction of ned from
nction of
ng linear
rrelation
s almost
nction is
that the
2
T
[71], fo
calculat
length.
T
(Egap in a
Tj100 an
structur
T
while th
the stab
that in t
T
also the
eo
2.3.Tetr
To study th
our series
ting the tota
The plot of
a.u.) is pres
nd Tj108, re
res were co
The results
he total ene
bility decrea
the case of
The neutra
ere is only a
Figure 27 Plenergy (Egap of the numb
rahedra
he effect of
of tetrahe
al energy a
the total e
sented in Fig
espectively.
ompared.
show that
ergy per nu
ase is confi
Tj52 junctio
l junctions
slight decr
lot of the toin a.u.) of th
ber of atoms,
aljuncti
f the attach
edral juncti
nd HOMO‐
nergy per n
gure 27 for
In case o
in all cases
umber of at
rmed both
n set the to
have a muc
ease along
otal energy he series of n, obtained fr
onswith
hed nanotub
ions (3,3)
LUMO gap
number of a
the set of s
f two serie
s the gap en
toms increa
kinetically
otal energy f
ch larger ga
the series.
per numberneutral and com PM6 sing
hattach
be length o
with open
energy for
atoms (Etot/
structures w
es (Tj52 an
nergy decre
ases with th
and thermo
first slightly
ap energy co
r of atoms (charged tetrgle point ene
hednano
on the stab
ends wer
structures
/N in kcal/m
with the cor
d Tj100) ne
eases in an
he size of t
odynamical
y decreases
ompared to
(Etot/N in kcahedral juncergy calculat
otubes
ility of the
re investig
with differe
mol) and gap
re junction T
eutral and
oscillating
the nanotub
lly. Notice h
within the
o the charge
cal/mol) andctions Tj52 asions.
16
junction
ated by
ent tube
p energy
Tj52, Tj84,
charged
manner,
be. Thus
however
series.
ed ones,
the gap s function
2
S
where t
T
length i
small. T
termina
oscillati
are the
j
2.4.Mul
Starting fro
the length o
The total e
in all four s
The stability
als), thus the
Figure 33
ng manner
four‐termin
Figure 32 Pljunctions winanotube le
ltitermin
om the C60
of the attach
energy per
series, how
y increases
e tetrahedr
presents th
r decreases
nal junction
ot of the totith one to fongth, obtain
naljunc
fullerene s
hed armcha
atom (pre
wever after
with the n
ral junctions
he gap ene
with the st
ns with the h
tal energy pour arms bued from PM
ctionsfr
set of junct
air nanotub
sented in
a certain tu
number of
s are the mo
ergy as fun
tructure siz
highest HOM
per number ouilt from the3 single poin
romC60f
tions with
e was varie
Figure 32)
ube length
the attache
ost stable o
nction of t
ze, once aga
MO‐LUMO
of atoms (Ete C60‐Ih fullernt energy cal
fulleren
one to fou
ed [71].
increases w
the energy
ed nanotub
one.
he tube le
ain the mos
gap values.
ot/N in kcal/rene as funcculations.
e
ur arms we
with the n
y incremen
bes (the nu
ength, whic
st stable st
.
/mol) of the ction of the
17
ere built,
anotube
t is very
mber of
ch in an
ructures
series of attached
2
T
the cor
series o
junction
correlat
the ene
junction
T
stability
opening
junction
Figure 33 Plobuilt from thPM3 single p
2.5.Con
Tetrahedra
rrespondenc
of octahedr
ns. All mole
tion was fou
ergy per at
ns.
The increas
y of the stru
gs has a st
ns. The stab
ot of the gaphe C60‐Ih fullpoint energy
nclusions
l fullerenes
ce with the
ral junction
ecules were
und to exist
tom increas
se in the le
ucture conf
trong effec
bility increas
p energy (Eglerene as fu calculations
s
s were gene
e derived t
ns were bui
e optimized
t between t
ses with th
ength of th
firmed both
ct on the s
ses with the
ap in eV) of tnction of ths.
erated and
tetrahedral
ilt using str
d at the B3
he total en
he junction
he attached
h kinetically
stability as
e number o
the series ofhe attached
classified b
junctions.
ructural fra
3LYP/6‐31G
ergies of th
size. Kinet
d armchair
y and therm
compared
f terminals.
f junctions wnanotube le
by the struc
A structur
agments fro
(d,p) level
he molecule
tic stability
(3,3) nanot
modynamica
within set
.
with one to fength, obtain
ctural fragm
rally corres
om the tet
of theory.
s. It was fou
favors the
tube decrea
ally. The nu
ts of multit
18
our arms ned from
ment and
sponding
rahedral
A linear
und that
neutral
ases the
umber of
terminal
3
T
several
with hy
For the
junction
Each stu
tubes a
studied
atoms. A
3.3.1.
Y
aa
3.3.Com
To study th
series of st
ydrogenated
closed stru
ns was stud
udied junct
re joined, th
In the case
, which are
All heptago
Opened
Yj(6,6)a ‐ TU(
Figure 46 That each opealong the th(right), respe
mputatio
he effect of
ructures we
d openings,
uctures two
ied, where
ion include
hey are dist
e of Yj(6,6)
different b
ons are isola
symmetr
(6,6,3)
hree symmetning an armhreefold axeectively.
onalstu
f attached c
ere built wh
, and close
o different
the chiralit
es six heptag
tributed in a
) nanotube
by the posit
ated, surrou
ricalarm
Yj
tric Y‐type jumchair nanotue C3 (left), tw
dyofar
carbon nan
here the len
d (by nano
nanotube c
ty of the op
gons, need
a symmetric
junctions
ion of hept
unded only
mchairYj(
j(6,6)b ‐ TU(6
unctions withube of lengtwofold axe C
rmchair
notube on t
ngth of the
otube caps)
caps was us
ening is (4,4
ed for the n
c way, each
three diffe
agons, and
by hexagon
(n,n)jun
6,6,3)
h D3h symmeth 3 is conneC2 top (cent
Yj(n,n) j
the stability
nanotube v
Y junction
sed. Only a
4), (6,6) and
negative cu
h structure h
erent juncti
also by the
ns.
ctions
Yj(6
etry with (6,6ected. The stter), and tw
junction
y of the Y j
varies. Both
s were con
armchair sy
d (8,8), resp
urvature wh
has D3h sym
ion topolog
e number o
,6)c ‐ TU(6,6
6) armchair otructures arofold axe C2
19
ns
unction,
opened,
nsidered.
mmetric
pectively.
here two
mmetry.
gies was
f carbon
6,3)
openings, e viewed
2 bottom
oa
being (6
heptago
figures
to the
preserv
shape.
T
armcha
shape, t
T
the PM
the tota
T
respect
total en
correlat
parame
energy
two cas
Yj(4,4
Figure 47 Sy(bottom), wof length 2 waxe C3 (left),
In Figure 46
6,6) and an
ons positio
the optimiz
junction ge
ves its origin
The Y junct
ir chirality.
the other tw
The geome
6 semiemp
al energy div
The results
ively. It can
nergy and
tion betwe
eters sort or
per atom i
ses the tube
4)a ‐ TU(4,4,2
ymmetric Y‐tith chirality with matchintwofold axe
6 the symm
n armchair
ns are diff
zed geomet
eometry th
nal circular s
tions relate
Only in the
wo junction
etry optimiz
irical level o
vided by th
obtained f
n be observ
the gap e
en the kin
rder match
n the case
e length incr
2)
ype armchai(4,4) and (8,ng chirality ise C2 top (cent
etric armch
nanotube o
ferent whic
tries are pre
he shape o
shape, whil
d to Yj(6,6)
e case of Yj
s deforms i
zation and
of theory. A
e number o
for the serie
ed that wit
nergy oscil
netic and t
es. In every
of Yj(4,4)a
reases the s
ir junctions w,8), respectivs attached, tter), and two
hair Y juncti
of length 3
ch gives the
esented (hy
of the attac
e in the oth
)a are show
(6,6)a the a
t into an el
single point
As stability
of carbon at
es of Y junc
th the incre
late with a
thermodyna
y case the g
increases w
stability of t
Y
with D3h symvely. To eachhe structureofold axeC2 b
ons are sho
3 is attache
e junction
ydrogen ato
ched nanot
her two case
wn in Figure
attached na
liptical geom
t energy ca
criterion tw
toms, and t
ctions Yj(4,4
ment of the
a periodicit
amic param
gap energy
with the len
the structur
Yj(8,8)a ‐ TU(
mmetry Yj(4,4h junction anes are viewedbottom (right
own, with th
d. As it can
a different
oms are not
tube chang
es the nano
e 47, they h
anotube pre
metry.
alculations w
wo paramet
he HOMO L
4)a, is pres
e attached
ty of three
meter, alon
decreases,
ngth of the
re.
8,8,2)
4)a (top) andn armchair nd along the tt).
he opening
n be observ
t geometry
t shown), ac
ges, in one
otube has a
have (4,4) a
eserves the
were perfo
ters were fo
LUMO energ
ented in Fig
nanotube b
e. There is
ng the seri
, however t
e tube. In th
20
d Yj(8,8)a nanotube threefold
chirality
ved, the
y. In the
ccording
e case it
n elliptic
and (8,8)
e circular
ormed at
ollowed:
gy gap.
gure 48,
both the
a good
es both
the total
he other
os
Figure 48 PlHOMO‐LUMPM6 semiem
Figure 51 Ploof the attachseries of ope
ot of the toO gap energmpirical meth
ot of the tothed nanotubened Y‐type j
otal energy (gy (Egap in auhod for the s
al energy (kcbe length, objunction Yj(4
kcal/mol) peu) as functionseries of ope
cal/mol) per btained with 4,4)a, Yj(6,6)a
er number on of the numned Y‐type j
number of cthe PM6 sema and Yj(8,8)
of carbon atmber of atomunction Yj(4,
carbon atommiempirical ma, respective
toms (Etot/N)ms, obtained ,4)a.
ms (Etot/N) asmethod for tely.
21
) and the with the
function the three
highest
series.
junction
the mos
T
series a
cannot
stable s
T
case of
the valu
oj
T
energy)
identic
In Figure 5
values corr
It can be c
n, therefore
st stable on
This sort or
s it can be
be related
tructures co
The periodi
opened or
ues of the to
Figure 52 Ploobtained at junctions: Yj
The three o
and Figur
along the
1 the total
respond to
oncluded t
e the juncti
e.
rder howev
seen in Figu
to the nan
orrespond t
ic oscillatio
closed arm
otal energy
ot of the HOthe PM6 se(4,4)a, Yj(6,6
opened arm
re 54 (tota
three serie
energy pe
the Yj(4,4)a
hat the nan
ion with th
ver is not p
ure 52. Not
notube leng
to a differen
n in the ga
mchair nano
per atoms
MO‐LUMO eemiempirica6)a and Yj(8,8
mchair Yj(6,
l energy),
es, and the
r atom bet
a series, wh
notube dia
he attached
preserved w
ice howeve
gth. It can b
nt nanotube
p energy w
tubes [31, 3
was observ
energy gap (El level of th8)a, respecti
,6) junction
respectivel
Yj(6,6)a h
tween the t
hile the low
meter cont
nanotube
when compa
er that the a
be observed
e length.
with the nan
35‐37, 97‐1
ved.
Egap in au) as eory, for thvely.
s stability i
y. The gap
as the high
three series
west on to th
tributes to
with larges
aring gap v
along one se
d that in ea
notube leng
103]. Howev
function of e three seri
is compared
p energy os
hest gap va
s is compar
he Yj(8,8)a
the stabilit
st diameter
values betw
eries the so
ach series t
gth was obs
ver no oscil
the nanotubes of opene
d in Figure
scillation is
alues. This
22
red. The
junction
y of the
r (8,8) is
ween the
ort order
he most
erved in
lation in
be length, ed Y‐type
53 (gap
s almost
stability
ordering
close at
decreas
possible
the stru
short na
oj
g is contrad
t a given na
ses with th
e that with
ucture. The
anotube is d
Figure 53 Ploobtained at junctions: Yj
dicted by th
anotube le
e tube len
a longer at
increase in
distorted, a
ot of the HOthe PM6 se(6,6)a, Yj(6,6
he next plot
ngth. Notic
gth, at lea
ttached nan
n energy can
nd strain is
MO‐LUMO eemiempirica6)b and Yj(6,6
, however t
ce that only
st in the s
notube, the
n be attribu
introduced
energy gap (El level of th6)c, respecti
the total en
y in the cas
tructured c
tube contr
uted to the
d.
Egap in au) as eory, for thvely.
nergy per at
se of Yj(6,6
considered
ributes mor
fact that th
function of e three seri
tom values
)a the tota
in this stu
re to the sta
he geometr
the nanotubes of opene
23
are very
l energy
udy. It is
ability of
ry of this
be length, ed Y‐type
os
3.3.2.
T
series Y
respect
pentago
nanotub
of the o
caps we
be seen
Figure 54 Ploof the attachseries of ope
Symmetr
To study th
Yj(6,6)a, Yj(6
to the jun
ons and ha
be, howeve
opening) [37
ere chosen
n in Figure 5
ot of the tothed nanotubened Y‐type j
riccappe
he cap effec
6,6)b and Y
ction (Figur
as a positiv
er a nanotub
7, 104, 105]
to close the
58.
al energy (kcbe length, objunctions Yj(
edYj(n,n
ct on the st
Yj(6,6)c wer
re 57). The
ve curvatur
be can be e
. To preserv
e structure,
cal/mol) per btained with 6,6)a, Yj(6,6
n)armcha
tability of t
re closed a
nanotube
re. One na
enclosed by
ve the junct
, their struc
number of cthe PM6 sem)b and Yj(6,6
airjuncti
the junction
the oppos
cap is a ha
anotube ca
y more than
tion high sy
cture viewe
carbon atommiempirical m6)c, respectiv
ions
n, the three
ing end of
alf fullerene
p can fit t
n one cap (d
ymmetry on
d along the
ms (Etot/N) asmethod for tvely.
e armchair
the nanotu
e, which inc
to only on
depends on
nly two sym
e symmetry
24
function the three
junction
ube with
cludes 6
kind of
the size
metrical
y axe can
ttcs
a
Yj(6
even nu
Figure 57 Coregion (centthree attachthat enclosecharacterizestructural ele
Figure 58 Arand three‐fopentagons w
6,6)a cap1
umber of row
Figure 59 YDepending orelative to th
onstruction otral) is charahed armchaire the oppositd by the poement must
mchair (6,6) old C3 (right)which gives a
ws
Y
odd n
Y‐type juncton the attache junction.
of a closed Yacterized byr nanotubes te send of thositive curvatbe the same
carbon nano) –cap 2 rota positive cur
Yj(6,6)a cap1
number of ro
tion Yj(6,6)ached nanotu
Y‐type junctioy negative cto the openhe nanotubeture introdue.
otube caps (ational symmrvature to th
ows eve
a closed at ube length
on Yj(6,6) frourvature intnings of the e with respecuced by the
half‐fullerenmetry axe, re structure.
Yj(6,6)a cap
en number of
each end (odd or eve
om three fratroduced byjunction, anct to the junpentagons.
nes), with sixrespectively.
p2
f rows o
with the sen) the cap
gments: they the heptagd the nanotnction, and wThe chirality
x‐fold C6 (leftEach cap co
Yj(6,6)a c
odd number
ame nanotuchanges its
25
e junction gons, the ube caps which are y of each
) – cap 1, ontains 6
ap2
of rows
ube cap. position
Y
even nu
tj
S
and the
atoms)
capped
Yj(6,6)c
os
Yj(6,6)b1
umber of row
Figure 60 Clthe same cajunction and
Series were
e cap. Depe
the cap ch
with both
closed with
Figure 61 Ploof the lengthsemiempiric
ws odd n
osed symmep. The cap pd cap (even a
e construct
ending on t
anges its po
nanotube
h cap 1.
ot of the toth of the (6,6)al method fo
Yj(6,6)b1
number of ro
etric Y junctiposition varieand odd num
ed by incre
the length
osition rela
caps, while
al energy (kc) armchair nor the series
ows eve
ions, Yj(6,6)bes dependingmber of atom
easing the l
of the nan
ative to the
e Figure 60
cal/mol) per anotube attaof closed Y‐j
Yj(6,6)c1
en number of
b and Yj(6,6)g on the leng rows).
length of th
notube (eve
junction. F
shows the
number of cached to thejunctions: Yj
f rows o
)c, capped atgth of the na
he tube bet
en or odd n
Figure 59 pr
geometry
carbon atome junction ob(6,6)aC, Yj(6
Yj(6,6)c
odd number
t each openanotube betw
tween the
number of
resents the
of the Yj(6,
ms (Etot/N) asbtained with ,6)bC, and Yj
26
c1
of rows
ing using ween the
junction
rows of
Yj(6,6)a
,6)b and
function the PM6 j(6,6)cC.
oj
T
in Figur
differen
nanotub
A
energy
the stru
nanotub
capped
howeve
the seri
Figure 62 Ploobtained at junctions: Yj
The plot of
re 61. The s
nce is that i
be.
As in the p
gap, as it c
uctures with
In the Figur
be length,
with cap 1
er they hav
es closed w
ot of the HOthe PM6 se(6,6)aC, Yj(6
the total en
stability ord
in all cases
revious stu
can be seen
h the largest
re 64 the to
respectivel
and 2. It ca
e almost th
with the high
MO‐LUMO eemiempirica,6)bC și Yj(6,
nergy per a
dering corre
the total e
dy the ord
n in Figure 6
t gap corres
otal energy
y, in the c
an be seen t
he same sta
her symmet
energy gap (Eal level of th,6)cC.
tom as func
esponds to
energy decr
ering of the
62. Once ag
sponds to t
y per atom
case of the
that althoug
ability. The
try cap is so
Egap in au) as heory, for th
ction of the
that of the
reases with
e three ser
gain the thr
he Yj(6,6)a
and the ga
e two serie
gh there is s
periodicity
omewhat m
function of he three ser
e nanotube
opened Y
h the increa
ies changes
ree curves a
series.
p energy is
s of closed
some minor
y in the gap
ore stable.
the nanotubries of close
length is pr
junctions. T
ase of the a
s in the cas
are almost
plotted ve
d Yj(6,6)a ju
r energy dif
p values is
27
be length, ed Y‐type
resented
The only
attached
se of the
parallel,
ersus the
unctions
fference,
present,
oY
3.3.3.
S
keeping
nanotub
Figure 6
geomet
Figure 64 Ploobtained at Yj(6,6)aC1 an
Closeda
Starting fro
g constant t
be’s length.
65 where t
tries were o
ot of the HOthe PM3 semnd Yj(6,6)aC2
asymmetr
om the Yj(
two of the
. Each nano
two arms a
optimized w
MO‐LUMO emiempirical 2, respective
ricYj(n,n
6,6)a junct
arms of the
otube was c
are of lengt
with the PM3
energy gap (Elevel of theoely.
n)armch
tion four s
e junctions,
closed at th
th 3, while
3 semiempi
Egap in au) as ory, for the t
airjunct
eries of ju
and increa
he end with
e the third
irical metho
function of two series o
tions
nctions we
asing only o
cap 1. An
nanotube
od.
the nanotubf capped Y j
ere constru
one of the a
example is
is of length
28
be length, unctions:
ucted by
attached
given in
h 8. The
c
C
and Fig
plotted
as was
with the
N
Only th
the ene
first two
of these
of
Figure 65 Anchirality, but
Calculation
gure 67, wh
as the func
previously
e tube lengt
Notice howe
e structure
ergy gap val
o members
e two series
Figure 66 Ploof the attachfour series o
n example oft the length o
results obt
here the to
ction of the
observed in
th in all fou
ever that in
s where the
ues. The tw
of the serie
s is a leapfro
ot of the tothed (6,6) nanof closed asym
f asymmetriof one arm is
tained at th
otal energy
e tube lengt
n the case o
r series.
n the second
e length of
wo series st
es. This cou
og structure
al energy (kcnotube lengtmmetric Y‐ju
c junction Yjs different fr
he same the
y per atom
th, respecti
of closed Y
d plot, not a
the tubes w
ability is alm
ld be expla
e [106‐109]
cal/mol) per th, obtainedunctions.
j(n,n): all attrom the othe
eoretical lev
and the H
vely. The fi
junctions, w
all series ha
was zero an
most the sa
ined by the
.
number of c with the PM
tached nanoters.
vel are pres
HOMO‐LUM
rst plot giv
which is the
ve the sam
nd three, sh
ame, with th
e fact, that e
carbon atomM3 semiemp
tubes have t
sented in F
MO gap ene
es a similar
e energy de
e stability o
how a perio
he exceptio
every third
ms (Etot/N) aspirical metho
29
the same
igure 66
ergy are
r answer
ecreases
ordering.
odicity in
on of the
member
function od for the
oa
3
T
series w
were clo
single p
that the
didn’t a
S
be obse
relation
increase
Figure 67 Ploobtained atasymmetric
3.4.Con
To study th
were mode
osed by a n
point energy
e nanotube
lter the sta
Similar to t
erved in th
nship could
e of the tub
ot of the HOt the PM3 Y junctions.
nclusions
he effect of
led, where
nanotube ca
y calculatio
length and
bility sort o
the armcha
he gap valu
d be found
be diameter
MO‐LUMO esemiempiric
s
the attache
the nanot
ap, to study
ns at the P
d diameter
order.
ir opened a
ues of each
d between
r decreases
energy gap (Ecal level of
ed nanotub
ube length
y the cap eff
M3 and PM
has a big in
and closed
h series as
the stabili
the total en
Egap in au) as theory, for
bes on the s
is increase
fect. Geome
M6 semiemp
nfluence on
nanotubes,
function o
ity and na
nergy.
function of the four s
stability of Y
ed. Also the
etry optimi
pirical level
n the stabil
, an oscillat
of the tube
notube len
the nanotubseries of clo
Y junctions,
e opened ju
zations follo
of theory s
ity. The clo
ting period
e length. N
ngth, howe
30
be length, osed and
, several
unctions
owed by
showed,
sing cap
icity can
o direct
ever the
4
T
our soft
second
t
A
face/rin
of ops o
of distin
A
were de
Omega,
tetrapo
4.2.Top
Tetrapodal
tware Nano
generation
First d
Figure 68 De
tetrapodal ju
As above m
ng eventual
of length c i
nct c expone
Analytical f
erived [119
, Sadhana p
dal junction
pologica
junctions w
o Studio) to
stages, res
dendrimer l
endrimer at f
unctions.
mentioned, a
ly forming a
s given by t
ents equals
formulas of
]; in every
polynomials
ns are prese
aldescrip
were conne
o build den
spectively, a
evel
first (left) an
a set of opp
a strip of ad
the polynom
s the numbe
f the Omeg
formula, m
s and deriv
ented in the
ptionof
ected by ide
ndritic mole
are illustrate
d second (rig
posite or top
djacent face
mial coeffic
er of equiva
ga and Sad
m represents
ed indices,
e tables belo
fnanode
entification
ecules. Nano
ed in Figure
Sec
ght) level bu
pologically
es, is called
ients of ter
alence classe
hana polyn
s the numb
in the case
ow.
endrime
(a procedu
o‐dendrime
e 68.
ond dendri
uilt from C84 (
parallel edg
d a qoc or a
ms at expo
es of oppos
nomial in te
ber of mono
e of dendri
ers
ure impleme
ers, at the f
mer level
( ( ( (Le Op Le L
ges within t
an ops. The
nent c; the
site edges in
etrapodal ju
omers. Exam
mers built
31
ented in
first and
( ))))Le T )
he same
number
number
n G.
unctions
mples of
up from
32
C84‐Td‐junction
11 1 11 11( ( ( ( ))))Le S Le Le T TU3.3.0
v =108 , f6 = 28, f7 = 12, e = 150
Omega Polynomial 1 3 712(2 1) 24 6m X mX mX
'( ,1) 138 12G m
CI index 2 2138 2778 132CI m m ; Example: 3; 179862m CI
Sadhana Polynomial 138 11 138 9 138 512(2 1) 24 6m m mSd m X mX mX
C84‐Td‐junction
11 1 11 11( ( ( ( ))))Le S Le Le T TU3.3.1
v = 132, f6 = 40 f7 = 12, e = 186
Omega Polynomial 1 2 3 96( 1) 12( 1) (30 6) 6m X m X m X mX
'( ,1) 174 12G m
CI index 2 2174 3366 144CI m m ; Example: 5; 773874m CI
Sadhana Polynomial 174 11 174 10 174 9 174 36( 1) 12( 1) (30 6) 6m m m mSd m X m X m X mX
C60‐td‐junction
11 1 21( ( ( )))Le S Ca T TU3.3.0
v = 84, f6 = 16, f7 = 12, e = 114
Omega Polynomial 1 2 312(2 1) 12 18m X mX mX
'( ,1) 102 12G m
CI index 2 2102 2214 132CI m m ; Example: 2; 46176m CI
Sadhana Polynomial 102 11 102 10 102 912(2 1) 12 18m m mSd m mX mX
C60‐td‐junction
11 1 21( ( ( )))Le S Ca T TU3.3.1
v = 108, f6 = 28 f7 = 12, e = 150
Omega Polynomial 1 2 36( 1) 12( 1) 6(6 1)m X m X m X
'( ,1) 138 12G m
CI index 2 2138 2934 144CI m m ; Example 4; 316584m CI
Sadhana Polynomial 138 11 138 10 138 96( 1) 12( 1) 6(6 1)m m mSd m X m X m X
33
C120‐td‐junction
11 1 11 2.0( ( ( ( ))))Le S Le Q T TU3.3.0
v = 144, f6 = 46 f7 = 12, e = 204
Omega Polynomial 1 3 4 6 1212(2 1) 12 6 12 3m X mX mX mX mX
'( ,1) 192 12G m
CI index 2 2192 3516 132CI m m ; Example: 2; 154620m CI
Sadhana
Polynomial
192 11 192 9 192 8 192 6 19212(2 1) 12 6 12 3m m m m mSd m X mX mX mX mX
C120‐td‐junction
11 1 11 2.0( ( ( ( ))))Le S Le Q T TU3.3.1
v = 168, f6 = 58 f7 = 12, e = 240
Omega
Polynomial
1 2 3 6 126( 1) 12( 1) 6(3 1) 18 3m X m X m X mX mX
'( ,1) 228 12G m
CI index 2 2228 4176 144CI m m ; Example: 7; 2576592m CI
Sadhana
Polynomial
228 11 228 10 228 9 228 6 2286( 1) 12( 1) 6(3 1) 18 3m m m m mSd m X m X m X mX mX
C76‐td‐junction TU3.3.0 v = 100, f6 = 24 f7 = 12, e = 138
Omega Polynomial 1 2 3 512(2 1) 12 6 12m X mX mX mX
'( ,1) 126 12G m
CI index 2 2126 2598 132CI m m ; Example: 2; 68832m CI
Sadhana Polynomial 126 11 126 10 126 9 126 712(2 1) 12 6 12m m m mSd m X mX mX mX
C76‐td‐junction TU3.3.1 v = 124, f6 = 36 f7 = 12, e = 174
Omega Polynomial 1 2 3 56( 1) 12(2 1) 6( 1) 18m X m X m X mX
'( ,1) 162 12G m
CI index 2 2162 3282 144CI m m ; Example: 3; 246186m CI
Sadhana Polynomial 162 11 162 10 162 9 162 76( 1) 12(2 1) 6( 1) 18m m m mSd m X m X m X mX
34
The first derivative of Omega polynomial gives the number of the edges in G. The first
derivative of Sadhana polynomials in x=1 is a multiple of the number of the edges, as shown
above. The Cluj‐Ilmenau CI index is a topological index, useful in correlating properties with
molecular structures.
By construction, the “negative” polygon is 7 (an odd ring, with no opposite edges), so
that the strip remains at the level of a single monomer unit and the polynomials are additive
in their coefficients. In the above, “negative” refers to polygons inducing the negative
curvature at the junction of tetrapodal units. The terms at c=1 (in Omega polynomial)
account for the non‐opposite edges or the odd rings.
Remark in these tables the simplicity at exponents of Omega vs Sadhana
polynomials; the polynomial order increases as the size of fullerenes increases. The first
derivative is 1st and 2nd order in m, for Omega and Sadhana, respectively. The value in x=1 is
the same for both polynomials and accounts for the total number of strips. For the first three
units, the same values ( ,1); ( ,1)G Sd G are obtained, telling about their structural
relatedness.
4.3.TopologicaldescriptionofnetworksbuiltfromP‐type
junctions
The studied P‐type networks are periodic graphitic arrangements with hexagonal and
heptagonal rings of carbon. They are a series of open networks made of sp2 carbon atoms
only. To the structures under study can connect six (4, 4) armchair carbon nanotubes,
therefore a genus three. The topology of such P‐type networks can be described by the aid
of some counting polynomial, called Omega polynomial. Analytical formulas for counting
polynomial, and derived indices, in P‐type networks are given.
The unit cell of the Schwarz P (primitive) triply periodic minimal surface can be
described as the intersection of six tubes coming through the faces of a cube, that meet into
a 6–coordinated octahedral junction (also known as the “plumber’s nightmare”).
Two networks of 6‐coordinated junctions were studied. The repeat units are
obtained by a sequence of map operations starting from the cube, Le(Op(Le(Le(C)))) and
Le(Op(L
the cha
resulted
bivalent
directio
became
armcha
coefficie
of equiv
c
c =
W
strip of
torus on
This stri
Le(Q(C)))) h
mfering Q t
d junctions
t vertex on
ons, indicati
e heptagons
ir (4,4) chira
In the P‐ty
ents of term
valence clas
c = 4; v = 288
= 8; v = 568;
Figure 71 Ex
When two
length 8.W
n the inside
ip is actually
ereafter no
transforma
have the o
n each edg
ng that it h
s, which ar
ality.
pe structur
ms at expon
sses of oppo
8; e = 408; f6 =
e = 812; f6 =
amples of O
P‐type jun
When four ju
e belt of the
y formed fr
otated P216
tions prese
octahedral
ge of the 4
has genus 3
re all rotate
res the num
nent c; the
osite edges
=92; f7=24
188; f7 = 48
mega polyno
ctions are c
unctions are
e torus the
om the fou
6 and P288
erve the sym
symmetry
4‐membere
3. By the ho
ed in the f
mber of op
number of
in G. Figure
c = 1
omial strips i
connected
e connected
two c = 8 s
r c = 4 strip
8, respectiv
mmetry of t
of the cub
ed rings th
omeomorph
inal step (L
s of length
f distinct c
e 71 illustra
16; v = 1120;
in case of Le
the two c
d in the shap
strips will re
ps of the fou
ely. Both t
the parent
e. By the in
he structure
h transform
Le), resultin
c is given
exponents
tes qoc/ops
e = 1616 ; f6
( ( ( (e Op Le Q C
= 4 strips f
pe of a squa
esult a new
ur junctions
he leapfrog
map, there
ntroduction
e is opene
mation 24 h
ng an open
by the pol
equals the
s.
6 = 384; f7 = 9
))))C .
forms a con
are they wi
w strip of le
.
35
g Le and
efore the
n of one
ed in six
exagons
ing with
lynomial
number
6
ntinuous
ll form a
ngth 16.
36
A complete network of P‐type junctions can be defined in terms of three integers a, b,
and c, their values represent the number of units along the three translation directions.
Assigning numbers to these parameters is completely arbitrary and describes the same
network (1×2×3 ≈ 2×1×3). A single unit is defined when all three integers are equal to one,
denoted 1×1×1. Depending on the values of the parameters one can distinguish three cases
of network types: 1. the structure is linear if only one parameter is greater than one (a > 1, b
= c = 1); 2. the structure is two‐dimensional if only one parameter equals one (a, b > 1, c = 1);
3. when all the parameters have a value larger than one (a, b, c > 1), a three‐dimensional
structure is described.
An attempt to derive the Omega polynomial solely from the number of units
(notated u) failed, because two distinct structures, although they have the same number of
units (for example 1×1×8, 1×2×4 and 2×2×2 are all built from eight units), their polynomials
are different. The case is similar when only the number of junctions (notated j) is taken into
account.
The number of units (u), junctions (j), and tori (t) can be defined in terms of the
structural parameters a, b, and c, the corresponding relationships are equations (8)‐(12):
u a b c (44)
1 1 1
3 3
j a b c a b c a b c
a b c a b b c a c u a b b c a c
(45)
3a b b c a c j u (46)
1 1 1 1 1 1
3 2 4 3
t a b c b a c c a b
a b c a b b c a c a b c j u a b c
(47)
4 3a b c t j u (48)
Analytical formulas of Omega polynomial in case of ( ( ( ( ))))Le Op Le Q C :
1 2 3 4 6 8 16 1872 20 8 24 4 3 2 24 4 4X u j X jX uX u j t X uX j t X tX uX
2 3 5 7 15 17' 72 20 16 72 16 3 2 144 32 16 72X u j jX uX u j t X uX j t X tX uX
' 1 408 4u j
2 4 6 14 16" 16 144 48 3 2 720 224 240 1224X j uX u j t X uX j t X tX uX
37
" 1 2232 144 64u j t
2408 4 2640 140 64CI u j u j t
Analytical formulas of Omega polynomial in case of ( ( ( ( ))))Le Op Le Le C :
1 2 3 7 7 772 20 8 48 4 4 4a b cX u j X j X u X bc X ac X ab X
2 7 1 7 1 7 1' 72 20 16 144 28 a b cX u j j X u X abc X X X
' 1 300 4u j
7 2 7 2 7 2" 16 288 28 (7 1) (7 1) (7 1)a b cX j u X abc a X b X c X
" 1 196( ) 204 16a b c u j
2300 4 196( ) 504 12CI u j a b c u j
4.4.Conclusions
Nano dendrimers built from two different tetrahedral junctions, and P‐type networks
built from octahedral junctions were characterized by means of Omega polynomial.
Analytical formulas were derived in terms of repeat unit in case of dendrimers, and using
three structural descriptors (number of units, junctions, tori) in case of the P‐type networks.
38
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Scientific communications
1. MolMod 2007, Molecular modeling in chemistry and biochemistry, July, 5‐8, Arcalia Diamond and dodecahedron architectures from carbon tetrapods (poster)
2. ChemMod 2007, Chemical Graph Theory and Molecular Modeling Workshop, October, 23‐26, Cluj‐Napoca Armchair [3,3] carbon nanotube junctions with tetrahedral symmetry (poster)
3. MolMod 2009, Molecular modeling in chemistry and biochemistry, April, 2‐4, Cluj‐Napoca Omega polynomials of carbon tetrapodal graphitic junctions (poster)
4. ICAM 2010 ‐ “Mathematical Chemistry in NANO‐ERA” Minisymposium, September, 1‐4, Cluj‐Napoca Y junctions from armchair carbon nanotubes(lecture)
Articles
1. Katalin Nagy, Csaba L Nagy, Gabriel Katona, Mircea V. Diudea Armchair [3,3] carbon nanotube junctions with tetrahedral symmetry Fullerenes, Nanotubes and Carbon Nanostructures, 18 (2010) 216‐223
2. Katalin Nagy, Csaba L Nagy, Mircea V. Diudea, Omega polynomial in diamond‐like dendrimers, Studia Univ. “Babes‐Bolyai” Chemia, 55 (2010) 77‐82
3. Mahboubeh Saheli, Mahdieh Neamati, Katalin Nagy, Mircea V. Diudea Omega polynomial in Sucor network, Studia Univ. “Babes‐Bolyai” Chemia, 55 (2010) 83‐90
4. Mahsa Ghazi, Modjtaba Ghorbani, Katalin Nagy, Mircea V. Diudea On Omega Polynomial of ((4,7)3) Network Studia Univ. “Babes‐Bolyai” Chemia, 4 (2010) 197‐200
5. Katalin Nagy, Csaba L Nagy, Mircea V. Diudea Omega and Sadhana polynomials of dendrimers designed from tetrapodal graphitic junctions MATCH Commun. Math. Comput. Chem, 65 (2011) 163‐172
6. Mircea V. Diudea, Katalin Nagy, Csaba L Nagy, Aleksandar Ilić Omega polynomial in puzzle zeolites MATCH Commun. Math. Comput. Chem, 65 (2011) 143‐152
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