純碳共軛系統之幾何、拓樸與物理性質的理論研究

Abstract

在本篇論文中，我們有系統地探討了sp2碳共軛體系可能的幾何、拓樸，其中包括環型碳管(Toroidal Carbon NanoTube, TCNT)、螺旋型碳管(Helical CNT, HCNT)、環結型碳管(CNT Torus Knot)、週期性最小曲面石墨(TPMS graphite, Schwartzite)、高虧格富勒烯(High-Genus Fullerene, HGF)等。對於高對稱的環型碳管我們進行了AM1量子化學的計算，並以此結果探討其幾何結構與化學穩定性的關係，並進一步利用彈性力學理論分析之。對於螺旋型碳管的幾何構型我們亦進行了數值實驗與結果分析，發現其幾何參數的相圖(phase diagram)相較於文獻中連續模型所得之結果大為不同。 本篇論文附有二附錄，一為本實驗室所開發之電腦程式，二則為一些在本篇論文中所探討的純碳分子之分子圖。

In this thesis I present a rather different aspect to the classification of complex graphitic structures. Among them numerous novel structures are proposed for the fist time. And new approaches to some complex molecules that had been discussed in the literature are also presented, which I believe are much easier for chemists to grasp since they contain minimal mathematics. In addition to the discussion of molecular structures, which is included in the part one of the thesis, the energetic of the proposed molecules is also analyzed both in traditional quantum chemical computational approach and by the theory of elastic membrane, which are presented in the second part of the thesis. In part one, I first make a thorough examination on the problem of finding possible isomers of highly symmetric toroidal carbon nanotubes (TCNT), which are isomorphic to torus. And the problem is attacked in two different ways each having its own advantages and drawbacks, namely the real space and the dual space approaches. The two different schemes are discussed in chapter 2 and 3, respectively. Starting from TCNT, complex graphitic structures with diverse geometries and topologies can be obtained by applying certain manipulations on the structure of parent TCNT, as shown in the following illustration.

Note that the shown molecular structures are categorized into two major groups: ones derived from the topological manipulations (LHS) and from the geometrical manipulations (RHS) of TCNT. In chapter 4 I address the construction of helically coiled carbon nanotube (HCCNT). Since the HCCNTs have helical symmetry, they are singly periodic extended structures contrary to their TCNT parent molecules. Along the line on the RHS of the above figure, tubular structures as complicated as carbon nanotube torus knot can be obtained by altering the nonhexagon distribution on the molecular surface. In general, carbon nanotubes with any desired geometry (space curve) can be approached with the geometric manipulation schemes discussed herein, where in chapter 6 I focus on torus knot in particular. Turning our attention to the LHS of the figure, a large family of porous graphitic structures, either closed (0D) or extended (2D, 3D), can be classified by assembling suitable sets of neck-like structures. The neck structure is obtained by peeling the outer-rim of a TCNT off leaving the central hole unchanged. Depending on the occasion, necks with different geometric features are used and the resulting porous molecules can be high-genus fullerenes (0D), doubly periodic supergraphene (2D), or triply periodic quasi-minimal surfaces (3D). These constitute chapter 5 of the thesis. The second part of the thesis is divided into two chapters. In chapter 7 I discuss the use of three different molecular mechanical (MM) potential forms in the context of graphitic molecules. The first two of them are new to the literature and are both derived from simple theories of structural stability, namely the framework rigidity theory and the valence-shell electron-pair repulsion (VSEPR) theory. By applying these MM potentials to the geometry optimization of above mentioned graphitic molecules, it is shown that they are essentially efficient and robust to give reliable results while costing minimal computational effort. Concerning the thermodynamic stability of these molecules more rigorously, a thorough AM1 quantum chemical computation on a large family (over 1,000) of TCNTs was carried out. The results are certainly more reliable than ones obtained by utilizing potentials in chapter 7, nevertheless are difficult and somewhat tedious to follow. Thus in chapter 8 I first review some of the key features of the calculation results, then I turn to a unique application of the theory of elastic membrane. In stead of continualizing the molecular structure and thereby obtaining a 3D surface for the analysis, here a discretized version of elasticity theory for membrane is proposed. By suitably defining Gaussian and mean curvature values for each face of a graphitic structure, the formation energy can be split into three parts: elastic bending energy of graphene sheet, formation energies of the nonhexagonal defects, and the residual energy originated from quantum effects. The main advantage of this approach is that it bypasses the complex arithmetic required to fit the molecular shapes into continuous surfaces, which is exclusively used in the literature, while gives us maximal information about the distribution of strain energy. In addition to the above mentioned, there are two appedices attached in the end of the thesis. One enclosing the computer codes (mainly Matlab) that are developed during the study. And the other includes numerous molecular figures of the molecular structures discussed in the thesis.