# Changes in State: Simulations of Aggregation and Ordering in Finite Systems translation missing: zh.hyrax.visibility.files_restricted.text

## Patel, Lara (Summer 2018)

Published

## Abstract

This dissertation presents three projects that use molecular dynamics (MD) simulations to investigate finite size effects on changes in state. The first project aims to explain the exponential decay during the sub-microsecond phase of melting kinetics of unilamellar vesicle lipid ordering observed in ultrafast IR temperature-jump experiments. MD simulations of small unilamellar vesicles of MARTINI coarse- grained DPPC lipids model the response to an instantaneous temperature jump from 280 K to final temperatures of 290 K to 310 K. Instantaneous heating led to partial or total melting, changes in vesicle shape, and the sizes and arrangements of the remaining gel-phase domains. At temperatures that produced partial melting, the gel-phase lipid content of the vesicles followed an exponential decay, consistent in form and timescale with experiment. The changing rate of melting results from the curvature stress arising from the expansion of the bilayer area upon melting competing with the confinement effect of a fixed internal volume. The subsequent projects employ a global fitting analysis method that obtains free energies of association from equilibrated cluster size frequency distributions of unbiased constant-temperature MD simulations. Simulated systems are typically too small for the law of mass action to accurately describe the aggregation statistics. This method relies on iteratively determining a set of cluster free energies that, using appropriately weighted sums over all possible partitions of N monomers, best reproduce the cluster size distributions. To showcase the method, a united-atom model of methyl tert-butyl ether is simulated in the vapor phase and in explicit water solution over a range of system sizes and concentrations. Bypassing the explicit generation of partitions significantly enhances the efficiency of this method and is named the Partition-Enabled Analysis of Cluster Histograms (PEACH) method. It is used to calculate the free energy surface of NaCl aggregation in MD simulations in four solvents (pure methanol, pure water, and two methanol/water mixtures). The presence of non-ideal crowding effects and the systematic concentration-dependent indicators in the results of the PEACH model fit are addressed. Insights into the proposed two-step mechanism of crystal nucleation and its dependence on solvent and degree of supersaturation are discussed.

## Table of Contents

List of Figures…

List of Tables…

1. Introduction … 1

2. Molecular dynamics … 5

2.1 Temperature and pressure coupling … 5

2.1.1 Berendsen thermostat … 6

2.1.2 Velocity rescaling thermostat … 7

2.1.3 Berendsen barostat … 7

2.2 Potentials and coarse graining … 8

3. Coarse-grained molecular simulations of the melting kinetics of small unilamellar vesicles … 10

3.1 Methods … 12

3.1.1 Bilayer patch simulations … 13

3.1.2 Vesicle construction and solvation … 13

3.1.3 Vesicle equilibration … 14

3.1.4 Temperature jump simulations … 15

3.1.5 Order parameter … 15

3.1.6 Relative shape anisotropy … 18

3.2 Results and discussion … 18

3.2.1 Vesicle equilibration … 18

3.2.2 Vesicle melting rates: Comparison with experiment … 21

3.2.3 Vesicle melting: Structural changes … 24

3.2.4 Vesicle melting rates: Interpretation ... 27

3.2.5 Topological evolution … 34

3.2.6 Refreezing … 35

3.2.7 Implications for dynamics beyond 500 ns … 38

3.3 Conclusions … 38

3.4 Acknowledgements … 40

4. PEACH method development … 41

4.1 What is a partition function? … 41

4.1.1 Volume scaling of the partition function … 44

4.2 The law of mass action … 45

4.2.1 The limitations of the law of mass action … 47

4.2.2 Finite size effects: What happens to the free energy surface in a simulation? … 50

4.3 PEACH: Partition enabled analysis of cluster histograms ... 54

4.3.1 Canonical ensemble partition function ... 55

4.3.2 Grand canonical ensemble partition function ... 56

4.3.3 ⟨ns⟩N from ⟨ns⟩λ … 58

4.3.4 Bypassing a sum over partitions in the calculation of ⟨ns⟩N … 59

4.4 Global fitting … 63

4.4.1 Evaluating the quality of the fit … 67

4.5 Free energy profile … 69

4.5.1 Classical nucleation theory (CNT) … 70

5. Cluster free energies from simple simulations of small numbers of aggregants: Nucleation of liquid MTBE from vapor and aqueous phases … 72

5.1 Methods … 75

5.1.1 General MD simulation methods … 75

5.1.2 Simulations of equilibrium cluster aggregation … 76

5.1.3 Calculation of surface tension … 77

5.2 Estimation of error in the global fit … 78

5.3 Results … 79

5.3.1 Cluster definition … 79

5.3.2 Global fitting … 82

5.4 Discussion … 86

5.5 Conclusions … 94

5.6 Acknowledgments … 95

6. Simulations of NaCl aggregation from solution: Solvent determines topography of free energy landscape … 96

6.1 Introduction … 96

6.2 Methods … 100

6.2.1 Cluster definition … 104

6.2.2 Order parameter definition … 104

6.2.3 Implementation of the PEACH method … 107

6.3 Results … 108

6.3.1 Pure methanol … 108

6.3.2 Mixed solvent simulation results … 111

6.3.3 Pure SPC/E water solvent simulation results … 116

6.3.4 Ordered clusters in water … 120

6.3.5 Classical nucleation theory and amorphous NaCl clusters ... 122

6.4 Conclusions … 124

6.5 Acknowledgments … 126

A. Derivations … 127

A.1 Ensemble averages … 127

A.1.1 Cluster co-frequency ⟨mjms⟩N … 127

A.1.2 Frequency of seeing two of the same cluster size simultaneously ⟨ mj^{2}−mj⟩N … 128

A.2 Defining the exact equilibrium association constant as a function of co-frequency … 130

A.2.1 Deriving ln[Kactual,j/KLoMA,j] … 131

B. Coarse-grained molecular simulations of the melting kinetics of small unilamellar vesicles … 134

B.1Truncatedicosahedronassembly … 134

B.1.1 Hexagonal faces … 135

B.1.2 Pentagonal faces … 135

B.2 Supplementary Figures … 137

B.2.1 Topological evolution … 137

C. Cluster free energies from simple simulations of small numbers of aggregants: Nucleation of liquid MTBE from vapor and aqueous phases … 145

C.1 Establishing an optimal cluster definition … 145

C.2 Cluster size distributions and global fitting for the aqueous phase simulations … 148

D. Simulations of NaCl aggregation from solution: Solvent determines topography of free energy landscape … 151

D.1 Cluster definition and the convergence criteria … 151

D.2 Implementation of the PEACH method … 152

D.3 Quality of the fits … 154

D.3.1 Pure SPC/E solvated NaCl aggregation … 154

D.3.2 NaCl aggregation in a solvent mixture of Methanol and SPC/E water … 159

Bibliography … 161

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