近20年來，薄膜揉皺的研究著重於力學性質(紙團大小和壓力的關係)、摺痕長度的機率分布（高斯或log-normal或其他）、所儲存的彎曲/拉伸位能比例、摺痕數目如何隨紙團大小改變、以及揉皺過程產生的噪音統計等，可資驗證的理論有適用在平面薄膜上的單一摺痕（亦即揉皺初期，摺痕彼此的作用還不重要時）之Witten風箏模型(kite model)。對於薄膜內在曲率和開放邊界的角色，並沒有深究，前者可以指汽車或血球，邊界的有無則可以指汽車的窗子有沒有打開。因此本研究室於前年以球殼為對象，採用分子動力學模擬研究，發現非零的雙向內在曲率，亦即所謂的非高斯曲面，和揉皺平面薄膜的性質不盡相同，例如：(1)摺痕的《彎曲與拉伸位能比值》與體積密度的關係。根據平面揉皺的風箏模型，如果摺痕彼此獨立，這個比值應該嚴格等於5，但是球殼的實驗結果卻給出1；至於揉皺晚期，當摺痕密度高到不容忽略摺痕之間與摺點之間的耦合時，這個原本大致不隨揉皺壓力改變的比值會突然明顯降低。(2)外力與紙團密度的關係，平面揉皺的情況可以透過平面面積與摺痕數目估算而得，但是球殼特有的「隕石坑」摺痕使得上述方式不再有效。最明顯的是平面揉皺的power law在球殼揉皺適用的壓力區間，變短到小於一個數量級。(3)摺痕所儲存的總位能與其長度的關係，對照風箏模型預測的早期1/3冪次，和我們實驗室先前的模擬發現晚期的正比關係，皆與球殼揉皺極為不同（早期得到1次方關係，晚期則增為2）。
本研究以真實球殼薄膜來做三維均向揉皺實驗，驗證了先前模擬的結果，發現大部份揉皺相關的性質皆和平面薄膜不同。總結目前發現的結果：
一、 與模擬吻合的部分，包括：平面揉皺所使用的風箏模型無法套用在球殼，這包括新的球殼揉皺的力學行為與機制、摺痕長度隨揉皺程度演化的冪次關係等。
二、 模擬沒有發現的現象，包括：摺痕的種類，除了做模擬的學長主張的（別於平面揉皺所產生之長條狀）「隕石坑(craters or indentations)」外，同時也觀察到許多長條摺痕、摺痕數目和平均長度平方的乘積大致和揉皺程度無關等。

In the past 20 years, crumpling membranes have been shown to exhibit many interesting mechanical and statistical properties. It remains an experimentally untested question whether these conclusions for flat sheets are applicable to an object exhibiting extrinsic curvatures, such as blood cells and automobiles.
Last year, one of our group members used Molecular Dynamics (MD) simulation to study ambient crumpling of a thin sphere in three dimensions. It came as a surprise and nightmare that most of the well-studied properties of flat-surface crumpling turned out to be drastically altered for spheres. Differences include:
(1) How the ratio of bending and stretching energies changes with volume density of the crumpled ball. According to Witten’s kite model, this ratio is a universal constant of 5, but becomes dependent on the thickness of sphere. When more crumpling drives the system into a more compact regime where many-body interactions are no longer negligible, the ratios in both cases are suppressed. Unfortunately we will not be able to explore energetic properties such as these in real experiments for obvious reasons.
(2) Similar to flat sheets, the relation between volume density and pressure also exhibits three different regimes: power-law, mixed, and compact. This is not surprising because the transition of these different behaviors is intimately correlated to the internal structure of the crumpled ball, which we expect to be insensitive to whether the membrane exhibit external curvature or not. Intuitively as the ball becomes smaller at high crumpling pressure, local layers tend to sacrifice their rotational degree of freedom to maximize transitional entropy. This is like the water molecules transit from gas (analogy to the power-law regime) to liquid (analogous to the scaling regime) with the mixed phase of gas and liquid at 0oC (analogous to the mix regime.) (3) Power-law relation between total energy stored in each ridge and its length remains valid in sphere. But, the exponent equals 1 in the power-law regime, instead of 1/3 for flat sheets. Into the scaling regime, it increases and saturates at 2, as opposed to 1 for flat sheets.
A short summary of our conclusions is:
(1) In line with the prediction by MD simulations: kite model for flat sheets is no longer applicable to spheres. This includes the mechanical behavior (i.e., how the ball radius shrinks with increasing pressure), the way ridge length evolves with crumpling, etc.
(2) Findings that MD did not predict: Actual deformations on crumpled sphere include not just “craters or indentations”, but also long ridges with vertices on both ends that are characteristic of flat sheets. The average ridge length and ridge number changes in a power-law fashion with the size of crumpled ball, but the product of number and square length turns out to be roughly a constant.

摘要 2
Abstract 3
目錄 5
第一章 簡介 7
1.1研究動機 7
1.2相關文獻 8
1.2.1 Witten’s lecture 8
1.2.2 Crumpling under an Ambient Pressure 11
1.2.3 Scaling relation for a compact crumpled thin sheet 14
1.2.4 Forced crumpling of self-avoiding elastic sheets 17
1.2.5 Effect of Ridge-Ridge Interactions in Crumpled Thin Sheets 18
1.2.6 Compression, crumpling and collapse of spherical shells and capsules 20
1.2.7 Power-law ansatz in complex systems: Excessive loss of information 22
第二章 實驗材料、實驗原理&相關操作流程 24
2.1實驗材料 24
2.2實驗原理 27
2.3實驗儀器 28
2.4實驗步驟 30
第三章 實驗結果與討論 34
3.1球殼受力與體積密度之關係 34
3.1.1假Power-law階段 35
3.1.2大小 37
3.1.3厚度 39
3.1.4材料 40
3.1.5有問題的數據之篩選及原因 43
3.2凹坑與擠壓半徑之關係 44
3.3揉皺噪音與出現機率實驗結果 49
第四章 結論 51
第五章 未來展望 52
參考資料 53

[1] A. J. Wood, Witten’s lectures on crumpling, Physica A 313, 83 (2002).
[2] Y. C. Lin, Y. L. Wang, Y. Liu, and T. M. Hong, Crumpling under an Ambient Pressure, Phys. Rev. Lett. 101, 125504 (2008).
[3] W. B. Bai, Y. C. Lin, T. K. Hou, and T. M. Hong, Scaling relation for a compact crumpled thin sheet, Phys. Rev. E 82, 066112 (2010).
[4] N. J. Wagner and J. F. Brady, Shear thickening in colloidal dispersions, Phys. Today 62, 27 (2009).
[5] G. A. Vliegenthart and G. Gompper, Forced crumpling of self-avoiding elastic sheets, Nature Mater. 5, 216 (2006).
[6] S. F. Liou, C. C. Lo, M. H. Chou, P. Y. Hsiao, and T. M. Hong, Effect of ridge-ridge interactions in crumpled thin sheets, Phys. Rev. E 89, 022404 (2014).
[7] G. A. Vliegenthart and G Gompper, Compression, crumpling and collapse of spherical shells and capsules, New Journal of Physics 13, 045020 (2011).
[8] S. T. Tsai et al., Power-law ansatz in complex systems: Excessive loss of information, Phys. Rev. E 92, 062925 (2015).
[9] MC Fokker, S Janbaz, and AA Zadpoor, Crumpling of thin sheets as a basis for creating mechanical metamaterials, RSC Advances 9, 5174 (2019).
[10] J. Baimova et al., Review on crumpled graphene: unique mechanical properties, Rev. Adv. Mater. Sci. 39, 69 (2014).
[11] J. Luo et al., Crumpled Graphene-Encapsulated Si Nanoparticles for Lithium Ion Battery Anodes, J. Phys. Chem. Lett. 3, 1824 (2012).
[12] J. Choi et al., Hierarchical, Dual-Scale Structures of Atomically Thin MoS2 for Tunable Wetting, Nano Lett. 17, 1756 (2017).