臺灣博碩士論文加值系統

本論文主要包含兩部分的研究工作，第一部份的研究工作，主要進行相平衡 (phase equilibrium )現象的計算與分析，第二部分的工作，則針對液態--玻璃相變（liquid-glass transition）的現象進行研究。我們所研究的對象雖然都是帶電膠體系統（charged colloidal dispersion），但是從統計力學的角度來看，第一部份的工作與第二部分的工作兩者極為不同，這兩部分各別涉及了「平衡統計」與「非平衡統計」兩個領域。

This thesis embodies two parts. The …rst part concerns with calculations of phase
equilibrium phenomena and the methodology and analysis we proposed for a thorough un-
derstanding of phase transition. The second part aims at liquid-glass transition phenomena
applying the microscopic mode coupling theory to study the dynamical behaviors and ex-
ploiting the factors leading to glass transition. Although the physical system of interest in
both parts is the charge-colloidal dispersion, the physical process is, however, fundamentally
di¤erent from the viewpoint of statistical mechanics. The …rst part describes equilibrium
scenario. In this part, we revisited the present status of thermodynamic theories applied
to phase-diagram calculations. We put forth in this thesis a novel means of phase-diagram
calculations and introduce free energy landscape analysis to gain insight into phase separa-
tion phenomena. The second part, on the other hand, deals with nonequilibrium processes.
Here we investigate how the colloidal particles embedded in disperse medium can be driven
to a glassy or non-ergodic state. The mechanism of glass transition and how the mode
coupling theory was used to predict and analyze its occurrence are tersely covered in the
present work. We describe below the essential problems that we have touched on in these
two parts.

List of Figures v
List of Tables vii
I Phase-diagram of a charged colloidal dispersion 1
1 Introduction: 3
1.1 Free Energy minimization method . . . . . . . . . . . . . . . . . . . . . . . 4
1.1.1 Derivation of FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Thermodynamic basis for FEM Method . . . . . . . . . . . . . . . . . . . . 7
1.2.1 Euler Theory: Criteria for understanding the interaction in subsys-
tems and the size of subsystems . . . . . . . . . . . . . . . . . . . . . 7
1.2.2 Thermodynamic basis for the mixed free energy . . . . . . . . . . . . 8
1.3 Phase rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.3.1 What is the phase rule? . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.4 Which is more stable: i-phases or (i+1)-phases in coexistence? . . . . . . . 9
1.5 Free energy landscape method: an alternative analysis for understanding
phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
1.5.1 Mixed free energy in the free energy landscape . . . . . . . . . . . . 10
1.5.2 Proof of 2-phases in coexistence . . . . . . . . . . . . . . . . . . . . . 11
1.5.3 Proof of 3-phases in coexistence . . . . . . . . . . . . . . . . . . . . . 13
1.6 Free energy landscape analysis . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Understanding phase diagrams by FEL analysis 19
2.1 van-der Waals-like theory . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.1 Gibbs-Bogoliubov inequality . . . . . . . . . . . . . . . . . . . . . . . 20
2.1.2 Helmholtz free energies of a liquid and a solid . . . . . . . . . . . . . 21
2.2 Numerical result and FEL analysis . . . . . . . . . . . . . . . . . . . . . . . 22
2.3 Paradox between FEL and phase rule . . . . . . . . . . . . . . . . . . . . . 28
3 Triple point problem by FEM method 29
3.1 Volume proportions in phase separation at triple point . . . . . . . . . . . . 29
3.2 Problem of coexisting 3-phases . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.3 Puzzle of phase equilibrium equations of three coexisting phases . . . . . . 34
4 Charge-colloidal dispersion induced at moderate and very low salt con-
centration: FEM method 38
4.1 Charge-colloidal dispersion induced at very low electrolyte concentrations:
Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Model Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.3 Density functional theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.3.1 Minimization of F0hc . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.3.2 Minimization of F0el . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.4 E¤ective colloid interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
4.5 Total free energy of the colloidal suspension . . . . . . . . . . . . . . . . . . 47
5 Domains of phase separation in a charged colloidal dispersion driven by
electrolytes 50
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2.1 Helmholtz free energy: general . . . . . . . . . . . . . . . . . . . . . 54
5.2.2 Helmholtz free energy: (s)
0 . M . . . . . . . . . . . . . . . . . . . . 56
5.3 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 57
5.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
6 Charge-colloidal dispersion at very low salt concentration 65
6.1 Free energy-landscape analysis for a two-component system . . . . . . . . . 65
6.2 The free energy landscape analysis of a real two component system . . . . . 71
6.2.1 Information hidden in the free energy curve . . . . . . . . . . . . . . 71
6.2.2 Di¤erence between ‡uid and solid free energies in a phase equilibrium
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.3 Comparison between di¤erent physical parameters (ns,Z) . . . . . . . . . . . 75
Bibliography 82
II Liquid-glass transition: Mode Coupling Theory 87
7 Introduction 88
7.1 Summary of works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
7.2 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
7.2.1 Mean spherical approximation . . . . . . . . . . . . . . . . . . . . . 91
7.2.2 Mode coupling theory . . . . . . . . . . . . . . . . . . . . . . . . . . 93
8 Rescaled mean spherical approximation for concentrated charge-stabilized
colloids 95
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
8.2 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 97
8.2.1 Rescaling of S(q): numerical method . . . . . . . . . . . . . . . . . . 97
8.2.2 Rescaling of S(q): analytical method . . . . . . . . . . . . . . . . . . 102
8.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
9 Liquid-glass reentrant behavior in a charge-stabilized colloidal dispersion105
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
9.2 Numerical results and discussion . . . . . . . . . . . . . . . . . . . . . . . . 106
10 Liquid-glass transition phase boundary for a monodisperse charge-stabilized
colloids in the presence of an electrolyte 112
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
10.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
10.3 Theoretical predictions and experiments . . . . . . . . . . . . . . . . . . . . 117
10.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Bibliography 120

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