二維氣墊床顆粒液體的普適擴散行為與微觀動力學

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王正宏(Chen-Hung Wang) 查詢紙本館藏

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論文名稱

二維氣墊床顆粒液體的普適擴散行為與微觀動力學
(Universal scaling laws of diffusion and microscopic dynamics of 2D granular liquid in an airtable granular system)

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摘要(中)

自從上個世紀五零年代左右, Enskog在假設碰撞只牽涉到兩個分子的前提下給出了波茲曼方程的近似解. 這是第一次科學家能夠對氣體分子擴散有一個普適與成功的描述. 除了Rosenfeld在1977年的先驅工作, 擴散理論上的進展停滯了將近40年之久. 其中的困難點主要是在於液體分子間的距離只比固體的多百分之十左右, 所以液體分子的擴散機制是迥異於氣體的(碰撞). 液體分子大部分的時間都被附近鄰居困在原地作微小的隨機震盪. 只有當其附近的鄰居在某個時刻出現較為鬆散的結構時, 該液體分子才可以離開原地而擴散. 在氣體中, 因為分子距離較遠, 因此碰撞的擴散機制較與分子間作用力無關. 然而, 在液體中分子的距離是如此的靠近, 因此形形色色的作用力, 也對描述液體結構造成了困難. 因此對液體分子的擴散一直缺乏一個普適性的描述.
1996年, Dzugutov基於上述機制, 提出了一個普適的, 經驗性的指數定律, 可以很好的解釋不同液體分子的擴散. 2004年Samanta等人, 更從Mode coupling theory出發得到了更完全的結果. 然而由於同時量測分子擴散與結構上的困難, 長久以來都沒有實驗能夠驗證這些關係.
在這篇論文裡, 我們建造了一個新的二維氣墊床顆粒系統. 在這個系統裡的顆粒能夠浮在一個通氣平板上. 依靠改變顆粒間的作用力, 密度, 以及風速, 這些鈕扣狀顆粒可以形成稀薄液體(模擬氣體), 稠密液體(液體), 與晶體. 這些顆粒的速度分佈與真實液體相同, 都是高斯分佈. 藉由此分佈, 我們定義了這些顆粒的”溫度”. 這個速度分佈只有在顆粒密度足夠高(依舊稀薄)的條件下才會發生, 所以我們推論其來源並不是氣墊床的氣體, 而是經由多體互相交互作用而來. 我們同時也量測了液體的結構, 擴散微觀運動與擴散行為. 最後將所量測到的擴散關係與至少三個上述的理論比較. 結果相符合的程度令人驚訝, 特別是這些理論是針對三維真實液體而不是二維顆粒流體. 這篇論文也闡述了這個新的系統的核心組成, 與這個系統的基本特性.

摘要(英)

In this thesis, a new 2D air table granular system has been built and demonstrated its advantages on ”simulating” such transportation and micro-dynamics of real liquids.
The main components of this air table system are a pinhole plate and a vertical wind tunnel.
The detail of how we made such pinhole plates and the design of the vertical wind tunnel are also given in this article.
Besides of the pinhole plate and the uniform airflow, the flexibility on the control of interactions and the shapes of particles, compared to traditional granular systems, also makes this system to be suitable on studying the topics in related condensed matter.
We found the statistics of these far from equilibrium granules is very close to the Boltzmann statistics and reveals Gaussian velocity distribution functions (GVDF).
The origin of the GVDF, unlike to previous works, is from many-body dynamics, which makes these granular liquids very similar to real liquid molecules.
We also found that the reduced dimensionless diffusion constant $D^*$ and excess entropy $S^*_2$ follow two distinct scaling laws D*~e^{S*2} for dense liquids and D*~ e^{3S^*_2} for dilute ones.
The scaling for dense liquids is very similar to the laws for 3D real liquids proposed previously [Nature (London) 381, 137 (1996); Phys. Rev. Lett. 92, 145901 (2004)].
In the dilute regime, a power law [J. Phys. Condens. Matter 11, 5415 (1999)] also fits our data reasonably.
In our system, particles experience low air drag dissipation and interact with each other through embedded magnets.
These near-conservative many-body interactions together with Boltzmann statistics, we believe, are responsible for the satisfied correspondence between our results to those laws for 3D real liquids.
The dominance of cage relaxations in dense liquids also leads to the two different scaling laws for dense and dilute regimes.
The phases diagrams with number densities and ”temperature” are investigated and the structures are identified by the pair correlation function.
A detail description on microscopic stringlike trajectories is also given in this work.
The stringlike trajectories are the trace of 7-fold defects accompanied by rotated (caged) order patches on both sides.

關鍵字(中)

★ 顆粒系統
★ 液體
★ 擴散

關鍵字(英)

★ granular system
★ liquids
★ diffusion

論文目次

1 Introduction 1
2 Granular systems 4
2.1 Dissipative natural of granular system . . . . . . . . . . . . . . 5
2.2 Typical granular systems . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Air table systems . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Self-diffusion for gas and simple liquid 10
3.1 Self-diffusion in gas . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.2 2D melting and defects . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Self-diffusion in simple liquid . . . . . . . . . . . . . . . . . . . . 14
3.4 Generalized Langevin equation and other theories for simple liquid 16
4 Setup and image processing 18
4.1 Pinhole plate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.2 Fan controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Low speed vertical wind tunnel . . . . . . . . . . . . . . . . . . 23
4.4 Granular discs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.5 Video processing . . . . . . . . . . . . . . . . . . . . . . . . . . 26
5 Results 29
5.1 Effect of isotropic pinholes, nite inter-hole distance and the rule
of magnetic force . . . . . . . . . . . . . . . . . . . . . . . . . . 29
i
5.2 Granular crystal, dense liquid and dilute liquid . . . . . . . . . . 32
5.3 Gaussian velocity distribution . . . . . . . . . . . . . . . . . . . 32
5.4 Diffusion dynamics and dynamic heterogeneous . . . . . . . . . 39
5.5 Diffusion constants . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.6 Pair correlation and excess entropy . . . . . . . . . . . . . . . . 44
5.7 Universal scaling laws of diffusion . . . . . . . . . . . . . . . . . 48
6 Conclusion 52
7 Bibliography 55

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指導教授

陳培亮

審核日期

2015-1-29

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