FA CULTY OF CHEMISTRY AND CHEMICAL ENG...

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

2

apoca

apoca

gy and

ra

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


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


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


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


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


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


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


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


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


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


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

References

1. HW Kroto, et al., Nature 318(6042), 162‐163 (1985). 2. W Kratschmer, et al., Nature 347(6291), 354‐358 (1990). 3. S Iijima, Nature 354(6348), 56‐58 (1991). 4. M Bockrath, et al., Science 275(5308), 1922‐1925 (1997). 5. YK Kwon, et al., Physical Review Letters 82(7), 1470‐1473 (1999). 6. SJ Tans, et al., Nature 393(6680), 49‐52 (1998). 7. BI Yakobson, et al., American Scientist 85(4), 324‐337 (1997). 8. H Terrones, et al., Carbon 30(8), 1251‐1260 (1992). 9. H Terrones, et al., Chemical Physics Letters 207(1), 45‐50 (1993). 10. D Ugarte, Nature 359(6397), 707‐709 (1992). 11. D Ugarte, Chemical Physics Letters 207(4‐6), 473‐479 (1993). 12. AL Mackay, et al., Nature 352, 762 (1991). 13. T Lenosky, et al., Nature 355, 333‐335 (1992). 14. M O'keeffe, et al., Phys Rev Lett 68(15), 2325‐2330 (1992). 15. H Terrones, et al., Carbon 30(8), 1251‐1260 (1992). 16. D Vanderbilt, et al., Phys Rev Lett 68(4), 511‐513 (1992). 17. JL Aragón, et al., Phys Rev B 48(11), 8409‐8413 (1993). 18. H Terrones, J Math Chem 15, 143‐156 (1994). 19. H Terrones, et al., Acta Metall et Mater 42, 2687‐2699 (1994). 20. H Terrones, et al., J Math Chem 15, 183‐195 (1994). 21. H Terrones, et al., Chem Soc Rev 24, 341‐350 (1995). 22. H Terrones, et al., Prog Crystal Growth and Charact 34(1‐4), 25‐36 (1997). 23. RB King, J Chem Inf Comput Sci 38, 180‐188 (1998). 24. H Terrones, et al., New Journal of Physics 5, 126.1‐126.37 (2003). 25. F Valencia, et al., New Journal of Physics 5, 123.1‐123.16 (2003). 26. N Park, et al., Phys Rev Lett 91(23), 237204(1‐4) (2003). 27. JM Carlsson, et al., Phys Rev Lett 96, 046806 (2006). 28. S Iijima, et al., Journal of Chemical Physics 104(5), 2089‐2092 (1996). 29. S Iijima, Nature (London) 354, 56‐58 (1991). 30. R Saito, et al., Phys Rev B 46, 1804–1811 (1992). 31. A Rochefort, et al., J Phys Chem B 103, 641‐646 (1999). 32. W Liang, et al., Journal of the American Chemical Society 122(45), 11129‐11137 (2000). 33. N Park, et al., Phys Rev B 65, 121405(1‐4) (2002). 34. R Tamura, et al., Phys Rev B 52, 6015–6026 (1995). 35. T Yumura, et al., J Phys Chem B 108, 11426‐11434 (2004). 36. S Reich, et al., Phys Rev B 72, 165423(1‐8) (2005). 37. SL Lair, et al., Carbon 44, 447–455 (2006). 38. LA Chernozatonskii, Physics Letters A 170(1), 37‐40 (1992). 39. BI Dunlap, Phys Rev B 46, 1933‐1936 (1992). 40. J Han, Chem Phys Lett 282, 187‐191 (1998). 41. L Liu, et al., Phys Rev B 64, 033412‐1(4) (2001). 42. M Sano, et al., Science 293(5533), 1299‐1301 (2001). 43. M Huhtala, et al., Comput Phys Commun 147(1‐2), 91‐96 (2002). 44. IG Cuesta, et al., ChemPhysChem 7(12), 2503‐2507 (2006). 45. K Sai Krishna, et al., Chem Phys Lett 433(4‐6), 327‐330 (2007). 46. C Feng, et al., Carbon 47(7), 1664‐1669 (2009).

39

47. G Brinkmann, et al., Match 36, 233‐237 (1997). 48. G Brinkmann, et al., Match 63(3), 533‐552 (2010). 49. CL Nagy, et al., Nano Studio, 2010: Babes‐Bolyai University. 50. JJP Stewart, J Mol Model 10(2), 155‐164 (2004). 51. JJP Stewart, J Mol Model 13(12), 1173‐1213 (2007). 52. J Tirado‐Rives, et al., J Chem Theory Comput 4(2), 297‐306 (2008). 53. AD Becke, Phys Rev A 38(6), 3098‐3100 (1988). 54. C Lee, et al., Phys Rev B 37(2), 785‐789 (1988). 55. R Ditchfield, et al., The Journal of Chemical Physics 54(2), 720‐723 (1971). 56. PC Hariharan, et al., Theor Chim Acta 28(3), 213‐222 (1973). 57. MJ Frisch, et al., Gaussian 09, Revision A.02, 2009: Gaussian, Inc., Wallingford CT. 58. M Stefu, et al., CageVersatile CVNet, 2005: Babes‐Bolyai University. 59. W Cheng, et al., Journal of Molecular Structure: THEOCHEM 489(2‐3), 159‐164 (1999). 60. W Cheng, et al., Journal of Molecular Spectroscopy 193(1), 1‐6 (1999). 61. W Cheng, et al., Journal of Electron Spectroscopy and Related Phenomena 107(3), 301‐308

(2000). 62. AY Li, et al., Physical Chemistry Chemical Physics 2(10), 2301‐2307 (2000). 63. AC Tang, et al., Chemical Physics Letters 258(5‐6), 562‐573 (1996). 64. AC Tang, et al., Theoretical Chemistry Accounts 102(1‐6), 72‐77 (1999). 65. A Hirsch, et al., Angewandte Chemie ‐ International Edition 39(21), 3915‐3917 (2000). 66. Z Chen, et al., Journal of Molecular Modeling 7(5), 161‐163 (2001). 67. Z Chen, et al., Theoretical Chemistry Accounts 106(5), 352‐363 (2001). 68. R Salcedo, et al., Journal of Molecular Structure: THEOCHEM 422(1‐3), 245‐252 (1998). 69. M Lin, et al., Journal of Molecular Structure: THEOCHEM 489(2‐3), 109‐117 (1999). 70. J Xiao, et al., Journal of Molecular Structure: THEOCHEM 428(1‐3), 149‐154 (1998). 71. K Nagy, et al., Fullerenes Nanotubes and Carbon Nanostructures 18(3), 216‐223 (2010). 72. GE Scuseria, Chemical Physics Letters 195(5‐6), 534‐536 (1992). 73. LA Chernozatonskii, Physics Letters A 172(3), 173‐176 (1992). 74. I Zsoldos, et al., Modelling Simul Mater Sci Eng 12(6), 1251‐1266 (2004). 75. I Zsoldos, et al., Diamond Relat Mater 14, 763‐765 (2005). 76. I Zsoldos, et al., Modelling Simul Mater Sci Eng 15(7), 739‐745 (2007). 77. GK Dimitrakakis, et al., Nano Lett 8(10), 3166‐3170 (2008). 78. E Tylianakis, et al., Chem Commun (Cambridge, U K) 47(8), 2303‐2305 (2011). 79. I László, Croatica Chemica Acta 78(2), 217‐221 (2005). 80. I László, Fullerenes Nanotubes and Carbon Nanostructures 13(SUPPL. 1), 535‐541 (2005). 81. I László, Croatica Chemica Acta 81(2), 267‐272 (2008). 82. I Ponomareva, et al., New Journal of Physics 5, 119.1–119.12 (2003). 83. LP Biró, et al., Diamond and Related Materials 11(3‐6), 1081‐1085 (2002). 84. LP Biró, et al., Materials Science and Engineering C 19(1‐2), 3‐7 (2002). 85. LP Biró, et al., Diamond and Related Materials 13(2), 241‐249 (2004). 86. D Zhou, et al., Chemical Physics Letters 238(4‐6), 286‐289 (1995). 87. GI Márk, et al., Physical Review B 58(19), 12645‐12648 (1998). 88. BC Satishkumar, et al., Applied Physics Letters 77(16), 2530‐2532 (2000). 89. WZ Li, et al., Applied Physics Letters 79(12), 1879‐1881 (2001). 90. J Li, et al., Nature 402(6759), 253‐254 (1999). 91. YC Sui, et al., Journal of Physical Chemistry B 105(8), 1523‐1527 (2001). 92. YC Sui, et al., Carbon 39(11), 1709‐1715 (2001). 93. AA Koós, et al., Materials Science and Engineering C 23(1‐2), 275‐278 (2003). 94. M Terrones, et al., Physical Review Letters 89(7), 075505/1‐075505/4 (2002). 95. M Terrones, et al., Microscopy and Microanalysis 9(SUPPL. 2), 320‐321 (2003). 96. M Terrones, et al., New Diamond and Frontier Carbon Technology 12(5), 315‐323 (2002). 97. L Stobinski, et al., Reviews on Advanced Materials Science 5(4), 363‐370 (2003).

40

98. J Cioslowski, et al., Journal of the American Chemical Society 124(28), 8485‐8489 (2002). 99. T Yumura, et al., Journal of the American Chemical Society 127(33), 11769‐11776 (2005). 100. T Sato, et al., Synthetic Metals 103, 2525‐2526 (1999). 101. T Yaguchi, et al., Physica B 323, 209‐210 (2002). 102. Y Matsuo, et al., Organic Letters 5(18), 3181‐3184 (2003). 103. T Yumura, et al., Chemical Physics Letters 386, 38‐43 (2004). 104. G Brinkmann, et al., Chemical Physics Letters 315(5‐6), 335–347 (1999). 105. G Brinkmann, et al., Discrete Applied Mathematics 116, 55–71 (2002). 106. PW Fowler, et al., J CHEM SOC, CHEM COMMUN, 1403‐1405 (1987). 107. PW Fowler, J CHEM SOC FARADAY TRANS 86(12), 2073‐2077 (1990). 108. P Fowler, et al., J CHEM SOC FARADAY TRANS 90(19), 2865‐2871 (1994). 109. KM Rogers, et al., J Chem Soc, Perkin Trans 2, 18–22 (2001). 110. MV Diudea, Carpathian Journal of Mathematics 22(1‐2), 43‐47 (2006). 111. MV Diudea, et al., Journal of Mathematical Chemistry 45(2), 316‐329 (2009). 112. DZ Djokovic, J Combin Theory Ser B 14, 263‐267 (1973). 113. PM Winkler, Discr Appl Math 8, 209‐212 (1984). 114. MV Diudea, et al., Croatica Chemica Acta 79(3), 445‐448 (2006). 115. MV Diudea, et al., Acta Chimica Slovenica 57(3), 565‐570 (2010). 116. MV Diudea, et al., Carpathian Journal of Mathematics 25(2), 177‐185 (2009). 117. MV Diudea, et al., Match 60(1), 237‐250 (2008). 118. F Gholami‐Nezhaad, et al., Studia Universitatis Babes‐Bolyai Chemia (4), 219‐224 (2010). 119. MV Diudea, et al., Match 65(1), 143‐152 (2011). 120. MV Diudea, et al., Carpathian Journal of Mathematics 26(1), 59‐66 (2010). 121. MV Diudea, Journal of Mathematical Chemistry 45(2), 309‐315 (2009). 122. MV Diudea, et al., Match 60(3), 945‐953 (2008). 123. M Saheli, et al., Studia Universitatis Babes‐Bolyai Chemia 1, 83‐90 (2010). 124. MV Diudea, Match 63(1), 247‐256 (2010). 125. K Nagy, et al., Studia Universitatis Babes‐Bolyai Chemia 1, 77‐82 (2010). 126. AE Vizitiu, et al., Match 60(3), 927‐933 (2008). 127. A Ilić, et al., Carpathian Journal of Mathematics 26(2), 193‐201 (2010). 128. MV Diudea, et al., Studia Universitatis Babes‐Bolyai Chemia 4(2), 313‐320 (2009). 129. MV Diudea, et al., Match 65(1), 131‐142 (2011).

41

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

FA CULTY OF CHEMISTRY AND CHEMICAL ENG...

Documents

Transcript of FA CULTY OF CHEMISTRY AND CHEMICAL ENG...