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研究生:陳淑孟
研究生(外文):Shu-Meng Chen
論文名稱:層狀液晶缺陷結構之彈性
論文名稱(外文):The Elasticity of the Smectic Onion Texture
指導教授:陸駿逸
指導教授(外文):Chun-Yi David Lu
學位類別:碩士
校院名稱:國立中央大學
系所名稱:物理研究所
學門:自然科學學門
學類:物理學類
論文種類:學術論文
論文出版年:2003
畢業學年度:92
語文別:中文
論文頁數:70
中文關鍵詞:洋蔥球液晶彈性層狀液晶
外文關鍵詞:elasticitysmecticliquid crystalonion
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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.
Contents
1 Introduction 6
2 Review of Spherulite Elasticicty 9
2.1 Smectic Onion Structure . . . . . . . . . . . . . . . . . . . . . 9
2.1.1 Smectic Texture Formed during the Shear . . . . . . . 10
2.1.2 Measured Elasticity of Onion Structure . . . . . . . . . 11
2.2 The Free Energy of Smectic . . . . . . . . . . . . . . . . . . . 12
2.3 Deformation of a lamellar droplet . . . . . . . . . . . . . . . . 14
3 Elasticity of Onion Texture 19
3.1 Symmetry Consideration . . . . . . . . . . . . . . . . . . . . . 19
3.2 The Free Energy of Smectic . . . . . . . . . . . . . . . . . . . 21
3.2.1 φ variable . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2.2 Interactions of FluidMembrances . . . . . . . . . . . . 21
3.2.3 Curvature Energy of FluidMembrances . . . . . . . . . 26
3.3 Equation ofMinimizing The Free Energy . . . . . . . . . . . . 27
3.3.1 Finite-Dierence Approximations to derivatives . . . . 29
3.4 Initial Trial Condition . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 30
3.5.1 Edges . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5.2 Core . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5.3 Cut . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5.4 Summary of the Parameters Used in the Onion Simulations
. . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.6 The ElasticModulus . . . . . . . . . . . . . . . . . . . . . . . 38
3.6.1 The Size Dependence of the ElasticModulus . . . . . . 38
3.6.2 The Bending Elastic Modulus Dependence of the ElasticModulus
. . . . . . . . . . . . . . . . . . . . . . . . 40
3.7 Energy Cost of Deformation . . . . . . . . . . . . . . . . . . . 43
4 Stress Distribution Analysis 47
4.1 Stress Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.1.1 Stress Tensor Expression . . . . . . . . . . . . . . . . . 47
4.1.2 Numerical Error . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Compression Force Balance . . . . . . . . . . . . . . . . . . . 51
4.2.1 The Distribution of the Compression Stress . . . . . . 52
5 Conclusion and Future Works 58
A The Finite Dieren Version of the Divergence Theorem 60
B The table of the data in our simulation 63
C The Bending Stress 65
1] J.Leng, F.Nallet, and D.Roux, Eur.Phys.J. E,4, 337-341(2001).
[2] P.-G.de Gennes and T.Prost, The Physics of Liquid Crystals, 2nd
ed.,(Oxford University Press, Oxford, 1993)
[3] C.Pozrikidis, Introduction to Theoretical and Computational Fluid Dynamics,
Oxford University Press, Oxford, 1997
[4] Samuel A.Safran, Statistical Theomodynamics of Surfaces, Interfaces,
and Membranes, Weizmann Institute of Science, Rehovot, Israel
[5] E.van der Linden and J.H.M Dr˝oge, Physics A, 439-447(1993).
[6] D.Roux, in Theoretical Challenges in the Dynamics of Complex Fluids,
edited by T.Mc Leish, NATO ASI Series E, Vol. 339 (Kluwer Academic,
Dordrecht, 1997), p. 203;in Soft and Fragil Matter: Nonequilibrium
Dynamics, Metastability and Flow, edited by M.E.Cates and
R.Evans(Institute of Physics Publishing, Bristol, 2000), p. 185
[7] L.D.Landau and E.M.Lifshitz, Theory of Elasticity, 3rd ed., (Pergamon
Press, London, 1986).
[8] M.Kleman and O.D.Lavrentovich, Soft Matter Physics:An Introfuction.
[9] D.Roux, F.Nallet, and O.Diat, Europhys.Lett.24, 53(1993).
[10] Lu C-Y D., Cates M.E.,to be published.
[11] M.Kleman and O.Parodi, J.Phys.Paris,36,671(1975).
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