薄膜揉皺的分子動力學模擬

Membrane Crumpling with Molecular Dynamics Simulation

羅鈞肇(Chun-Chao Lo)

指導教授 ： 洪在明

國立清華大學/理學院/物理學系/碩士(2012年)

https://doi.org/10.6843/NTHU.2012.00379

摘要

我研究的是薄膜揉皺的分子動力學模擬。揉皺力學是個新的物理領域，我有興趣的部分在模擬薄膜裝進一個球型的球殼，並施以均勻壓力，觀察薄膜的力學反應和統計行為。我們用的模型是三角晶格，在晶格點擺上粒子，形成一張二維的薄膜。

有外國研究團隊做電腦模擬發現如果對薄膜球施以均勻的外力，外力與半徑會形成簡單的冪次關係~(Power law)~，而且冪次與薄膜的材質、厚度與大小都沒有關係。先前學長已經用實驗證實半徑與外力的確會形成冪次關係，但是冪次會與材質有關，所以我們希望用電腦模擬確認他的實驗結果。

模擬容許我們研究許多實驗中難以量到的物理量，例如摺痕能量的分布以及內部結構，也可以觀看紙球揉皺的過程，內部結構如何變化，從中了解紙球是如何在內部形成支柱來抵抗外力。我們也有模擬混合揉皺，並與實驗結果對照。

為了分析摺痕的長度與能量分布，我寫了一個連線的程式，能夠輕易地使用滑鼠連出摺痕所在，並分析長度的分布，排除之前的研究使用水淹法不準的問題，而且能夠得知摺痕能量的分布，從結果中我們發現，在紙球揉皺過程的摺痕能量幾乎是與長度成正比，而不是與前人所預測的長度的1/3成正比。

我們也發現摺痕長度的平均值會與密度倒數成正比，並提出模型和理論來解釋這個結果。

關鍵字

揉皺 ； 薄膜 ； 分子動力學模擬

並列摘要

In this thesis, we use molecular dynamics (MD) simulation to study crumpled membranes. Although it exists in our everyday life,
there are still many mysteries in the properties related to crumpling
that are starting to be unraveled in recent years. We use MD to study the
mechanical properties of
a thin sheet being crumpled under an ambient pressure. The way we
proceed is to confine the fictitious sheet
by a spherical shell under an external pushing force. In
our MD, we use a triangular lattice model in which particles
are simulated by lattice points and form a two-dimensional membrane.

Two previous Nature-Material papers have confirmed a scaling law between the
radius and external force
and concluded that the power of the relation was universal - independent of
materials, thickness and size of the sheet. Through real experiments, our
group have upheld the scaling law, except that we found the exponent to vary
with different made of the sheet. Why did the previous renowned workers fail
to find this? If they had made a mistake, is there any other conclusion of
theirs that will be cast into doubt? To explain all these questions,
we set out to redo the MD simulation ourselves.
Simulations allow us to
measure and observe many physical quantities that is hard or impossible to
obtain in
experiments. For example, we can know (1) how the ridges corroborate in
their formation to resist against the external force, (2) the statistical
distribution of ridge length, (3) how the stored energy varies with the
ridge length, (4) all the questions above for the case when two sheets of
different made are crumpled together (we have a lab mate, Ming-Han Chou, who
is in charge
doing this experiment so that we can compare our results). To obtain the above
mentioned distributions, I wrote a program that enables me to build
the net of ridges fast and easily. With the help of this program, the
inaccuracy of the locations and lengths of ridges from
Watershed Algorithm was minimized. In the end, we found the
energy to be linearly proportional to the ridge length, instead of having a
one-third power as Professor Witten claimed. Besides, the
average ridge length is found to be proportional to the inverse of
the density. We proposed a simple model which can satisfactorily explain all
our findings.

並列關鍵字

Crumpling ； Membrane ； Molecular Dynamics Simulation

參考文獻

[1] E. Sultan and A. Boudaoud, Phys. Rev. Lett. 96, 136103 (2006).

[2] G. A. Vliegenthart and G. Gompper, Nat Mater 5, 216 (2006).

101, 125504 (2008).

[4] Y.-C. Lin et al., Phys. Rev. Lett. 103, 263902 (2009).

[5] A. Wood, Physica A 313, 83 (2002).