我們以理論分析及分子動力學模擬的方式研究了高分子鏈從空腔中釋
放出來的過程。我們發現整個過程可以分成兩個階段，第一階段是球形膨
脹階段，此時整個分子鏈的形狀像是一顆球，而整個系統的自由能則遵
循F ∼ (b/R)^(3/3ν−1)×N^(3ν/3ν−1)。藉由平衡自由能變化率和能量耗散率，我們推得鏈的尺寸隨時間的變化公式為R(t) = R0×(1+ t/τs)^(3ν−1/6ν+1)，其中τs是球形膨脹階段的特徵時
間，其形式遵守τs ∼ R0^(6ν+1/3ν−1)N^(−1/3ν−1)。在模擬中，我們發現實際的τs大約正比於N。
當分子鏈尺寸R變成正比於Nν時，球形膨脹階段結束，第二階段開始。第二階
段叫做線圈鬆弛階段。此時整個分子鏈的形狀像是一團線圈。如果高分子鏈是
在易溶溶液中，其系統自由能會變為F ∼ −logR+ R^2Nb^2+vN2/R^3。藉由平衡能量變
化率，我們可得出分子鏈尺寸隨時間變化的情形:R(t) ≃ (Rrx^5 − C1 exp(−(t−tb)/τc) )^(1/5)，其中Rrx是分子鏈在鬆弛情況下的尺寸，τc是這個階段的特徵時間。根據模擬結
果，我們得知第二階段佔整個釋放過程的大部分時間，因此高分子鏈的鬆弛時
間應該會正比於τc。τc理論預測會正比於N^2，然而模擬的結果告訴我們τc大略正
比於N^2.4，因此整個高分子鏈的鬆弛時間也大約正比於N2.4。

We investigate single chain releasing from a cavity via theoretical analysis and
molecular dynamics simulation. We discover that the releasing process can be
divided into two stages. In the first stage, the compressed chain expands like a
sphere. The free energy of the system is calculated. By balancing the rate of
free energy change with the rate of energy dissipation, we derive the variation
of chain size to be R(t) = R0×(1+ t/τs)^(3ν−1/6ν+1)where τs is the characteristic time.
Our simulations show that τs is about proportional to N. The sphere expansion
stage ends at the moment when R turns to scale like N^ν and the second stage
starts, called coil relaxation stage. In the stage, the chain relaxed like a coil.
The free energy of the system becomes F ∼ −logR+ R^2Nb^2+vN2/R^3. The balance
equation is solved, and the chain size respects R(t) ≃ (Rrx^5 − C1 exp(−(t−tb)/τc) )^(1/5) . τc
is the characteristic time for the coil relaxation stage and behaves like N^2.4. We
find that a releasing process spends most of the time on the coil relaxation stage.
Therefore, the total relaxation time is determined by τc.

Abstract (Chinese) I
Abstract II
Acknowledgements (Chinese) III
Contents IV
List of Figures VI
List of Tables X
1 Introduction 1
1.1 Structure of virus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Infection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Paper Review and Motivation . . . . . . . . . . . . . . . . . . . . . 8
2 Theory 11
2.1 Sphere Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.1.1 Zimm Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.1.2 Rouse Model . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2 Coil Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Method and Simulation 19
3.1 LAMMPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Interaction Between Monomers . . . . . . . . . . . . . . . . . . . . 20
3.3 Capsid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.4 Lagevin Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.5 Chain Length and Volume Fraction . . . . . . . . . . . . . . . . . . 21
3.6 Simulation Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6.1 Loading Phase . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.6.2 Equibration Phase . . . . . . . . . . . . . . . . . . . . . . . 23
3.6.3 Releasing Phase . . . . . . . . . . . . . . . . . . . . . . . . . 24

4 Results and Discussion 25
4.1 Chain Size v.s. Time . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4.2 R ≃ Nγ(et) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.3 Domination of Coil Relaxation . . . . . . . . . . . . . . . . . . . . . 31
4.4 Sphere Expansion Stage . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 Coil Relaxation Stage . . . . . . . . . . . . . . . . . . . . . . . . . . 36

5 Conclusion 41
Bibliography 43

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