層狀排列之液晶分子(smectic liquid crystal)放在流變儀中受到剪切力而形成多層封閉的洋蔥球結構，它被稱之為『洋蔥球液晶』(onion liquid crystal)。在這篇論文中，我們所要研究的是這種特殊結構的彈性。洋蔥球液晶的能量以兩種方式來儲存：一種是層與層之間的壓縮程度來決定，另一種是由層的彎曲程度來決定。我們透過相似於實驗所得到的洋蔥球的結構，在二維系統下模擬一對稱四邊形的多層結構，以數值模擬的方式用疊代法計算此結構之自由能。給予洋蔥球拉伸的形變，由形變前後自由能的變化量去計算此結構的彈性模數。數值計算的結果，得到洋蔥球液晶的彈性模數與其洋蔥球大小無關，而此結果與J.Leng， F.Nallet， 和D.Roux的實驗所得到的結果一致。而另一個數值結果是洋蔥球液晶的彈性模數與材質本身的彎曲彈性模數K(bending elastic modulus)也無關。 In this thesis, we discuss the elastic properties of the Smectic texture of the special shape which consists of the close packed multilayered vesicles (the so-called onion texture) which is formed by the applied shear. The free energy of the texture is stored in terms of the compression energy and the bending energy. The free energy of the texture is calculated numerically by the iteration method to approach the minimum free energy. Applying the elongation stress and calculating the free energy stored, we calculate the elastic modulus of the 2D onion crystal with the square symmetry. We simulate the onion with the grain boundary structure to understand the size dependence of the elastic modulus. Our numerical result indicates that the elongation modulus of the onion is approximately independent of the onion size. This result agrees with the result of the experiment by D.Roux et al.. Our result also indicates that the elongation modulus is independent of the bending modulus K, which can be checked by the further experiments.
