Theoretical and Quantitative Structural Relationship Study on Fullerenes Polarizabilities on the basis of Monopole-Dipole Interactions Theorem  

Introduction

The science and technology of nano-scale materials was applied in studies of nanotechnology and the results have been applied in many areas of sciences such as computers, microchips, sensors, actuators and machines fall. The role of computational nanotechnology has become important in the sustainable development of nanotechnology. The rapid increase in computing power has been used for large-scale and high-accuracy simulations. It makes increasingly possible for nano scale modeling to also be predictive. Fullerenes are a class caged molecules containing only hexagonal and pentagonal inter-atomic bonding networks (1–17). Study of the electronic properties of these materials such as the dipole polarizability has made highly useful and effective results in applications in different areas of science (1, 2, 17, 18).

Since the discovery of fullerenes [19], many studies have been undertaken to characterize the physical properties of these molecules [20]. Among these properties, the dielectric response of fullerenes and carbon onions is of great interest for their possible use in composite materials for the coating of electronic circuits.[20] The lattice and electronic structures of fullerenes will greatly change with the increase of their carbon number and thus will have a large effects on their physicochemical  properties.

There are important theoretical studies on geometry of fullerenes “C_n” in the chemical literatures. The geometry optimization of fullerenes C₂₀ through C₂₄₀ had been performed with the second generation reactive empirical bond order potential energy, before [21,22,26]. The molecular polarizability of spherical and icosahedral fullerenes was found to vary linearly with the cube of their mean radius.[21,26] Many articles deal with the polarizability and optical properties of hyper fullerenes. Gueorguiev et al. give results for icosahedral fullerenes based on linear combination of atomic orbitals (LCAO) approximation, classic, and semi-classic calculations.[21,23,26] The results obtained give approximately a linear behavior versus R³ where R is the averaged radius for optimized fullerenes.[21,23,26] Mayer et al. had been applied the Renormalized Monopole–Dipole Model (RMDM) developed to hyperfullerenes, carbon onions and fullerene-based molecules. [21,24,26] In addition to dipole excitation, the model included a free-charge response characteristic of metallic structures or the delocalized π-electrons and so provides a significant improvement with respect to the classical dipolar interaction model, while keeping reasonable the calculation time.[21,26]

The linear uniform field electric dipole polarizability tensors of the fullerenes C₂₀ through C₂₄₀ were calculated by the Olson-Sundberg atom monopole-dipole interaction (AMDI) theory, using the monopole and dipole polarizabilities of the carbon atom found previously to fit polarizability tensors of aromatic hydrocarbons.[26,27] The molecular structures were taken to be the predicted results by Zhang and co-workers with respect to the molecular dynamics energy optimization.[ 26,28-30] The isotropic mean polarizabilities calculated for C₆₀ and C₇₀ are comparable to experimental data from solid film studies and quantum mechanical calculations.[26,31-43] Polarizability tensors were calculated for conducting ellipsoidal shells which have the same moment of inertia tensor as the corresponding fullerenes by Shanker and Applequist.[26] The results of the polarisability calculations were substantially smaller than the AMDI-polarizabilities for the smaller fullerenes, but the two calculations tend to converge for the larger molecules.[26-30, 34-44]

Graph theory has been found to be a useful tool in assessing the QSAR (Quantitative Structure Activity Relationship) and QSPR (Quantitative Structure Property Relationship).[45-53] Numerous studies in the above areas have also used what are called topological indices (TI).[54-57] It is important to use effective mathematical methods to make good correlations between the data corresponding to several chemical properties. One of the useful numerical and structural values in unsaturated compounds like fullerenes is the number of carbon atoms (C_n). To establish a good structural relationship between the structures of fullerenes (n) with the structural mechanical properties here was utilized the number of carbon atoms of the fullerenes “n.” The relationship between this index with diameter (h_o, A°), volume (V_o, A°³) molecular energy, endurance load and compressive stiffness of empty fullerenes (n = 20, 60, 80, and 180) were presented before. The results were extended for other empty fullerenes up to C₃₀₀. [46,58]

In this study, the relationship between the number of carbon atoms of fullerenes (C_n) index and polarizability of the fullerenes are presented on the basis of the Olson-Sundberg AMDI theory [27], Zhang et al. [28-30], Shanker and Applequist studies.[26-30] Here, were extended the calculation of the parameters concern to atom monopole-dipole moment such as AMDI (in ų, Atom monopole-dipole interaction theory; α₁ to α₃ and ā), Ellipsoid (in ų, α₁ to α₃ and ā) and semi-axes a,b,c of a thin ellipsoidal shell of uniform thickness (in Å, abc α₁ to α₃) for C₁₂₀ through C₃₀₀, 6-15.

 Graphing and mathematical method

The number of carbon atoms of the fullerenes (C_n) was utilized as a structural index for fullerenes. All graphing operations were performed using the Microsoft Office Excel-2003 program. The number of carbon atoms of these fullerenes (C_n) seems to be a useful numerical and structural value for the empty fullerenes. For modeling, both linear (MLR: Multiple Linear Regressions) and nonlinear (ANN: Artificial Neural Network) models were examined in this study. Some of the other indices were examined and the best results and equations for extending the physicochemical data were chosen.

Results and Discussions

The fullerenes possess a large number of conjugated π-electrons, but are composed of only carbon atoms and therefore do not have any residual infrared absorption that the polymers have.[59] Naturally, these properties make the fullerenes very exciting materials for their potential applications in nonlinear optical devices. The lattice and electronic structures of fullerenes will greatly change with the increase of their carbon number and thus will have a large effect on their polarizability properties. [59-61] Therefore, it is interesting to investigate theoretically this effect from the viewpoint of AMDI (in ų, Atom monopole-dipole interaction theory; α₁ to α₃ and ā), Ellipsoid (in ų, α₁ to α₃ and ā) and semi-axes a,b,c of a thin ellipsoidal shell of uniform thickness (in Å, abc α₁ to α₃) for C₂₀ through C₃₀₀.[26] Study on the comparison of the AMDI results with the conducting ellipsoid and monopole interaction treatments by Olson-Sundberg AMDI theory [27], Zhang et al. [28-30], Shanker and Applequist studies.[26-30], had given some insight into the behavior of the AMDI model.[26] The near parallel behavior of the three models is consistent with the major role of charge mobility in all three.[26] The values for the higher members of the series of the fullerenes had made this suggestion that charge mobility becomes the dominant mechanism of polarization with increasing size of the fullerenes in the AMDI model.[26] However, it was shown that the role of induced dipoles is substantial, especially for the lower members of fullerenes. It can be seen from the comparison of the AMDI and monopole interaction data (see Table-1 from ref.[26]). Shanker and Applequist have investigated AMDI, ellipsoid and semi-axes a,b,c of a thin ellipsoidal shell of uniform thickness for fullerenes C₂₀ through C₂₄₀.[26] The AMDI model had treated the molecule (here fullerene) as an array of point atoms which interact with each other by way of the electrostatic potentials and fields of their induced electric monopoles and dipoles in the presence of an external field.[26] The molecular polarizabilities were calculated from the charges and dipole moments induced in the atoms with respect to a uniform external field, by Shanker and Applequist.[26] In their model the fullerene molecule was approximated by an equivalent thin shell ellipsoid whose thickness was uniform.[26] To obtain the moment of inertia of such an ellipsoid, they had investigated that it is important to note the moments of inertia of a solid ellipsoid of mass M.  These values were obtained as: [26,62]

I₁=M (b²+c²) ⁄5                                          Eq.-1

I₂=M (a²+c²) ⁄5                                          Eq.-2

I₃=M (a²+b²) ⁄5                                          Eq.-3

where a,b,c are the semi-axis lengths along principal axes 1,2,3, respectively. The character of ρ is the density, so that M=4πρabc/3. The moment of inertia of a thin shell of uniform thickness and mass m was then obtained by taking differentials of I₁, I₂, I₃ and M at equal differentials da, db, dc.[26,62] Here, by utilizing the relationship between number of carbon atoms of the fullerenes and the parameters concern to atom monopole-dipole moment such as AMDI (in ų, Atom monopole-dipole interaction theory; α₁ to α₃ and ā), Ellipsoid (in ų, α₁ to α₃ and ā) and semi-axes a,b,c of a thin ellipsoidal shell of uniform thickness (in Å, abc α₁ to α₃) for C₂₀ through C₂₄₀ were extended the results for the bigger fullerenes C₁₂₀ through C₃₀₀ 6-15.

In Table-1 was demonstrated the values of AMDI (in ų Atom monopole-dipole interaction theory; α₁ to α₃ and ā), Ellipsoid (α₁ to α₃ and ā) and Semi-axes of a thin ellipsoidal shell of uniform thickness (in Å, abc, α₁ to α₃) of the selected fullerenes 1-5 from the results that was obtain for the fullerenes C₂₀ to C₂₄₀ byOlson-Sundberg AMDI theory [27], Zhang et al. [28-30], Shanker and Applequist studies.[26-30]. These data were extracted from reference [26]. Table-2 shows the equations of polarizabilities and semi-axes of the fullerenes in accordance with the values of AMDI, Ellipsoid and Semi-axes of a thin ellipsoidal shell of uniform thickness which they have been reported in reference [26] In Table-3 has shown the calculated values of AMDI (in ų Atom monopole-dipole interaction theory; α₁ to α₃ and ā), Ellipsoid (α₁ to α₃ and ā) and Semi-axes of a thin ellipsoidal shell of uniform thickness (in Å, abc, α₁ to α₃) of fullerenes C₁₂₀, C₁₃₂, C₁₄₀, C₁₄₆, C₁₅₀, C₁₆₀, C₁₆₂, C₂₇₆, C₂₈₈ and C₃₀₀ (6-15) by using equations 4 to 14. The data of the fullerenes 1-5 which they have demonstrated again from reference [26] in Table-1, and the obtained results for fullerenes 6-15 by the use of equations 4-14 (see Table-2), show increasing the values of AMDI, Ellipsoid and Semi-axes of a thin ellipsoidal shell of uniform thickness values of the fullerenes C₂₀ to C₂₄₀, by increasing the number of carbon atoms at the structures of the fullerenes.[26-30] All of the equations 4-14, the lines of best fit are Nieperian logarithmic behavior curve. Equations 4-7 describe Fig.-1 and show the Nieperian logarithmic behavior of the relationship between the numbers of carbon atoms versus the AMDI (in ų Atom monopole-dipole interaction theory; α₁ to α₃ and ā) values in the fullerene structures C₂₀ to C₂₄₀. The R-squared values (R²) for these four graphs are equal to 0.9989, 0.9932, 0.9991 and 0.9993 for AMDI α₁ to α₃ and ā, respectively. By utilizing the equations it is possible to achieve a good approximation for extending the determination of the AMDI polarizabilities (α₁ to α₃ and ā) for the fullerenes [26-30], 6-15 and other fullerenes C_n. See Figure 2, Tables 1 and 3.

Figure 1: The relationship between the number of carbon atoms “n” and the values of polarizabilities in AMDI (in Å3, Atom monopole-dipole interaction theory; α1 to α3 and ā) method for fullerenes C20 to C240. The values of AMDI of C20 to C240 is from Ref.[26]. Click here to View figure

 

Figure 2: The relationship between the number of carbon atoms “n” and the values of polarizabilities in Ellipsoid (in Å3, α1 to α3 and ā) method for fullerenes C20 to C240. The values of Ellipsoid of C20 to C240 is from Ref.[26]. Click here to View figure

 

Figure 3: The relationship between the number of carbon atoms “n” and the values of polarizabilities in Semi-axes of a thin ellipsoidal shell of uniform thickness (in Å, abc, α1 to α3)  for fullerenes C20 to C240. The values of Semi-axes of C20 to C240 is from Ref.[26]. Click here to View figure

 

Table 1: The values of AMDI (in ų Atom monopole-dipole interaction theory; α₁ to α₃ and ā), Ellipsoid (α₁ to α₃ and ā) and Semi-axes of a thin ellipsoidal shell of uniform thickness (in Å, abc, α₁ to α₃) of fullerenes C₂₀ to C₂₄₀. These data were extracted from reference [26].

No	Fullerenes	AMDI (Å3)				Ellipsoid (Å3)				Semi-axes (Å)
		α1	α2	α3	ā	α1	α2	α3	ā	α1	α2	α3
1	60	60.75	60.75	60.75	60.75	44.74	44.74	44.74	44.74	3.55	3.55	3.55
2	70	72.39	72.42	76.46	73.76	51.81	51.93	62.47	55.40	3.54	3.55	4.13
3	76	79.73	82.26	86.92	82.97	54.61	63.48	74.72	64.27	3.50	3.98	4.54
4	82	88.96	90.79	94.88	91.54	64.85	70.67	80.05	71.86	3.82	4.09	4.55
5	86	93.06	97.20	102.56	97.61	64.74	77.55	89.28	77.19	3.68	4.29	4.81

 

Table-2: The equations of polarizabilities and semi-axes of Fullerenes; AMDI (in ų Atom monopole-dipole interaction theory; α₁ to α₃ and ā), Ellipsoid (α₁ to α₃ and ā) and Semi-axes of a thin ellipsoidal shell of uniform thickness (in Å, abc, α₁ to α₃).

Equations	Y-value*	R2	Y= a(n)b
			a	b
Eq.-4 Eq.-5	-AMDI1α -AMDI2α	0.9989 0.9932	0.3251 0.2869	1.2755 1.3139
Eq.-6	-AMDI3α	0.9991	0.3232	1.2891
Eq.-7	-AMDIā	0.9993	0.3214	1.2843
Eq.-8	-Ellipsoid1α	0.9953	0.0991	1.4698
Eq.-9	-Ellipsoid2α	0.9960	0.1087	1.4743
Eq.-10	-Ellipsoid3α	0.9955	0.1087	1.4937
Eq.-11	-Ellipsoid ā	1	0.1058	1.4793
Eq.-12	–abc1α	0.9694	0.4474	0.4854
Eq.-13	–abc2α	0.9755	0.4844	0.4888
Eq.-14	–abc3α	0.9739	0.4840	0.5049

Figure 2 demonstrates four Nieperian logarithmic behavior curves relevant to relationships between the numbers of carbon atoms and the values of Ellipsoid (α₁ to α₃ and ā) of C₂₀ to C₂₄₀. Equations 8–11 describe Fig.-2 and show the Nieperian logarithmic behavior of the relationship. By the equations it is possible to achieve to the good approximations for extending the determination of Ellipsoid (α₁ to α₃ and ā) in the bigger fullerenes C₁₂₀ to C₃₀₀, 6-15 and other fullerenes C_n. In Fig.-2, the R-squared values (R²) for these four graphs are equal to 0.9953, 0.9960, 0.9955 and 1.0 for Ellipsoid α₁ to α₃ and ā, respectively. The predicted values of the α₁ to α₃ and ā for 6-15 were calculated by the use of equations 8–11 (see Figure 2 and Tables 1 and 3).

Similar to Figures 2 and 3, in Fig.-4 the lines of best fit are Nieperian logarithmic behavior curve. Four Nieperian logarithmic behavior curves relevant to relationships between the numbers of carbon atoms and the values of Semi-axes of a thin ellipsoidal shell of uniform thickness (in Å, abc, α₁ to α₃) were shown in Fig.-3. Equations 12-13 describe Fig.-3. By the equations it is possible to achieve to the good approximations for extending the determination of Semi-axes (in Å, abc, α₁ to α₃) of the fullerenes C₁₂₀, C₁₃₂, C₁₄₀, C₁₄₆, C₁₅₀, C₁₆₀, C₁₆₂, C₂₇₆, C₂₈₈ and C₃₀₀ (6-15) and other bigger fullerenes C_n. In Fig.-3, the R-squared values (R²) for these four graphs are equal to 0.9694, 0.9755 and 0.9739 for Semi-axes (in Å, abc, α₁ to α₃), respectively. The predicted values of the α₁ to α₃ for 6-15 were calculated by the use of equations 12–14 (see Figure 3, Tables 1 and 3). The number of carbon atoms in the structures of the fullerenes (C_n) shows a good relationship with their AMDI (in ų Atom monopole-dipole interaction theory; α₁ to α₃ and ā), Ellipsoid (α₁ to α₃ and ā) and Semi-axes of a thin ellipsoidal shell of uniform thickness (in Å, abc, α₁ to α₃) polarizability values on the basis of the Olson-Sundberg AMDI theory, Zhang et al., Shanker and Applequist studies.[26-30] Table 3 show the calculated values of AMDI, Ellipsoid and Semi-axes of a thin ellipsoidal shell of C₂₀ to C₂₄₀ on the basis [26-30] that the selected data were shown in Tables 1, 2 and figures 1-3. The explained polarizability data for 6-15 and other fullerenes C_n were obtained by extension values of C₂₀ to C₂₄₀ fullerenes which they reported before.[26]

Table-3: The calculated values of AMDI (in ų Atom monopole-dipole interaction theory; α₁ to α₃ and ā), Ellipsoid (α₁ to α₃ and ā) and Semi-axes of a thin ellipsoidal shell of uniform thickness (in Å, abc, α₁ to α₃) of fullerenes C₁₂₀, C₁₃₂, C₁₄₀, C₁₄₆, C₁₅₀, C₁₆₀, C₁₆₂, C₂₇₆, C₂₈₈ and C₃₀₀ (6-15) by using equations 4 to 14. See Table-2.

 

No	Fullerenes	AMDI (Å3)				Ellipsoid (Å3)				Semi-axes (Å)
		α1	α2	α3	ā	α1	α2	α3	ā	α1	α2	α3
6	120	145. 90	154. 73	154. 79	150. 43	112. 73	126. 35	138. 64	125. 96	4. 57	5. 03	5. 43
7	132	164. 74	175. 37	175. 03	170. 02	129. 69	145. 41	159. 86	145. 03	4. 77	5. 27	5. 69
8	140	177. 60	189. 46	188. 82	183. 37	141. 40	158. 59	174. 54	158. 22	4. 92	5. 42	5. 87
9	146	187. 35	200. 20	199.31	193. 52	150.40	168. 71	185. 83	168. 35	5. 03	5. 53	5. 99
10	150	193. 92	207. 44	206. 38	200. 35	156.49	175 .57	193. 49	175. 22	5. 09	5. 61	6. 07
11	160	210. 56	1034. 08	224. 29	217. 67	172.06	193. 09	213. 07	192. 77	5. 25	5. 79	6. 28
12	162	213. 92	229. 52	227.91	221. 17	175. 23	196. 66	217. 06	196. 35	5. 29	5. 82	6. 32
13	276	422. 08	3222. 11	452. 95	438. 43	383. 46	431. 38	481. 08	431. 84	6. 85	7. 56	8. 26
14	288	445. 63	488. 79	478. 49	263. 07	408. 21	459. 32	512. 65	459. 90	6. 99	7. 71	8. 44
15	300	469. 44	515. 73	504. 35	487. 99	433. 46	487. 81	544. 89	488. 53	7. 13	7. 87	8. 62

Conclusion

The science and technology of nano-scale materials was applied in studies of nanotechnology and the results have been applied in many areas of sciences such as computers, microchips, sensors, actuators and machines Study on the useful applications of graph theory have made the different mathematical methods for finding the relationship between several data of the material properties. In this study, the relationships between the number of carbon atoms of fullerenes (C_n) index and AMDI, Ellipsoid and Semi-axes of a thin ellipsoidal shell polarizability of the fullerenes C₂₀ to C₂₄₀ are presented on the basis of the Shanker and Applequist studies were demonstrated.  By the use of the results and the equations 4-14 of this model it is possible to achieve the good approximations for extending the AMDI, Ellipsoid and Semi-axes of a thin ellipsoidal shell polarizabilities values for bigger fullerenes C_n such as 6-15.

Acknowledgments

The corresponding author gratefully acknowledges his colleagues in the Chemistry Department of The University of New England (UNE)-Australia for their useful suggestions. He is also thankful from the Research Council of Science and Research Campus of Islamic Azad University and Arak branch of I.A.University.

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