title: Molecular dynamics simulations of polymer electrolytes in the bulk and at interfaces: methods and atomistic insights
Creators: name: Thum, Andreas
Publisher: name: University of Muenster, Institutional Repository MIAMI
Year: 2025
Subjects: subject: 540 Chemistry and allied sciences

description: Polymerelektrolyte haben das Potenzial, die Sicherheit heutiger
Lithium-Ionen-Batterien zu erhöhen und neue Zellchemien mit höheren
Energiedichten zu ermöglichen.
Diese Arbeit liefert atomistische Einblicke in die Koordination und den
Transport von Lithium-Ionen in gebräuchlichen PEO-LiTFSA-Elektrolyten im
Volumen und an Elektrodengrenzflächen mittels Molekulardynamik-Simulationen
(PEO = Polyethylenoxid,
LiTFSA = Lithiumbis(trifluormethylsulfonyl)amid).
Der Einfluss unterschiedlicher Kettenlängen, Salzkonzentrationen und
Elektrodenoberflächenladungen auf die (Grenzflächen-)Struktur und Dynamik der
Elektrolyte wird untersucht.

Darüber hinaus deckt diese Arbeit Schwachstellen und Fehlinterpretationen von
zwei etablierten und weit verbreiteten Nachbearbeitungs- und Analysemethoden
für Molekulardynamik-Simulationen auf, und zwar bei der Trajektorienentfaltung
und bei der Bestimmung von Lebenszeiten (z.B.\ Koordinationszeiten oder
Schichtverweilzeiten) auf der Basis von Autokorrelationsfunktionen.
Verbesserungen und Alternativen werden diskutiert.
description: Polymer electrolytes have the potential to increase the safety of today's
lithium-ion batteries and to enable new cell chemistries with higher energy
densities.
This thesis provides atomistic insights into the lithium-ion coordination and
transport in common PEO-LiTFSI electrolytes in the bulk and at electrode
interfaces using molecular dynamics simulations
(PEO = poly(ethylene oxide),
LiTFSI = lithium bis(trifluoromethanesulfonyl)imide).
The impact of different chain lengths, salt concentrations and electrode
surface charges on the (interfacial) electrolyte structure and dynamics is
explored.

Moreover, this thesis unveils flaws and misinterpretation of two
well-established and widely used post-processing and analysis methods for
molecular dynamics simulations, namely trajectory unwrapping and lifetime
estimation (e.g. of bond lifetimes or layer residence times) from
autocorrelation functions.
Improvements and alternative methods are discussed.
description: Abstract ..... i
Kurzfassung ..... ii
Contents ..... iii
1 Introduction ..... 1
I Theoretical Background ..... 5
2 Molecular Dynamics Simulations ..... 7
2.1 Introduction ..... 7
2.2 Basic Procedure ..... 7
2.3 The Force Field ..... 9
2.3.1 Non-Bonded Interactions ..... 9
2.3.2 Bonded Interactions ..... 10
2.3.3 Accounting for Polarization: The Charge-Scaling Approach ..... 12
2.4 Integration Scheme ..... 13
2.5 Temperature and Pressure Control ..... 15
2.5.1 Thermostats ..... 15
2.5.1.1 Langevin Thermostat ..... 15
2.5.1.2 Berendsen Thermostat ..... 16
2.5.1.3 Nosé-Hoover Thermostat ..... 17
2.5.2 Barostats ..... 18
2.6 Periodic Boundary Conditions ..... 18
2.6.1 Minimum Image Convention ..... 19
2.6.2 Truncation and Ewald Summation ..... 21
2.7 Limitations of Molecular Dynamics Simulations ..... 22
3 General Observables and Analysis Methods ..... 23
3.1 Radial Distribution Function and Coordination Number ..... 23
3.2 Free-Energy Profile or Potential of Mean Force ..... 24
3.3 Mean Squared Displacement and Self-Diffusion Coefficient ..... 26
II Method Development and Evaluation ..... 29
4 Unwrapping Constant-Pressure Molecular Dynamics Trajectories ..... 31
4.1 Introduction ..... 31
4.2 Mathematical Analysis of the Unwrapping Algorithms ..... 33
4.2.1 Lattice-Based Methods ..... 34
4.2.1.1 Heuristic Unwrapping Method ..... 34
4.2.1.2 Hybrid Unwrapping Method ..... 38
4.2.1.3 Box-Coordinate Unwrapping Method ..... 40
4.2.2 Toroidal Methods ..... 42
4.2.2.1 Displacement Unwrapping Method ..... 42
4.2.2.2 Improved Displacement Unwrapping Method ..... 44
4.2.2.3 Interaction-Displacement Unwrapping Method ..... 45
4.2.3 Summary of the Unwrapping Methods ..... 47
4.3 (Re-)Wrapping Algorithms ..... 48
4.3.1 Lattice-Based (Re-)Wrapping ..... 48
4.3.2 Toroidal (Re-)Wrapping ..... 49
4.3.2.1 Rewrapping the Displacement and Improved Displacement Method ..... 49
4.3.2.2 Rewrapping the Interaction-Displacement Method ..... 49
4.4 Conclusions ..... 50
5 Calculating Lifetimes of Discrete States From Discrete Time Series ..... 51
5.1 Introduction ..... 51
5.2 Fundamentals of Survival Analysis ..... 54
5.2.1 Basic Definitions ..... 55
5.2.2 Common Model Distributions and Their Relations ..... 56
5.3 Mathematical Formalism of the Autocorrelation-Function Method ..... 58
5.3.1 The State Operator ..... 58
5.3.2 ACF of the Intermittent State Operator ..... 59
5.3.3 ACF of the Continuous State Operator ..... 59
5.4 Simulation Details ..... 63
5.5 Results and Discussion ..... 65
5.5.1 The Convergence Issue of the Intermittent ACF ..... 65
5.5.2 Impact of Trajectory Size ..... 66
5.5.3 Impact of Lifetime Distribution ..... 67
5.5.4 Real-Life Example ..... 70
5.6 Conclusions ..... 71
III Molecular Dynamics Simulations of PEO-LiTFSI Electrolytes ..... 73
6 Force Field Validation ..... 75
6.1 Introduction ..... 75
6.2 Simulation Details ..... 76
6.3 Results and Discussion ..... 77
6.3.1 Densities ..... 77
6.3.2 Radial Distribution Functions ..... 78
6.3.3 Cation-Anion Coordination Numbers ..... 80
6.3.4 Cation-Polymer Coordination Numbers ..... 81
6.3.5 Mean Squared Displacements and Self-Diffusion Coefficients ..... 83
6.4 Conclusions ..... 85
7 Bulk Properties of PEO-LiTFSI Electrolytes ..... 87
7.1 Introduction ..... 87
7.2 Simulation Details ..... 88
7.3 Results and Discussion ..... 91
7.3.1 Lithium-Ion Coordination ..... 91
7.3.2 Renewal of the Lithium-Ion Coordination Environment ..... 94
7.3.2.1 Autocorrelation Functions ..... 94
7.3.2.2 Correlation Times ..... 95
7.3.2.3 Renewal Times ..... 96
7.3.2.4 MSD During the Renewal Time ..... 97
7.3.3 Self-Diffusion Coefficients of the Electrolyte Components ..... 98
7.3.4 Effect of Salt Concentration ..... 100
7.3.5 Molar Volumes of the Electrolyte Components ..... 102
7.3.6 Specific Volumes of the Electrolyte Mixtures ..... 105
7.4 Conclusions ..... 107
8 Interfacial Behavior of PEO-LiTFSI Electrolytes ..... 109
8.1 Introduction ..... 109
8.2 Simulation Details ..... 110
8.3 Results and Discussion ..... 112
8.3.1 Layering at the Electrode-Electrolyte Interface ..... 112
8.3.2 Lithium-Ion Coordination ..... 117
8.3.3 Free-Energy Profiles ..... 119
8.3.4 Free-Energy Peak Positions and Barrier Heights ..... 121
8.3.5 Layer Residence Times ..... 122
8.3.6 Ligand Substitution Upon Barrier Crossing ..... 126
8.3.7 Renewal Times ..... 129
8.3.8 Density Distribution Within the First Lithium-Ion Layer ..... 131
8.3.9 Free-Energy Barriers on the Electrode Surface ..... 132
8.3.10 Residence Times on the Electrode Surface ..... 133
8.3.11 Lithium-Ion Displacement ..... 134
8.4 Conclusions ..... 137
8.5 Outlook ..... 139
9 Summary ..... 141
IV Appendices ..... 145
A Fluctuations of the Kinetic Energy in the Canonical Ensemble ..... 147
A.1 Variance of the Kinetic Energy ..... 147
A.2 Variance of the Instantaneous Temperature ..... 149
A.3 Distribution of the Kinetic Energy ..... 150
A.4 Probability That the Kinetic Energy of a Particle is Higher Than a Threshold ..... 151
B Additional Information to “Unwrapping Constant-Pressure Molecular Dynamics Trajectories” ..... 153
B.1 Minimum Image Convention in the Box Coordinate System ..... 153
B.2 Alternative Derivations ..... 153
B.2.1 Alternative Derivation of the Hybrid Method ..... 153
B.2.2 Alternative Derivation of the Box-Coordinate Method ..... 154
B.3 Analytical Application of the Unwrapping Algorithms to a Stochastic MD Model ..... 155
B.3.1 Lattice-Based Methods ..... 156
B.3.1.1 Unwrapping the Model With the Heuristic Method ..... 156
B.3.1.2 Unwrapping the Model With the Hybrid Method ..... 156
B.3.1.3 Unwrapping the Model With the Box-Coordinate Method ..... 157
B.3.2 Toroidal Methods ..... 157
B.3.2.1 Unwrapping the Model With the Displacement Method ..... 157
B.3.2.2 Unwrapping the Model With the Improved Displacement Method ..... 157
B.3.2.3 Unwrapping the Model With the Interaction-Displacement Method ..... 158
B.4 Generalization of the Newly Proposed Methods to Triclinic Boxes ..... 158
B.4.1 Generalization of the Box-Coordinate Method ..... 159
B.4.2 Generalization of the Improved Displacement Method ..... 159
B.4.3 Generalization of the Interaction-Displacement Method ..... 159
B.5 Differences When Using the Alternative Equations for the Interaction and Scaling Displacement . . 159
C Additional Information to “Calculating Lifetimes of Discrete States From Discrete Time Series” ..... 161
C.1 Alternative Time Averaging When Calculating the ACF ..... 161
C.2 Relations of the Exponential Distribution to Some Other Probability Distributions ..... 162
C.2.1 First-Order Decay With a Single Decay Rate ..... 162
C.2.1.1 Case I: Constant Decay Rate (Exponential Distribution) ..... 162
C.2.1.2 Case II: Time-Dependent Decay Rate ..... 163
C.2.2 First-Order Decay With a Discrete Distribution of Decay Rates ..... 163
C.2.2.1 Example I: Serial Decay Paths (Hypoexponential Distribution) ..... 163
C.2.2.2 Example II: Parallel Decay Paths (Hyperexponential Distribution) ..... 167
C.2.3 First-Order Decay With a Continuous Distribution of Decay Rates ..... 168
C.2.3.1 Example I: Gamma Distributed Decay Rates (Lomax Distribution) ..... 170
C.2.3.2 Alternative Interpretation of the Lomax Distribution ..... 171
C.2.3.3 Example II: Decay Rates Distributed According to a One-Sided Lévy Distribution
(Weibull Distribution) ..... 171
D Additional Information to “Bulk Properties of PEO-LiTFSI Electrolytes” ..... 173
D.1 Relation Between the Li/EO Ratio r and the Salt Weight Fraction w ..... 173
D.2 Radial Distribution Functions ..... 174
D.3 Lithium-Ion Coordination Numbers ..... 182
D.4 Lithium-Ion Coordination Motives ..... 191
D.5 Chain Dimensions: End-to-End Distance and Radius of Gyration ..... 194
D.6 Autocorrelation Functions ..... 195
D.7 Renewal Events ..... 199
D.8 MSDs and Self-Diffusion Coefficients of the Electrolyte Components ..... 200
D.9 Specific Volumes of the Electrolyte Mixtures ..... 207
E Additional Information to “Interfacial Behavior of PEO-LiTFSI Electrolytes” ..... 209
E.1 Layering at the Electrode-Electrolyte Interface ..... 210
E.1.1 Density Profiles ..... 210
E.1.2 Trajectory Snapshots ..... 221
E.1.3 Charge Density, Electric Field and Electric Potential ..... 225
E.2 Lithium-Ion Coordination ..... 237
E.3 Free-Energy Landscape for Lithium-Ion Transport ..... 248
E.3.1 Free-Energy Profiles ..... 248
E.3.2 Peak/Layer Identification and Simulation-Box Discretization ..... 252
E.3.3 Peak/Layer Clustering ..... 254
E.3.4 Free-Energy Peak Positions and Barrier Heights ..... 257
E.4 Layer Residence Times ..... 264
E.4.1 Mean Residence Times and Distribution Characteristics ..... 264
E.4.2 Residence-Time Histograms ..... 271
E.5 Ligand Substitution Upon Barrier Crossing ..... 278
E.6 Renewal Events (Interchain Transfers) ..... 306
E.7 Density Distribution Within the First Lithium-Ion Layer ..... 313
E.8 Free-Energy Barriers on the Electrode Surface ..... 319
E.9 Residence Times on the Electrode Surface ..... 323
E.10 Mean Displacement and Displacement Variance ..... 324
E.10.1 Lithium-Ion Mean Displacement vs. Time ..... 324
E.10.2 Lithium-Ion Displacement Variance vs. Time ..... 330
E.10.3 Lithium-Ion Mean Displacement: Cross Section ..... 337
E.10.4 Lithium-Ion Displacement Variance: Cross Section ..... 339
E.10.5 Ether-Oxygen Mean Displacement: Cross Section ..... 341
E.10.6 Ether-Oxygen Displacement Variance: Cross Section ..... 343
E.10.7 Lithium-Ion Displacement per Lithium-Ion Layer ..... 345
E.10.8 Ether-Oxygen Displacement per Lithium-Ion Layer ..... 352
Bibliography ..... a
Data and Code Availability ..... q
Declaration on Collaborations ..... s
Acknowledgments ..... u
Curriculum Vitae ..... w
Publications ..... y