Interfacial Potentials in Ion Solvation
A dissertation submitted to the
Graduate School of the University of Cincinnati
in partial fulfillment of the requirements
for the degree of
Doctor of Philosophy
in the Department of Physics
of the McMicken College of Arts and Sciences
by
Carrie Conor Doyle
B.S. in Physics, Rutgers, the State University of New Jersey, 2013
June 2020
supervised by
Dr. Thomas L. Beck
Committee Co-Chair: Dr. Carlos Bolech, Physics
Committee Member: Dr. Rohana Wiedjewardhana, Physics
Committee Member: Dr. Leigh Smith, Physics Abstract
Solvation science is an integral part of many fields across physics, chemistry, and biology.
Liquids, interfaces, and the ions that populate them are responsible for many poorly understood natural phenomena such as ion specific effects. Establishing a single-ion solvation free energy thermodynamic scale is a necessary component to unraveling ion-specific effects.
This task is made difficult by the experimental immeasurability of quantities such as the interfacial potential between two media, which sets the scale. Computer simulations provide a necessary bridge between experimental and theoretical results. However, computer models are limited by the accuracy-efficiency dilemma, and results are misinterpreted when the underlying physics is overlooked. Classical molecular dynamic techniques, while efficient, lack transferability. Quantum-based ab initio techniques are accurate and transferable, but their inefficiency limits the accessible simulation size and time. This thesis seeks to determine the physical origin of the interfacial potential at the liquid-vapor interface using classical models.
Additionally, I assess the ability of Neural Network Potential (NNP) simulation methods to produce electrostatic properties of bulk liquids and interfaces. Complicating factors are minimized through a simple water model (SPC/E) free of experimental parametrization and a
finite droplet simulation free of Ewald effects. Multipolar decomposition of the potential in the region of zero charge density provides a direct method for determining the potential felt by ions near interfaces. Non-aqueous solvents are studied through an OPLS-AA based model of the organic liquid ethylene carbonate (EC) using the same approach to compare aqueous and non-aqueous solvents. Neural network potentials may be a step towards the higher-level
1 need for predictive models, but they require further testing. Using an existing NNP framework, I train models for dynamics, as well as multipole electrostatics. Combining the two in both a bulk and interfacial system allows for the calculation of interfacial electrostatic properties. My results for water elucidate the reason for widely varying net potentials calculated for various models with similar dipole but differing quadrupole moments.
Near-cancelling dipole contributions between the droplet interface and the cavity interface of a solvated ion leaves the quadrupole as the dominant contribution to the net potential.
Molecular density profiles and potential profiles show that a length scale of 5 Å from cavity boundary is needed for a convergent potential. A theoretical argument for the radial dependence of each contribution is made, which supports my results. EC also has this radial dependence but has a different length scale of convergence. Differences in the molecular size, orientation, hydrogen-bonding capabilities, and multipole moments results in solvent-specific net potential contributions. This is evidenced by the results for charged cavities and an orientational analysis of EC. NNPs are shown to provide excellent agreement with ab initio electrostatic properties. This is encouraging evidence for the use of NNPs in the calculation of thermodynamic properties and in force field development.
2
Acknowledgements
I would like to express my gratitude to all those who were a part of my PhD studies. The
guidance and support of my advisor Dr. Thomas L. Beck was an essential part of my thesis work. His vast interdisciplinary knowledge and encouragement of independent research was a
constant source of motivation. I thank Dr. Bolech, Dr. Wiedjewardhana, and Dr. Smith for
their contributions as my committee members, and as professors, along with others at the
University of Cincinnati and Rutgers University who prepared me for rigorous scientific
investigations.
Lab group members Zohre Gorunmez, Mimi Liu, and Andrew Eisenhardt were always there
to commiserate and grow with. Extensive discussions and collaboration with Yu Shi were an
important part of the evolution of my research and a fond memory. I am forever grateful for
the friends I met along the way.
Finally, I thank my family, Lorraine, William, and Jessie Doyle for the constant support and
encouragement, no matter the distance between us.
4 Yes, as everyone knows, meditation and water are wedded for ever. —Herman Melville
Dedicated to William Doyle
5 Contents
1 Introduction 13
1.1 Preface...... 13
1.2 Evolution of the Computer Simulation of Liquids...... 16
1.2.1 History...... 17
1.2.2 Connection with Theory and Experiment...... 18
1.3 Solvation Science...... 20
1.3.1 Mixtures and Phase Separation...... 20
1.3.2 Proteins and Membranes...... 22
1.3.3 Specific Ion Solvation: Hofmeister Series...... 24
1.4 Theories and Simulation of Ion Solvation...... 27
1.4.1 Past...... 27
1.4.2 Present...... 30
Classical Simulation...... 30
Ab Initio Simulation...... 32
Neural Network Potentials...... 33
1.5 Interfacial Potential Effects...... 35
1.5.1 Thermodynamic Scale...... 35
1.5.2 Are they measurable?...... 38
6 CONTENTS
1.5.3 What are we measuring?...... 41
1.6 Summary...... 45
2 Theory 48
2.1 Preface...... 48
2.2 Classical Molecular Dynamics...... 48
2.2.1 Equations of Motion...... 49
2.2.2 Thermodynamic Ensembles and Equilibration...... 50
2.2.3 Classical Force Fields...... 50
2.2.4 SPC/E Force Field...... 52
2.2.5 OPLS-AA...... 53
2.2.6 Boundary Conditions...... 53
2.3 Ab Initio Molecular Dynamics...... 54
2.4 Neural Network Potentials...... 56
2.5 Thermodynamics of Ion Solvation...... 59
2.5.1 The Potential Distribution Theorem...... 60
2.5.2 Quasichemical Theory...... 61
2.5.3 Interfacial Potentials...... 63
2.6 Macroscopic Interfacial Electrostatics...... 65
2.6.1 Multipole Expansion of a Charge Distribution...... 66
2.6.2 Coordinate Systems: Cartesian and Spherical...... 67
2.6.3 Molecular Multipole Expansion of Electrostatic Potential...... 70
3 Water Liquid-Vapor Interfacial Potential Shifts 71
3.1 Preface...... 71
7 CONTENTS
3.2 Computational Methods...... 73
3.3 Results and Discussion...... 74
3.4 Conclusion...... 84
4 Ethylene Carbonate Liquid-Vapor Interfacial Potential Shifts 87
4.1 Preface...... 87
4.2 Computational Methods...... 90
4.3 Results and Discussion...... 91
4.3.1 Multipole moments of EC and Water...... 91
4.3.2 Electrostatic Potential Analysis: Neutral Cavity...... 92
4.3.3 Electrostatic Potential and Molecular Orientation...... 97
4.3.4 Electrostatic Potential Analysis: Charged Cavities...... 103
4.4 Conclusion...... 108
5 Electrostatic Properties from Neural Network Potentials 112
5.1 Preface...... 112
5.2 Computational Methods...... 114
5.2.1 Ab Initio Simulation...... 114
5.2.2 DeepMD reconstruction of the PES...... 117
5.2.3 DNN for determination of Multipole Moments: DeePD, DeePM, DeePPM 119
5.2.4 DNN for determination of DeePMD multipole moments...... 120
5.3 Results and Discussion...... 120
5.4 Conclusion...... 129
6 Conclusions 131
8 CONTENTS
A Appendix 153
A.1 Molecular Multipole Definition...... 153
A.2 Supporting Information: Chapter 3...... 155
A.3 Supporting Information: Chapter 4...... 159
9 List of Figures
1.1 Diagram of computational simulations, theory, and experiment...... 19
1.2 Schematic of boundary conditions and interfacial potential shifts...... 45
2.1 Schematic of basic neural network...... 57
2.2 Schematic of studied system- a liquid solvent droplet and interfacial potential
shifts...... 64
3.1 SPC/E electrostatic potential profile for different cavity sizes...... 76
3.2 SPC/E electrostatic poential multipole expansion and RDF for differnet cavity
sizes...... 78
3.3 SPC/E net potential dependence on cavity size: multipole decomposition.... 83
4.1 EC electrostatic potential profile for different cavity sizes...... 93
4.2 EC electrostatic potential multipole expansion and RDF for different cavity sizes 95
4.3 EC net potential dependence on cavity size: multipole decomposition...... 96
4.4 Diagram of EC orientation...... 99
4.5 EC distribution of dipole vector projection, neutral cavity...... 101
4.6 EC distribution of planar angle, neutral cavity...... 103
4.7 EC electrostatic potential profile for different cavity sizes, charged cavities... 106
4.8 EC distribution of planar angle, positive charge cavity...... 107
10 LIST OF FIGURES
4.9 EC distribution of planar angle, negative charge cavity...... 108
5.1 Schematic of DNN framework for electrostatic properties...... 115
5.2 Loss functions for DeePMD, DeePD, DeePM, DeePPM...... 121
5.3 Radial Distribution Function of AIMD and DeepMD bulk water...... 122
5.4 Dipole moment distribution for AIMD and DeePMD bulk water...... 123
5.5 Eigenvalues of the primitive quadrupole moment for AIMD and DeePMD bulk
water...... 124
5.6 Trace of the primitive quadrupole moment for AIMD and DeePMD bulk water. 125
5.7 Distribution of the double dot product of the primitive quadrupole moment for
AIMD and DeePMD bulk water...... 126
5.8 Radial Distribution Function of AIMD and DeepMD interfacial cavity system.. 127
5.9 Average dipole moment of for AIMD and DeePMD Interfacial Cavity System in
shells around cavity...... 128
5.10 Average traceless quadrupole double dot product for AIMD and DeePMD Inter-
facial Cavity System in shells around cavity...... 129
11 List of Tables
3.1 Electrostatic Contributions to Net Potential at Different Cavity Radii. Poten-
tials Computed at the Cavity Center with Contributions from Cavity-Liquid and
Liquid-Vapor Interfaces a ...... 79
3.2 Electrostatic Potential Shifts from the Cavity-Liquid and Liquid-Vapor Interfaces
at Different Cavity Radii. a ...... 80
4.1 Quadrupole Components of ethylene carbonate (EC) and water (SPC/E) a ... 92
4.2 Electrostatic Contributions to Net Potential at Different Cavity Radii. Poten-
tials Computed at the Cavity Center with Contributions from Cavity-Liquid and
Liquid-Vapor Interfaces a ...... 98
4.3 Electrostatic Contributions to Net Potential for Charged Cavities. a ...... 104
12 Chapter 1
Introduction
1.1 Preface
Life would not exist without the fascinating compound known as water. Society would not have developed without the ability to harness energy from the natural world for machines. Water and energy are unequivocally important for understanding the physical phenomena on earth, in our bodies, and to power civilizations, and yet there is still a lot to be learned about them.
Water, which composes 99% of the molecules in the human body,1 and 71% of the Earth’s surface2 is still a mystery. Energy, and improving ways to generate and store it, is a scientific problem which has grown exponentially as demands for energy increase.
It is hard to separate the study of water or the improvement of energy storage devices (bat- teries) from salts. These ion pairs exist in varying concentrations in almost all of the water on earth from seawater to the cytoplasm in your cells. They drive biological functions and help to regulate earth’s atmosphere. Since the advent of galvanic cells, energy has been stored in batteries through ions. Quantitative and predictive models of ion solvation are critically impor- tant to the improvement of energy storage devices and understanding of physical phenomena
13 1.1. PREFACE in a wide range of scientific research.
The study of liquid solvation was forever changed by the incorporation of computer sim- ulations in the mid-20th century. By introducing a new category of science— simulations— theories could be tested and experiments validated, while giving access to physical parameters which could not be achieved through experiment. Now, with the newest revolution, that of machine learning, computers are again poised to change the way that we study liquid systems.
With physically grounded principles these new methods can be used to access length scales and time scales which are drastically limited in quantum ab initio simluation methods. With these tools, I tackle the problem of ion solvation with the goal of improved physical understanding of molecular liquids. While accuracy in measurement is important for some purposes, I aim to answer the fundamental question of the origin of the interfacial electrostatic potential and whether or not neural network potentials can be used to study the electrostatics of ion solvation.
In the third chapter of this thesis I use the simplest toy model for a water molecule. Stripping away the decades of refinement in various water models which are fit to experimental results, I use classical MD to investigate the multipolar origin of the interfacial potential at the surface of liquid water. I implement a new way of analyzing the components that is more intuitive than previous methods, and leads to a quick realization that the quadrupolar potential close to the solvated ion is the dominant contributor to the potential which the ion feels. This result solves the long-standing question of why various types of water models produce varying interfacial potential values. Additionally, this analysis established a length scale at which electrostatic potentials are physically meaningful— where the charge density averages to zero.
I then extend this analysis to ethylene carbonate, an organic liquid relevant to energy stor- age, and which has a range of academic interest behind it. A popular solvent in lithium-ion batteries, this carbonate and the related propylene carbonate and dimethyl carbonate are in-
14 1.1. PREFACE teresting because with just a small structural change in the atomic constituents, many different chemical properties are changed. Using similar methods to those I used for water, I investi- gate whether additional insights can be gained. Are there any general properties of interfacial potentials that exist in these differing solvent molecules? Indeed, I find that there are length scales at which the interfacial potential is physically meaningful in ethylene carbonate, though the specific multipolar contributions to the potential are quite different. Using an orientational analysis, I find that the sign of the quadrupolar potential is dependent on the quadrupole mo- ment, the orientation of the molecule, and surface geometry. While general principles exist, knowledge of charge symmetries contained within a molecule’s unique multipole moments and the orientation of the molecule at the interface produce the interfacial potential.
Throughout this work, I highlight the limitations of classical and quantum simulation meth- ods. While I employ classical MD in my first two projects, it is used to extract understanding of general physical principles, not to calculate highly accurate results. In my final project, I used the emerging field of Neural Network Potentials to simulate liquid systems of ab initio accuracy at classical cost. Validating these methods for use in the study of the electrostatic properties of liquids is a necessary step in using these models for the calculation of physical properties.
The meaningful impact of this thesis is to establish and clarify the physical origin of the interfacial potential in solvents. While theoretical and experimental methods have started to zero in on a value of -0.4V for liquid water, the questions remain of how this value is generated, and what will it be for other solvents. The importance of system size is demonstrated, as it establishes a zeroing of the potential, a concept that is largely ignored in many accepted works.
New methods of Neural Network Potential are shown to be successful and exciting new methods in the science of liquid simulation.
15 1.2. EVOLUTION OF THE COMPUTER SIMULATION OF LIQUIDS
1.2 Evolution of the Computer Simulation of Liquids
Physics is the study of the universe through substances and their interactions. We learn about these interactions though the equations that describe them. Prior to the 20th century, knowl- edge of the physics of materials and many-body systems was restricted to two categories.
One category sought to solve these equations analytically and exactly. In this case, there are huge limits on what systems can be modeled. Newton’s Equations of Motion, for example, are restricted to solving for two interacting objects. Materials are composed of many more than two objects. Avogadro’s number tells us that one mole of a substance contains 6.022 x 1023 molecules. For these many-body systems there are, again, a very limited amount of problems for which equilibrium properties can be solved for exactly. Namely, the Ising Model of Ferromagnetism, the Ideal gas, and a harmonic crystal.3 The Ideal Gas Law can be derived directly from the Kinetic Theory of Gases and the Maxwell-Boltzmann Distribution using statistical mechanics.
The other category uses approximate equations that describe the macroscopic, large-scale behavior of a substance. These theories ignore the intricacies of the underlying microscopic behavior of the constituent atoms and molecules and their interactions, in favor of a more qualitative description of how a substance behaves. Limited knowledge of the intermolecular interaction leads to equations which attempt to describe the physics. Examples include the
Van der Waals Equation for dense gases, Debye-Hückel theory, and the Boltzmann Equation for intermolecular interactions.3 While these theories expand the number of systems we can study, they are still approximations. If theoretical and experimental results disagree, we are left with the question of whether the theory itself is wrong, or our assumptions about the intermolecular interactions are wrong. This question is where computer simulations had a fortuitous entrance in the scientific timeline.
16 1.2. EVOLUTION OF THE COMPUTER SIMULATION OF LIQUIDS
1.2.1 History
The simulation of liquids using computers arose out of scientific necessity and chance. Prior to 1905, there was no physical explanation for the seemingly random motion of dead pollen molecules suspended in a liquid, termed Brownian motion after the botanist who studied the phenomena. Unexplained by liquid currents, evaporation, or Maxwell-Boltzmann Kintetic
Theory of Gases, Einstein and Smoluchowski independently reached similar conclusions about
Brownian Motion. That is, large suspended molecules (pollen) experience motion due to their collisions with the small molecules of the liquid (solvent)— and further that mean displacement and observation time are proportional to the Diffusion Constant.4 This novel result not only settled the idea that liquids have a non-continuous behavior, but also marked the first time that an experimental observable in liquid could be connected with theory. Later works by Morrell and Hildebrand (1936) compared liquid mercury x-ray diffraction data to their 3D gelatin ball model, while examining factors such as temperature on intermolecular distance.5
There are obvious limitations in these types of experiments. Suspending particles in liquids or building physical ball models while simple and intuitive, are not suitable to handle the large scale collective properties that must be calculated for liquids. The integration of programmable digital computers into military and rare scientific operations by the 1930s and 1940s was an auspicious occurrence which has altered the trajectory of liquid science. Computers, which have the capacity to store large quantities of information that well exceeds that of the human brain, perform operations on data, and incorporate randomness, are perfect candidates to handle liquid experiments. This led to the first computer simulation of a liquid state in 1953 by Metropolis et al.6 at Los Alamos National Laboratories7 on the MANIAC (Mathematical
Analyzer Numerical Integrator and Computer)— known as "Monte Carlo" simulation.
The original Monte Carlo algorithm works to choose configurations from a probability distri-
17 1.2. EVOLUTION OF THE COMPUTER SIMULATION OF LIQUIDS bution. After this landmark paper, the field developed to include Molecular Dynamics (MD)— which aims to solve Newton’s classical equations of motion for a system of particles. Throughout the 1970s and 1980s, researchers incorporated increasingly complicated molecules and methods into MD. It is important to remember for what reason these methods were being developed and how simulations are situated in the hierarchy of science.
1.2.2 Connection with Theory and Experiment
The underpinning of Molecular Dynamics and other liquid simulation methods lies in the idea that many desired physical quantities can be calculated if the dynamics of many interacting particles can be determined. The entire system’s coordinates ~x and momenta ~p (or velocities), once solved for, allow us to determine energies, entropies, and other thermodynamic or transport properties. Statistical Mechanical Theory allows us to use an array of trajectories to calculate averaged quantities. However, just as a computer program is only as correct as the computer programmer, these results are only as physically correct as the models used in simulation.
The usefulness of simulations is their ability to connect experimental results with theoretical predictions as shown in Figure 1.1. Simulation results can be compared to results of approximate theories, providing tests for theories, or to experimental results, providing evidence for or against the model used. If the model does work well, it can be used to access timescales (both large and small), temperatures, pressures, and any other parameter that is impossible to measure with an experiment.
18 1.2. EVOLUTION OF THE COMPUTER SIMULATION OF LIQUIDS
Liquids Model Liquids
Do Computer Make Approx. Simulations Theories
Experimental Exact Results Theoretical Results for Model Predictions
Compare Compare
Tests of Tests of Models Theories Figure 1.1: Diagram relating computational simulation to theory and experiment in the science of liquids. Figure inspired by Ref.7, diagram otherwise original.
In this thesis I use simulation largely as a tool for theoretical prediction. As I will explain below, electrostatic properties of liquids are difficult to access with both experiment and theory.
Simulation models are extremely useful in electrostatics for a few reasons. First, the additive nature of charge, fields, potentials, etc. is perfectly suited for simple ensemble averaging.
Additionally, the basic theory of electrostatics has been understood for centuries. Finally, it is possible to extrapolate simple toy-models of interacting charges to slightly more complicated or more realistic molecular structures—allowing for predictions beyond the original simulation model.
19 1.3. SOLVATION SCIENCE
1.3 Solvation Science
The science of liquids and simulations have been intertwined since the advent of computers.
One field is Solvation Science, or the study of ionic and molecular liquids and interfaces. En- compassing much of physical chemistry, biochemistry, atmospheric science, and other diverse areas, the study of Solvation Science has wide ranging applications. Here, I will discuss a few of the major scientific questions. After presenting the broad goals of the fields, I will discuss how current simulation models are proving unequipped to solve problems outside of the areas for which they were originally developed. Connecting to Figure 1.1, the comparison of model results to experiment is not favorable. In section 1.4, I will discuss why these models fail.
First, I will discuss fluid mixtures and phase separation. Then I will summarize some relevant research in membrane proteins. Finally, I will introduce the study of specific ion solvation. While these are only a select few areas of research, they are all difficult to model with existing theories, thus providing adequate motivation for this thesis.
1.3.1 Mixtures and Phase Separation
Classical simulation models for water are generally parametrized for unique states of a liquid at ambient temperature. Complicated many-body effects are squeezed into a few parameters within a given model. The experimental gas phase dipole moment of a water molecule is 1.85 D, but the accepted values for water and ice range from 2.5 to 3.0 D. The simplest computational water models do not incorporate this fluctuation between phases, instead they use average values— the SPC/E classical model of water has a dipole of 2.36D. Alternative polarizable models such as dipole-polarizable or fluctuating-charge models incorporate polarizability as a response to the local electric field. These models offer improvement for a slightly higher computational cost. Polarizabilty is just one element needed for an accurate model of different
20 1.3. SOLVATION SCIENCE phases of a solution.
One application affected by these shortcomings is phase diagrams. Phase diagrams show the thermodynamic parameters or concentrations at which different phases of matter exist. At phase boundaries, the two phases coexist. The classic example is the Pressure v. Temperature plot for solid, liquid, and gaseous water. For systems of more than one substance, the Liquid-
Liquid Equilibrium defines the concentrations and conditions at which two or more liquids are at equilibrium. Knowledge of these concentrations are beneficial to advance the understanding of pharmokinetic properties of drugs in different living biophases, or pollutant molecule behavior in organic environments such as soil and groundwater.89 Environmental chemistry, drug design, and energy storage are some areas of research where phase separation is vitally important.
Simulations are a large component of phase separation research, because determining phase boundaries for particular mixtures at certain conditions requires extensive experiments. Ap- proaches such as SAFT10 (Statisical Associating Fluid Theory) rely on experimental data. The difficulty is being able to predict these boundaries for previously undetermined combinations of liquid compounds at specific conditions, without experiment. Some studies have proven the ability of certain force field models to accurately produce phase behavior for some liquids— wa- ter and alcohol (n-butanol) for example.11 Despite this, the challenges of simulating adequate conformational space and obtaining properly parametrized force fields for the liquids remains.
One study of LLE for Dipropylene glycol dimethyl ether and water pointed to the possibility of the system not reaching equilibria, inadequate sampling of compositional fluctuations, and small system size for deviation of mutual solubilities with experiment.12
One attempt to improve the modeling of water in phase separation added coupling of
Lennard-Jones repulsion parameters to the magnitude of the fluctuating charge in order to model the charge dependence of the overlap integrals in quantum mechanics. Although certain
21 1.3. SOLVATION SCIENCE properties such as vapor and liquid densities, heats of vaporization, and dielectric constants showed better agreement with experiment, the models still failed to produce the vapor-liquid coexistence curve at room temperature.13
Clearly there is a need for improved liquid simulation models in order to find predictive mod- els for Liquid-Liquid Equilibria. Mixtures of organic carbonates such as Ethylene Carbonate
(EC) and Dimethyl Carbonate (DMC) are commonly used to solvate LiP F6 in Lithium-ion bat- teries.14 Energy technology benefits from detailed knowledge of the chemistry of these mixtures in order to optimize important metrics such as electrolyte viscosity, melting point, and conduc- tivity. Simulations can provide insights and resolve discrepancies between experiments such as the number of EC or DMC molecules in the solvation shell around the lithium-ion, which differs between electrospray ionization mass spectroscopy and NMR spectroscopy.14 Models which are transferable and incorporate higher-level theory such as polarization and many-body effects are expected to produce accurate phase boundaries.
1.3.2 Proteins and Membranes
All cells and the organelles within them are composed of liquids, the cytoplasm, surrounded by a membrane. This membrane is made of a lipid bi-layer embedded with special proteins which allow the inside of the cell to communicate with it’s environment.15 Signal transduction through ions, catalysis of enzyme reactions, and transport of small molecular nutrients are among some functions of these proteins. These processes are essential to nearly all physiolog- ical activities such as muscle contractions through nerve and muscle excitation, brain activity
(learning and memory), immune response, blood pressure regulation, and hormone secretion.15
Defects and mutations of ion channel proteins are responsible for many different diseases. They provide insights into the functions of ion channels. For example, Cystic Fibrosis is caused
22 1.3. SOLVATION SCIENCE by a reduction of the epithelial Cl- transport. Familial hemiplegic migraines, LQT syndrome
(cardiac disorder leading to sudden death in young people), and Diabetes Mellitus are just a few types of clinical diagnoses that result from genetic mutations generating faulty membrane proteins. Understanding of the physics of these proteins can help researchers develop targeted therapies.16
Membrane proteins are loosely defined by three categories: ion channels, pumps, and trans- porters. Channels are gated pores whose opening and closing is governed by "chemical’ means, voltage gradients, or "mechanical means" through ligand binding. The gating of these channels leads to the ionic composition outside of cells differing from that within cells. Transporters change diffusion of an ion or substrate through existing gradients across the membrane.17 This creates an electrical signal that travels rapidly in excitable cells. The characterization of trans- port mechanisms is an essential step towards understanding their function and in developing therapeutic substances. Thermodynamics governs the transport of ions and the binding prop- erties of ligands which allow for transport.
Simulations are commonly used to understand the thermodynamics of this process after the protein’s crystal structure is determined. In one study of the Gramicidin channel, which could be used in fuel cell technology, the free energy barrier of K+ agreed with experimental results, but the entropy and enthalpy failed. This failure can be attributed to problems with the simulation model used.18 A later study looked at the free energy profiles of K+ and Na+ through the channel using two different models— one including polarization of the solvating molecules using the AMOEBA model, and one without. Including polarization and multipole terms on the atoms, brought the simulation results much closer to experimental free energy results.19
One final example is the CLC chloride channel/transporter, specifically CLC-ec1 of E. coli.
23 1.3. SOLVATION SCIENCE
This channel exchanges a proton for two chloride ions. In order to better understand the mechanics of ion transport Chen et. al.20 used simulations to calculate the absolute hydration free energy of ions in water and on the protein binding sites. It was found through analysis of free energies for gate opening and closing that proton binding is necessary before Cl– can bind to the central binding site, in agreement with experiments. One outstanding question with this channel is the perplexing experimental result that the substitution of F- for Cl- in this transporter effectively stops the flow of protons.21 Classical simulation results do not explain this ion selectivity, although quantum chemistry calculations suggest that the higher proton affinity and smaller size of F- make the passage of the proton more difficult.20 This unresolved issue illustrates the need for more advanced, predictive models for liquid simulation. Additionally, this is a classic example of ion specificity. A "small" change in ion identity generates a substantial change in protein function. The next section will highlight a few specific ion solvation problems.
1.3.3 Specific Ion Solvation: Hofmeister Series
A brief summary of the field of specific ion effects is difficult because of it’s breadth. Historically, the field developed 150 years ago after Franz Hofmeister, a pharmacologist, and his laboratory in
Prague, Czechoslovakia, discovered and published many papers on the ordered ability of certain salts to affect many different physical chemistry phenomena such as dissolving, precipitating, and lycotropic swelling of proteins. The most well-known example is the ordering of anions species (of like charge) by increasing ability to precipitate egg-white protein from solution.
At that time, the scientific fields of pharmacology, botany, biology, and physical chemistry were closely related.22 Scientifically, while these fields have since become segregated, these effects apply to just as many diverse fields as in Hofmeister’s time. Fundamental and applied chemistry, biology/ biophysics, biochemistry, materials science, atmospheric/marine science,
24 1.3. SOLVATION SCIENCE energy, climate science, physiology, colloid/surfactant science, and many other fields could all benefit from improved understanding of specific ion effects and solvation. If the field relates to aqueous (or nonaqueous) liquids, there are related specific ion effects, which in light of practicality and the lack of theoretical understanding, were largely ignored until recently.23
Specific ion effects are the varying abilities of certain ionic species to generate physical phenomena. Extensive theoretical work has not completed the puzzle. Beyond empirically derived rules, there has not been a quantitatively predictive theory developed for the phenom- ena. Questions such as whether ion size, charge density, charge distribution, geometry, and interaction beyond simple electrostatics such as polarizability play a role, all still persist. The goal is to extract generalities and understand the interactions that generate the effects so that we can comprehend the processes of the living systems and develop technologies.23 Examples of aqueous ion specific effects include:
• the alteration of ionic surfactant phase behavior with counter ion identity changes24 25
• accumulation of ions at the water liquid-vapor interface26 27 28
• activity coefficients23 29
• pH measurements29
• protein denaturation30
• phospholipid aggregation23
• surface tension31 32 33
Non aqueous examples, relevant to this thesis in organic carbonates include:
• salts effecting structuredness in liquids34
25 1.3. SOLVATION SCIENCE
• solubility35
• Hofmeister series ordering (not consistent with that in water)36
• bubble-bubble coalescence
Atmospheric science is one area where specific ion effects, in particular those which contain interfacial effects, are wide-spread. The study of the physics and chemistry of clouds, gases, and aerosols is important for understanding the climate of earth and other planets, cloud formation, and the weather.26 The natural world is full of aqueous interfaces which contain ions (like seawa- ter) and the delicate interactions thereof are responsible for many natural phenomena. Liquid and solid aerosols such as fog or clouds and ice clouds or particulates, respectively, are colloids.
The potential of mean force between two colloidal particles in a salt solution has been shown to contain ion-specific dependence37 in addition to charge and size dependence through dispersion forces. Similar ion-specificity has been found in the stability and formation/deformation of amphiphiles such as micelles and biological membranes.38
Theoretical formulations of ion specific effects use ions embedded in a dielectric continuum with ion charge, size, and dielectric constant as variables in the free energy of hydration. In
1.2 I discussed the limitations of macroscopic theories in describing what is clearly microscopic.
Here, the granular nature, local electrostatic effects, and quantum forces near an ion necessitate the use of model liquids beyond approximate models.26 In the next sections I will discuss both past and present limitations to theories of ion solvation, as well as the importance of interfacial potential effects in single-ion solvation free energy calculations necessary for unraveling specific- ion effects.
26 1.4. THEORIES AND SIMULATION OF ION SOLVATION
1.4 Theories and Simulation of Ion Solvation
Early theories of ion solvation developed to explain a deviation from some idealized problem which is formed to make the problem tractable. They ascribe qualities to the system which allow for approximate physical models, such as the solvent as a simple dielectric continuum and the ions as charged point particles or hard spheres. The addition of computer simulations into the picture in the early 1960s, as I discussed in section 1.2, allowed for these theories to be tested using MD and MC. The advantages of simulations is that they do not use macroscopic parameters that overlook the microscopic properties of the system— orientations, ion distribu- tions, hydrogen bonding, etc.— but derive properties from atomistic interactions. These "past" theories are still in use and can be accurate under certain conditions. There are three main categories of simulation in the present day: those using classical MD, ab initio theories, and
Machine Learning derived potentials. Various combinations also exist, but I will focus on them separately, providing a quick introduction to the benefits and drawbacks of each.
1.4.1 Past
Electrolytes are any ionic solid that can dissolve in polar liquids (like water), dissociate into ions and carry an electric current.39 The textbook definition of the electrochemical potential for an ideal electrolyte (neutral salt) in solution with activity coefficient of 1 is,
µ(x) = µ0 + kT ln c(x) + zeψ(x)
where µ0 is the chemical potetial, x is position and ψ(x) the potential at that location. This description is incomplete if the ions are near large charged surfaces such as colloids or have unequal concentrations. Possion-Boltzmann and Debye-Hückel Theory were developed as a
27 1.4. THEORIES AND SIMULATION OF ION SOLVATION theoretical explanation for this deviation, using electrostatic forces as the origin. These theories use the Boltzmann relation— where local average electric potentials define the difference in potential energy between ions— and a Boltzmann distribution in the Poisson Equation,
ρ ∇2φ = e (1.1) r0
where ρe is a local charge density, r is the solvent dielectric constant. Ion motion is governed by Boltzmann statistics and ion density is given by,
−Wi 0 k T ci = ci e B (1.2)
where the Wi = qφ is the work to move an ion towards the surface, ci is the local ion concen-
0 tration, and ci is the bulk ion concentration.
Debye-Hückel theory gives the activities of fully dissociated electrolyte solutions. It relies on several assumptions: The ions are spherical with no polarization, the solvent medium is a dielectric continuum, and the solute is completely dissociated.39 The derivation involves using the Boltzmann distribution for the charge density in Poisson’s Equation. The extra
(electrostatic) free energy per ion is then
κq2 µ = − i (1.3) es 2
−1 2 4π P 2 where κ is the inverse Debye wavelength and κ = kT i qi ci. Essentially, the ions are treated as non-interacting, screened point charges. Basic physics such as dispersion, solvent interaction and polarization are completely ignored. Setting this equal to kT ln(γi) where γi is the activity coefficient gives a measurable way to compare to experiment.
28 1.4. THEORIES AND SIMULATION OF ION SOLVATION
Even so, this theory is only valid at low concentration (100mM/L),40 whereas many specific ion effects emerge beyond this limit. In addition, many systems in the natural world exceed
+ + 2+ 2+ 2+ − 2− − this. Ions such as Na , K , Mg , Ca , CO3 , HCO3 , SO4 , Cl are responsible for maintain- ing the processes of life. In seawater, Na+ and Cl− concentrations often exceed 500 mM/L. In
Mammalian cells (intracellular and extracellular), blood plasma, and interstitial fluid composi- tions are often higher than 100 mM/L.41 Furthermore, recent experiments of aqueous NaCl and other liquids, have shown the opposite of what Debye-Hückel predicts— showing an increase in screening length with increasing concentration.42
Derjaguin-Landau-Verwey-Overbeek (DLVO) theory describes the interaction between charged surfaces (colloids) in a liquid through a screened Coulomb (Yukawa) potential. The interaction is a balance between repulsive entropic and attrative Van der Waals dispersion. Like Debye-
Hückel theory it assumes ions are charged hard spheres in a dielectric continuum. Because polarization of the electron cloud and ion size are ignored, ion specificity is explicitly ignored.43
Other assumptions, such as an ideally smooth surface have been revised to include surface roughness,44 but the general assumptions remain. This model was revised by Onsager and
Samaras33 in 1934 to predict surface tension.
Onsager-Samaras Theory predicts that ions will be expelled from the interface,26 demon- strating how continuum models fail in some theoretical predictions. Beginning with the Gibbs-
Duhem equation for a flat surface and and dividing by surface area A while holding T constant leads to the Gibbs adsorption isotherm where ni is the excess number of moles for species i,
ni Γi = A and surface tension is γ:
dγ = −Γ1dµ1 − Γ2dµ2 (1.4)
For inorganic salts (alkalide haldide series), as measured by Heydweiller31 in the 1910s, there
29 1.4. THEORIES AND SIMULATION OF ION SOLVATION is an increase in surface tension with increased concentration which leads to a negative Gibbs surface excess. This means there should be a depletion of these ions at the interface. Theories of
Wager32 and others33 conceptualized this depletion as a simple fictitious image charge force.26
In reality, this depletion is not seen by simulations or experiments of large polarizable ions, which calls in to question the validity of the entire theory.
The Born model45 of ion solvation is another continuum solvation method used to evaluate the free energy change from vapor (vacuum) to bulk (can be generalize to other mediums). For
46 ion charge q, radius r0, and the dielectric constant of the medium r, the Born Equation is,
N q2 1 µ = − A 1 − (1.5) 8π0r0 r
This approximation is moderately successful, especially for distant ion-solvent interactions. It does account for ion size, although it is assumed they are perfectly spherical, and can be used to judge general behavior of ions in solvents. Still, it ignores the free energy changes of nearby solvent, solvent reorganization, and polarisation and is unsuccessful once dispersion forces are included.40
1.4.2 Present
Classical Simulation
Classical molecular dynamics simulations (MD) use point-particles to approximate atoms and molecules and an empirically derived interaction potential energy. Atomic/molecular motion is determined through integration of Newton’s Equations of Motion. I will discuss this theory more in section 2.2. They are the most widely used, as they are theoretically simple, easy to implement, and the oldest form of simulation. Perhaps the greatest benefit is the speed:
30 1.4. THEORIES AND SIMULATION OF ION SOLVATION the computational cost of MD simulations ranges from O(n2) to O(nlog(n)) if electrostatic approximations like Ewald summation are used. With these classical models it is possible to simulate approximately 100,000 atoms for 1 µs while for polarizable models 10,000 atoms can be simulated for 10 ns. For context, the timescale of protein diffusion is seconds, protein folding is ms, and ion channel gating is µs. MD has been successful in unraveling the biology of protein- protein interaction and conformational change. The field of allostery— the study of protein
(enzyme) binding has benefited as well. The conformational shift of Bacillus stearothermophilus lactate dehydrogenase, for example can be modelled in 50 ns of simulation.47 Drug Design and
"docking" of a compound to a macromolecule can be achieved through MD.
However, using Classical Mechanics to describe inherently quantum interactions has pre- dictable drawbacks. Past models do not include all of the interaction types and length scales that are needed to produce accurate results. The length scales of nanometer-scale materials behave drastically differently from what we experience in daily life. Although quantum mechan- ics tells us how molecules behave, we cannot intuitively understand the interactions of atoms and molecules at the nanometer scale. Designing proper experiments, theoretical models, and computational models to learn about these systems often suffers do to our lack of understand- ing. Even though the well-understood electromagnetic force is the fundamental force which generates atomic interactions, there are many types of forces under this umbrella which char- acterise interactions between different media. Interactions can be either short or long-ranged and their distinctions are less rigid at the nanoscale. In solvation, short-ranged interactions between ion and solvent are dominated by multipolar electrostatics (coulomb), induction, and exchange-repulsion. Long range-forces, which were overlooked in the past, are now proving to be important in solvation science48 49 as well as nanometer-scale device design, self-assembly and colloid science.48 Primary long-range forces include the Electrostatic (Coulombic), Polar
31 1.4. THEORIES AND SIMULATION OF ION SOLVATION
(acid-base), and Electrodynamic forces (van der Waals): Dispersion (London- instantaneously induced multipoles), Induction (or Debye) (permanent multipole and induced multipole), and
Keesom (permanent multipoles). Secondary long range forces are material specific, a sub- set of the above interactions, are macroscopic in nature, and include the hydrogen bonding, hydrophobic, hydration, and osmotic interactions, etc.48
There are many different ways to incorporate polarization. Point multipoles, fluctuating charge models, and harmonic spring models such as Drude oscillators are the main classical
MD options to model the above electrodynamic/electrostatic forces. Reference 50 modeled po- larization and charge transfer with a Drude oscillator model, and produced more accurate den- sities with temperature.50 In general, over-polarization is a major problem with many classical models, producing large errors.51 For example, modeling of DNA and RNA is largely inaccurate because the phosphate backbone is surrounded by ions in a highly polarized environment52 53
Understanding the folding process is dependant on understanding these interactions.
Non-electrostatic forces are also modeled with crude approximations which have varying degrees of success. Modeling of dispersion and exchange-repulsion is particularly rough.54 Van der Waals forces are modeled with a Lennard-Jones Potential55 which is treated as catch-all for corrections after simulation results are fit to a particular experimental property.
Ab Initio Simulation
Ab Initio Molecular Dynamics (AIMD) is the most realistic method of simulation because it uses the quantum mechanical Schrödinger equation instead of a crude empirical approximation for the interaction energy. Some basic theory will be discussed in section 2.3. Furthermore, these physically derived interaction potentials will be more transferable than classical models which are parametrized for specific conditions. It overcomes many of the drawbacks of classical
32 1.4. THEORIES AND SIMULATION OF ION SOLVATION
MD discussed above, as it is able to produce accurate condensed phase simulations capable of modeling electronic polarization effects and chemical reactivity56 (bond breaking/formation) that is sorely needed in problems of solvation science, among others. Some common methods include Born-Oppenheimer MD, Hartree-Fock MD, Kohn-Sham MD, Car-Parinello MD, and
Path Integral MD57
As ideal as this sounds, any simulation is restricted by the large computational cost of
AIMD— making applications of these methods very limited in size and time. An AIMD sim- ulation can only be done for approximately 100 atoms for 100 ps. This makes simulations of interfacial systems extremely difficult because of the number of atoms necessary. Electrostatic potentials, for example, must be determined in systems with a bulk-like region in order to find the convergence of the potential.
Neural Network Potentials
So far I have discussed the most common forms of molecular liquid simulation. Classical MD often represents a molecular charge distribution as a set of (often rigid) partial charges which approximate the multipole expansion, or as a set of point multipoles. While they are adequate approximations for some purposes, they fail to accurately represent the reality of the electron density. They are very computationally efficient— able to simulate the longest timescales of several nanoseconds. AIMD is far more accurate, but inefficient, as nuclei and electron density are treated quantum mechanically.
Recent advances in Neural Network Potentials (NNP) have created a new category of sim- ulation protocol where quantum accuracy can be achieved at the low cost of classical MD.
Neural networks are an advanced technique used to develop machine learning (ML) potentials.
Theoretically, these potentials must (1) express the analytic structure-energy relationship using
33 1.4. THEORIES AND SIMULATION OF ION SOLVATION machine learning, (2) use a first-principles training set of energies and forces, and (3) not con- tain any ad hoc assumptions.58 The general steps to construct an NNP are as follows. First, an initial set of electronic structure calculations is generated. This is often the most computation- ally expensive element which establishes the bottleneck. Next, the data needs to be transformed into appropriate input features that satisfy conditions of rotationally and translationally invari- ant system energy and permutation symmetry with the exchange of like atoms. Parameters are then varied to agree with the training data energies and forces in order to construct a Potential
Energy Surface. Finally the model is tested and simulations follow.
One such NNP is the DeePMD potential of Car and coworkers which has been shown to correctly represent the PES58 59 60 and produce accurate simulations of condensed phase systems.61 62 Metrics such as RDFs, packing parameters in water,63 organic liquids,59 and metal alloys64 have shown to very accurately reproduce AIMD reference data. In a subsequent study,
Zhang et al. 65 were able predict the dielectic response and the infrared spectra of liquid water, which was previously unattainable at the timescales of AIMD simulations. Drawbacks of NNP’s include many of the difficult aspects of neural networks in general. For example, the predictive power of the model is limited to the sample space (configuration space) on which it is trained.66
Additionally, reference data is obtained through the computationally expensive AIMD data we seek to avoid. Work to alleviate this issue includes active learning techniques to generate more sample space.64 Also, the NNP is essentially a black box: extracting physical principles from the simulation requires the thoughtful selection of the NNP and understanding of the input data, the training process, what it is trained for, and the output data.
Nonetheless, the benefits of NNPs outweigh the cost, and they represent a groundbreaking new tool in answering the outstanding questions of Solvation Science. Beyond the simple use of
NNPs for MD, they can be used for the calculation of various properties and incorporated into
34 1.5. INTERFACIAL POTENTIAL EFFECTS alternative models. There is further research needed to ensure that these models are physically grounded. In the context of molecular liquids, this includes whether or not the electrostatic forces are correct.
1.5 Interfacial Potential Effects
Motivation for the study of interfacial potentials and their effects is most directly linked to the establishment of a single-ion solvation free energy scale. Besides understanding the fundamental physics of interfacial potentials, they are essential for determining the thermodynamic scale that will help us to understand specific ion effects. First I will discuss this scale before introducing some of the more fundamental questions.
1.5.1 Thermodynamic Scale
Establishing the solvation free energy for an ion pair (salt) is simple because the total charge is zero. Thermodynamic and electrochemical experiments can only be applied to neutral macro- scopic systems. When the ion pair is separated, the ion has a charge and no single experiment can determine the solvation free energy. In order to determine single-ion solvation free energies, a conventional scale must be created, where everything is measured relative to the proton (H+) solvation free energy. Once, the proton value (or any single ion) is determined, all other ions can be determined from experimental bulk thermodynamic data.67 68 For a positive ion (P),
ex,conv ex ex ex ex µP = µP − µH+ = µP,b − µH+,b (1.6)
35 1.5. INTERFACIAL POTENTIAL EFFECTS and for a negative ion (N),
ex,conv ex ex ex ex µN = µN + µH+ = µN,b + µH+,b (1.7)
Different experiments have obtained highly accurate and agreeable results for both the pair free energies and conventional scale free energies,69 but not for the single-ion values.70 Beyond es- tablishing this scale relative to the proton, there must be an "extrathermodynamic assumption" that is used to rationalize the single-ion values.70 This is essentially an additional theoretical input that must be applied to deduce the single-ion quantities.
One commonly used assumption is the TATB hypothesis of Marcus71 — that two specific
(TA+/TB-) ions of opposite charge have the same absolute free energies in every solvent. This is because TA+ (tetra-phenyl arsonium) and TB- (tetra-phenyl borate) are large molecules of similar size, nearly spherical, and have an inert periphery.68 Because of this, no charge-specific solvent ordering will occur, and the solvation is nearly Born-like. The TATB hypothesis is similar to other methods such as those of Latimer72 73 where an interfacial-potential-free model is used to set the thermodynamic scale by partitioning bulk pair thermodynamic data into individual contributions.
Methods such as the Cluster-Pair Approximation (CPA), used by Tissandier,74 instead uses ion-water cluster data with an increasing number of waters (n) and bulk conventional scale ion hydration free energies to estimate the free energy to insert a proton into water. Free energies are calculated for small ion-water clusters relative to H+ and OH- in order to estimate the proton hydration free energy and set the scale. The original authors claimed this method is free of extrathermodynamic assumptions. As suggested by our lab75 and others76 77 78 79there is an extrathermodynamic assumption: that the conventional scale pair solvation free energies for small clusters converge to the bulk value as n increases. In other words, adding more water
36 1.5. INTERFACIAL POTENTIAL EFFECTS molecules to an anion cluster or cation cluster has the same binding energy, if they are large enough. In fact, there are effects from the net potential of water and hydration effects which contradict this assumption.
The other difficulty is that the real free energy in dragging an ion from gas phase to bulk liquid creates a contribution from the electrostatic potential of the vapor-liquid interface— the interfacial potential. I will discuss this contribution extensively throughout my thesis, particularly how it is created by the anisotropy of molecules at the liquid interface. Determining the value of this "phase potential" is essential for establishing this free energy scale, because it
ex ex differentiates the real free energy µi from the bulk value µb,i
ex ex µi = µb,i + qiφnp, (1.8)
Enthalpies and Entropies can be calculated similarly from temperature derivatives of the inter- facial potential. The real free energy can be experimentally measured from a combination of conventional ion hydration free energies, referenced to the proton, and cluster free energies of formation for ions.
Pollard and Beck80 asserted how the difference between the free energy values obtained with the Marcus and Latimer methods differed from the CPA method of Tissandier by approximately
+10kcal/mol. Noticing that the difference in simulation methods is largely due to one being interface-free and the other is not, they reasoned that the effective electrochemical surface potential of water should be -0.4 V. In another study,75 using classical and quantum methods, this value was found again by examining experimental results of ion distributions at interfaces and a re-analysis of CPA without any extrathermodynamic assumptions.
Shi and Beck81 compounded the confidence in this result by a quantum DFT simulation of bulk hydration free energy for Na+ and F- ions. By averaging the cation and anion bulk
37 1.5. INTERFACIAL POTENTIAL EFFECTS hydration free energy term that includes interfacial potential effects, the potential contribu- tion cancels and just the bulk value is found. Comparing the simulated bulk value with the experimentally obtained real free energy, results in an effective surface potential of -0.41 V.
Although evidence is accumulating for this liquid water effective surface potential value, we are still left with the question of it’s physical origin and value in other solvents. How does the anisotropy around an interface lead to this potential? Does the geometry of the interface affect this value?
Another important point which has been debated is the effect of a "local" interfacial potential due to anisotropic molecular arrangement around the ion. Pollard and Beck simulated TA+ and
TB- ion models in a surface potential-free system and showed the existence of a local interfacial potential. The free energies of these ions were similar, even in different solvents (water and dimethyl sulfoxide) after this local potential was removed. The existence and importance of the local potential cannot be overstated. It holds a key to understanding ion specific phenomena.
But first, I will discuss some subtle theoretical points of the interfacial potential.
1.5.2 Are they measurable?
The indeterminate nature of interfacial potentials goes back to the Gibbs-Guggenheim princi- ple which states that the electrostatic potential difference between two phases cannot be mea- sured.82 83 The principle further critisizes the use of classical electrostatics in electrochemical theories based on thermodynamic violations. Gibbs first wrote about the impossibility of mea- suring the potential difference between an electrolyte and electrode in 1889, and Guggenheim reformulated this in 1929 to say that the, "electric potential between two points in different media can never be measured and has not yet been defined in terms of physical realities. It is therefore a concept which has no physical significance.".82 He did state that the potential
38 1.5. INTERFACIAL POTENTIAL EFFECTS difference does have significance in electrostatics, where the electric field is an imaginary con- struct that determines the flow of electrons and ions, and their equilibrium is not static, but thermodynamic.
This debate over measurability has continued with various interpretations.83 84 85 86 Many widely used theories discussed in section 1.4.1 explicitly chose to ignore this principle by us- ing electrostatic theories in thermodynamic processes, which is nonphysical. For example, researchers still use theories like Debye-Hückel in biological studies of mammalian cells beyond the concentrations limits of the theory.83 Distributions of ions in solutions and colloid dispersion still make use of these theories.
Molecular dynamics methods, however, do not violate this principle and avoid the nonphys- ical assumptions of Poisson-Boltzmann theory and it’s derivatives. While there is no thermo- dynamic basis for electrostatic and non-electrostatic forces to be additive, it has not yet been conclusively disproven83 and remains the best method for investigation of interfacial potentials.
It is our aim that through a thoughtful combination of theoretical input and simulations, as well as comparison with experiment, that interfacial potential shifts felt by ions near interfaces can be determined.
There are three main reasons that make the study of interfacial potentials so difficult.
First, different experiments probe different physical effects.87 Each experiment probes a dif- ferent length scale, and each probe reacts differently to the potential gradients. For example, electrokinetic potentials, or the zeta potential ζ, are commonly measured by colloid scientists.
This is done by measuring the electrophoretic mobility of a colloid particle, and using the
Helmholtz-Smoluchowski law. It describes the electric potential at the plane where the mobile
fluid is separated from the attached fluid ("slipping plane"), and is not the same as the interfa- cial potential at two points deep in the phases, relevant to ion solvation. Second, as explained
39 1.5. INTERFACIAL POTENTIAL EFFECTS above, there is no direct way to measure the potential shift experienced by an ion crossing an interface.83 84 85 86 Third, the interfacial potential is not directly measurable. There is a slew of ambiguity in it’s definition accepted by experimental, theoretical, and simulation researchers.
It is of the utmost importance that a systematic definition across fields is developed. In the next section, I will discuss the definition further.
Even though the interfacial potentials are ambiguous and formally immeasurable, there are detectable physical consequences which support their existence and help in their determination.
One effect is the measurable ion distributions at interfaces caused by driving forces. In one experimental study, Conboy and Richmond88 investigated the interface of water and dichloro- ethane (DCE) and the distribution of TA+ and TB- ions. Using Total Internal Reflection
Second Harmonic Generation (TIR SHG), an optical technique, they found that ions were mainly located in the DCE phase, until an external potential of 0.4 V was applied, and the cation of the TA/TB pair nearest to the interface was replaced by the anion TB-. Beyond the confirmation that ion distributions at interfaces are affected by the interfacial potential, this result agrees with our value in section 1.5.1 obtained through analyzing thermodynamic data.
The TB anion has been shown to be more strongly hydrated than the TA cation.89 Another study by Faurudo et. al90 found that the anion TB moves towards hydrophobic surfaces, opposite to what is expected, and strongly suggests the influence of interfacial electrostatic driving forces. The force in this study agrees with the sign we predicted.
Another class of experiments are acid-base shifts which lead to the charging of neutral gas bubbles in water.91 92 93 Behaviour of carboxylic acids at the interface of water and other mediums is relevant to the transport of nutrients through cellular membranes, aerosol emissions from water and atmosphere, and origins of life. Creux et al91 studied the pH of the water surface when exposed to a low dielectric medium (oil, teflon, air) through electrophoresis and
40 1.5. INTERFACIAL POTENTIAL EFFECTS other methods, and found that anions were always drawn to the interface, creating a basic air/water interface. This is despite numerous MD results which showed the this interface is
93 acidic. Eugene et. al. exhibited the different pKa values for some of these acids near the water/vapor interface. The observed shifts are again consistent with an effective interfacial potential of -0.4 V.
Finally, there is the very recent discovery, by Zare’s group, that chemical reactions are spon- taneously induced by electric fields near the surface of a liquid.94 After accidentally finding that hydrogen peroxide (H2O2) was created in water microdroplets (micron-sized), they suggested the mechanism of hydroxyl (OH) radicals being created by OH- losing an electron. This fas- cinating result, would not be expected for inert, catalyst-free water, and suggests the role of an intrinsic interfacial potential at the surface of the droplet. This discovery showed a new method to produce hydrogen peroxide in an environmentally friendly way and a new possible method of disinfecting surfaces which could have profound impacts on society. Beyond that, it shows how an often overlooked physical quantity, the interfacial potential, can impact chemical synthesis, an area previously not considered.
This experimental evidence clearly shows that the pursuit of understanding and measuring interfacial potentials is not simply an intellectual exercise, but an important emerging field which can have a dramatic effects on a diverse range of fields— from green chemistry to the origins of life. However, the problem of studying this elusive quantity remains, and for that there must be a clear definitions between theory, experiment, and simulation.
1.5.3 What are we measuring?
In electrochemistry, potentials are often referred to as the Galvani (inner) potential φ and Volta
(outer) potential ψ. The Galvani potential is equal to the Volta potential plus the surface dipole
41 1.5. INTERFACIAL POTENTIAL EFFECTS contribution χ; φ = ψ + χ. The Galvani potential φ is the immeasurable quantity at the center of the material of interest. The Volta potential ψ, measurable in principle, is that outside of the condensed phase, close enough to pick up long-range electrostatic effects. The χ potential is often referred to as the "interfacial potential jump" going from one medium to another.87
Pratt argued that the separation of φ and ψ is not necessary after restricting the problem to conducting materials where there are non-existent macroscopic electric fields (and constant electrostatic potentials). Fluid dielectrics, although not conductors, can be determined to have macroscopic electric fields of zero.
Pratt’s work, which we follow closely, views the interfacial potential (or surface potential) as a contact potential as described by Laudau, Lifshitz, and Pitaevskii.95 The idea is introduced through the work done to remove a charged particle through the surface of a conductor— the work function. It must be a thermodynamically reversible process that depends on particle charge, thermodynamic state of the conductor, and the state of the surface itself. The relation
dφ of the work function to the charge density of the surface ρ(x) is given by the relation: dx =
R ∞ −4π −∞ ρdx. Another integration leads to,
Z ∞ φ(−∞) − φ(∞) = 4π xρdx (1.9) −∞
This relation shows that the difference in the work potential between the surfaces of a conductor is dependent on the dipole moments of the charges near the surfaces. The contact potential is the potential difference between the two conductors which prevents the flow of charges between them.95 From this it is clear that determining the contact (surface) potential is simply a matter of determining the the charge density. Further information can be gained through this formula and a substitution of the multipolar expansion of charge density as will be shown in sections
2.6-2.6.2. This theoretical definition provides a sensible way to determine this potential through
42 1.5. INTERFACIAL POTENTIAL EFFECTS simulation.
Landau et al. also discuss the chicken/egg aspect of the electrostatic potential difference in their discussion of the galvanic cell.95 The total contact potential in a closed circuit of conductors is nonzero, and called the electromotive force (e.m.f). They show that the total e.m.f of the cell is equal to both the sum of the interfacial potential shifts crossing all of the interfaces or as the differences of bulk ion chemical potentials between the phases (in electrodes and electrolytes). Despite this ambiguity, there are measurable physical effects which confirm the existence and physical importance of this potential.
Perhaps the trickiest part of understanding interfacial potentials beyond the theoretical definitions is that they differ depending on what experiment is used to measure them and what simulation is used to model them. In reality, the charge distribution around a molecule is composed of a dense nuclear core and a diffuse electron cloud. Ab initio simulation through
Density Function Theory approximates this distribution, while classical MD methods use a crude model of finite point charges. Clearly, the electrostatic potential measured with a test particle through a simulated media will differ based on the level of theory chosen as well as the nature of the test particle. Furthermore, an experimental comparison must account for the choice of probe.
In classical simulation, interfacial potentials are often calculated from the charge density.
The charge density is constructed from an ensemble average of the partial charges at spatial points through the entire system. In a spherical droplet system, this is accomplished through the the partial charge values and a radial distribution function (RDF) of all positions. Subsequent integrations using Maxwell’s Equation, ∇ · E = 4πρ, and the definition of electric potential,
E = −∇φ, results in electrostatic potential profiles through the liquid. Alternatively, the interfacial potential can be determined from constructing point dipole and quadrupole moment
43 1.5. INTERFACIAL POTENTIAL EFFECTS densities. Using these methods, the "probe" inherently penetrates into the molecular charge distribution. Similarly, in DFT calculations, the interfacial potential probe penetrates the interior of the molecule.
Examining interfacial potentials with these methods gives rise to an unexpected bulk po- tential value known as the Bethe Potential. Known under various terms as the "mean inner potential"96 or "exclusion potential",85 it describes the average electrostatic potential inside of a molecule. It is a transitionally invariant, bulk liquid property, which exists in nature. Both classical and ab initio methods contain the Bethe potential, but it is highly dependant on the
4π P R 3 2 2 theory level and model used. In general it is given by φB = − 6V i d rρ(r)r . The r term hints at it’s intrinsic relation to the quadrupolar terms in electrostatics. It tends to have a large positive value for quantum mechanical methods (∼ +3 V) and a small negative value for classical methods (∼ -0.5V).96 97 The bulk nature of the Bethe Potential leads to the un- surprising property that it cancels when crossing the two interfaces from vapor, to bulk, and then into a local cavity. Hence, it is common for a single surface potential estimate to contain this contribution, but a net potential containing two interfaces will not (no matter what theory level or model is used). Figure 1.2 shows a schematic of these interfaces for various systems.
Whether or not the Bethe potential is picked up in experimental studies is dependant on the probe used.98 85 99 96 If the probe is small and has high enough energy to penetrate the molecular center, there will be a contribution from the Bethe Potential. Experiments using high-energy electron diffraction or high-energy electron holography do penetrate into the molecular center.
These measurements should be most similar to the AIMD calculation of the surface (or net) potential of the solvent.98 100 In contrast, electrochemical experiments determine interfacial potentials by calculating the difference between the chemical potential of hydration of an ion and the real free energy change during transfer of an ion from vapor to liquid phase.101 102 103
44 1.6. SUMMARY
(a) (b) (c)
bulk solvent bulk solvent bulk solvent
Figure 1.2: A schematic representation of the possible simulated systems. (a) and (b) are slab systems, usually modeled with PBC Ewald boundary conditions. (c) is a droplet system with a cavity at the center. Both (b) and (c) contain two interfaces and thus contain the same Bethe potential cancellation, while (a) does not have this Bethe potential cancellation.
Here, the common probe is a proton, which does not penetrate the molecule due to the Van der Waals force. These experiments are most similar to classical MD, using a test particle as a probe, because the artificial Lennard-Jones potential exclusion prevents intrusion into the molecular center.
In summary, a sensible comparison between simulation and experimental interfacial poten- tial values is dependent upon understanding 1) Whether your experimental or simulation probe penetrates the molecular center (and picking up a Bethe potential contribution) and 2) how many interfaces are crossed, affecting your result (Bethe potential cancellation).
1.6 Summary
Many solvent models fail to produce accurate results for systems requiring physical effects be- yond their parametrization. While these models can perform for a small set of circumstances, they do not contain the proper physical theory to be transferable or reproduce experimental results for systems with slightly different specifications. Models that are capable of chemical and quantum physics can only be used for a limited number of systems due to their highly
45 1.6. SUMMARY computationally expensive nature. Ion solvation is one such problem which is critically impor- tant for a vast range of scientific areas. It is a complex problem where interpreting specific ion effects in experiment is difficult. Compounding on that, translating those observed effects and studying them with computational and theoretical models must be done in a methodical way which accounts for all underlying physical principles. A well-thought approach to studying ion solvation and the subtleties of the physics is essential. Only recently has the community begun to understand the profound effect that interfacial potentials can have on a range of physical phenomena, including the determination and unified scientific agreement on single-ion solvation
ex free energy values through the equation µ = µb + qφnp. Until recently, it was widely accepted that only a distant electrochemical surface potential φsp had an effect on free energies. Further, the electrochemical surface potential of water was determined from an averaging of various experiments, without regard for theory level or system details. This is clearly not the right ap- proach to measuring the interfacial potential shifts felt by ions near an interface. Recent work in our group provided further thermodynamic evidence for a water-vapor surface potential of -0.4
V and for the existence of a local potential φlp near an ion. But how is this potential generated by the molecules at the surface, and how does it differ from other non-aqueous solvents?
The problem of unraveling the relation between experiment, theory and simulation in Sol- vation Science is difficult for two reasons; 1) There are so many variables of simulation details, theory level, experimental methods, etc. in each of the three areas, that translation between them can be difficult and 2) Simulation models all suffer from the cost-accuracy problem which limits the calculation of properties like free energies and interfacial potentials with an accept- able accuracy. My thesis addresses the ion solvation problem by attempting to overcome these challenges with physically grounded analysis as follows:
• Chapters 3 & 4: What is the physical origin of φnp?
46 1.6. SUMMARY
– Using a simple model of classical water, eliminating variables such as boundary
conditions and complicated parametrization, how does this potential arise?
– Using a more complex organic molecule, are there general physical properties of the
potential shift which emerge? How do water and organic solvents differ in their ion
solvation properties?
• Chapter 5: Model development of ab initio accuracy at classical cost
– Can Neural Network Potentials produce electrostatic properties of liquid water?
47 Chapter 2
Theory
2.1 Preface
This chapter presents the theoretical background for the work which follows. The first section discusses classical MD and statistical mechanics. The next section provides a brief review of
DFT and AIMD techniques, focusing on those used in this thesis. Then I introduce the theory behind the Neural Network Potential methods that I use in my final project. Finally, I outline the thermodynamic techniques related to ion solvation, including the important contributions from interfacial potentials.
2.2 Classical Molecular Dynamics
Molecular Dynamics (MD) is a technique that uses the classical laws of motion to describe nuclear motion and can be used to calculate the equilibrium and transport properties of a many- body system.3 Like experiment, an individual atomic trajectory is not the sought after quantity, instead the average collective properties in time and space are determined. It is important to distinguish between time and space averaging, as they are important to the concept of ergodicity
48 2.2. CLASSICAL MOLECULAR DYNAMICS which is the underpinning of classical MD theory. For a short time simulation to produce real statistical mechanical results, it is necessary that the system satisfy the ergodic hypothesis3
ai(r) = hai(r)iNVE (2.1)
where a is some physical observable, the bar indicates a time average, and the brackets indicate an ensemble average. The ensemble average means an average over the phase space of the system for a given energy E. This hypothesis, which holds for the systems of interest here, says that the average of a over the time evolution of the system is equivalent to the ensemble average. Hence, a simple time evolution can be used to produce a full thermodynamic average.
Here I outline some of the tools and techniques used accomplish this.37
2.2.1 Equations of Motion
The most basic procedure is as follows. The first step in an MD system is to initialize with a set of atomic coordinates and velocities. Forces on each atom are computed and Newton’s equations of motion are integrated in order propagate a system trajectory. System averages can be calculated at each step. The Verlet method104 is one of the most popular methods of integration, with each new position is evaluated by,
f(t) r(t + δt) = 2r(t) − r(t − δt) + δt2 δt2 (2.2) m and the velocities constructed with,
r(t + δt) − r(t − δt) v(t) = (2.3) 2δt
49 2.2. CLASSICAL MOLECULAR DYNAMICS
The kinetic energy and system temperature can immediately be constructed from this knowl- edge of system coordinates and velocities at the microscopic level and can be translated to the macroscopic level through statistical mechanics. Once the thermodynamic state is speci-
fied, thermodynamic properties such as density, chemical potential, heat capacity, structural quantities, and time correlations can be determined.
2.2.2 Thermodynamic Ensembles and Equilibration
Equilibration of the system ensures that a statistical mechanical ensemble is maintained. The
Microcanonical, NVE, Canonical, NVT, and Isothermal-isobaric, NPT, are commonly used.
Constant Volume, V , and number of particles, N, are easily achieved in simulation. Constant temperature is maintained through use of a thermostat such as Nose-Hoover or Berendsen.
The Nose-Hoover thermostat’s instantaneous kinetic energy (and velocities) fluctuate while temperature is held constant due to introduction of a heat bath to the Hamiltonian. Constant pressure can be achieved with a barostat, which scales unit-cell vectors in response to the stress tensor.
In practical applications, one or more of these ensembles may be used depending what system is being modeled. Simulation of interfaces between different phases or chemical species may involve separate equilibration procedures before combining them.
2.2.3 Classical Force Fields
Perhaps the most important choice in MD simulation is deciding how to model the poten- tial energy. Commonly referred to as the force field, it determines the functional form and parametrization of the atomic/molecular interaction energy. The potential energy function is usually divided into bonded and non-bonded interactions. The former typically meaning inter-
50 2.2. CLASSICAL MOLECULAR DYNAMICS molecular (chemical) bonds between atoms and the later meaning intramolecular interaction.
Bonded terms can be simplified through use of "rigid" models in which molecules are seen as ball-and-stick entities where molecular vibrations can be ignored. Alternatively, harmonic springs approximate the atomic bonds, out-of-plane angles, and torsion of molecular motion.
Non-bonded interactions model the physical forces of dispersion, repulsion, and electrostat- ics. The commonly used Lennard-Jones potential uses a repulsive r−12 term to imitate the effects of the Pauli exclusion principle of overlapping electronic orbitals, and an r−6 term for the attractive, long-range dispersion (van der Walls force).7 In the Lennard-Jones potential shown below, parameter has units of energy, describing the depth of the potential well, and
σ has units of distance and describes the distance at which the potential is zero.
"σ 12 σ 6# U = 4 − (2.4) LJ r r
Each atomic species is assigned an i and σi with interaction between species determined by mixing rules. Electrostatic interactions are determined by a simple Coulomb interaction energy,
X qiqj Ues = (2.5) i6=j rij
where charges qi are the partial charges of each atom.
In practice the parameters and partial charges above are determined either by comparison with quantum chemical calculation or through experimental properties. Simulations are run with initial guesses to calculate properties of solids, liquids, and gases, and then compared with reference values. Parameters like i and σi are adjusted until satisfactory agreement. Because of this approach, classical force fields are only successful in the subset of systems for which they are parametrized. As a result they are non-transferable.
51 2.2. CLASSICAL MOLECULAR DYNAMICS
2.2.4 SPC/E Force Field
Modeling water is integral to many bio-molecular, condensed matter, and chemical systems, leading to intense efforts in force field development. The detailed solvent interactions are critically important and are not correctly described by continuum models. Classical water models generally consist of three to five atomic sites with various partial charge values and locations with rigid or non-rigid Lennard-Jones parameters. The SPC/E (simple-point charge, extended) water model105 106 is popular for it’s simplicity and computational efficiency. It consists of 3 atomic sites: oxygen and two hydrogen atoms, with a partial charge on each site. SPC/E refined the SPC model by adding a term to correct for average polarization which improves the density and diffusion constant. The SPC/E model uses partial charges qoxygen = −0.8476 and qhydrogen = +0.4238 with the equilibrium bond length of 1.0 Å and an
H-O-H angle of 109.47◦.
The SPC/E model inter-atomic potential is given by the force field,
1 h i 1 U = k (r − req )2 + (r − req )2 + k (θ − θeq )2 + U + U (2.6) SP C/E 2 b OH1 OH OH2 OH 2 a HOH HOH LJ ES
where rOH are bond lengths and θHOH is the H-O-H angle. Here, we use the rigid model, where spring constants ka and kb are very large. For the purposes of theory, this model is ideal as it is one of the most basic models, yet it performs remarkably well in comparison with a wide range of experimental properties.107 108 109
52 2.2. CLASSICAL MOLECULAR DYNAMICS
2.2.5 OPLS-AA
The OPLS-AA force field was parametrized for experimental properties of liquids. It is non-rigid and the functional form of the potential is,110
X 2 X 2 X UOP LS−AA = Kr(r − r0) + kθ(θ − θ0) + f(dihedrals) + ULJ + UES (2.7) bonds angles dihedrals
q q where the Lennard-Jones potential uses mixing rules Aij = AiiAjj and Cij = CiiCjj.
2.2.6 Boundary Conditions
The boundaries of the simulation region depend on the system being modeled. In the case of a finite system, a bounding potential is often used. In my droplet study, I use a simple
1 2 half-harmonic bounding potential wall U = 2 k(r − r0) to ensure that molecules do not leave the simulation box. If the system is properly equilibrated, the number of molecules interacting with this potential is minimal.
In the case of extended systems, such as bulk liquids or flat interfaces, a more complex treatment is needed using periodic boundary conditions (PBC). The number of molecules that would be needed for these large systems is computationally unfeasible, so a relatively small system box is replicated in each dimension with conditions on each boundary. The Ewald potential is commonly used to implement PBC. For a box of size V = L3 both the Coulomb interaction potential and the energy between an ion and it’s periodic images is expressed through
Ewald summation: k2 4η2 X erfc(η|r + nL|) X 4π e ikr ψewald = + 3 2 e (2.8) n |r + nL| k6=0 L k
η is the convergence parameter, n is the real-space lattice vector, and k is the Fourier-space
53 2.3. AB INITIO MOLECULAR DYNAMICS lattice vector. The electrostatic potential energy is then:
N N N X X ξ X 2 U = qiqjψewald(rij) + qi (2.9) i=1 i=1 2L i=1
The last term in this equation is the self-energy correction term, which must be evaluated for the specific system geometry (ξ = −2.837297 for cubic geometry). This correction arises out of the need for a neutralizing background and accounting for the image charge effects.1117
While this method drastically improves computational efficiency, there are some issues.
This artificial system produces artifacts outside of physical reality.112 113 111 Correction terms for solvent polarization, finite-size effects, thermodynamic single-ion solvation properties (enthalpy, entropy, heat capacity, etc.) have to be included.112
2.3 Ab Initio Molecular Dynamics
Highly accurate ab initio molecular dynamics (AIMD) methods create finite-temperature dy- namic trajectories by performing electronic structure calculations during the simulation. Ap- proximate solutions to the Schödinger Equation are achieved through Density Functional The- ory (DFT). For a system of N nuclei and Ne electrons, the total Hamiltonian is given by
H = Te + Vee + VeN + TN + VNN (2.10)
In classical MD, this Hamiltonian is reduced to a force field approximation as in sections 2.2.3-
2.2.5. In DFT, the Hamiltonian is slightly simplified with the Born-Oppenheimer Approxima-
114 tion, which allows for the electronic and nuclear coordinates to be separated. That way, VNN is a constant and TN is equal to zero. There are many AIMD methods and codes available
54 2.3. AB INITIO MOLECULAR DYNAMICS to researchers. I will narrow the scope to the methods of Density Function Theory115 (DFT) and the Gaussian Plane Wave approach (GPW)116 117 used in the open source software package
CP2K.118
According to DFT, the electronic ground state structure is determined through the electronic density distributions n(r) and ground state energy E. This is based on the Hohenberg-Kohn
119 Theorems which state that (1) The external potential Vext and energy E are functions of density n(~r) and (2) The exact density n(~r) is that which minimizes the total energy. So, there is an exact one-to-one correspondence between ground-state electron density and the wave function and between the wave function and the many-body Hamiltonian. The Kohn-Sham ansatz115 56 makes the problem solvable by replacing the interaction electron potential with an effective non-interacting electron potential Veff . The reduced Hamiltonian is then:
H = T [n] + V [n] + U[n] = T [n] + Veff [n] (2.11)
R e2 0 where Veff [n] = V (~r)+ |~r−~r0| dr +Vxc[n] and Vxc[n] is a chosen exchange-correlation functional.
The accuracy of the electronic structure calculation depends on a good approximation to Vxc.
This reduced problem leads to the Kohn-Sham Equations which must be solved self-consistently.
The Gaussian Plane Wave method uses atom-centered Gaussian orbitals to represent the wave functions (the Kohn-Sham matrix) and auxiliary plane waves to represent the electronic density.
The energy is efficiently solved for with Fast Fourier Transform Methods (FFT).118
Once the ground state energy is determined, the dynamics are solved for using Born-
Oppenheimer Molecular Dynamics.120 The potential energy surface is the ground state energy and the atomic coordinates are treated classically and propagated using Newton’s Equations.
Like classical MD, it is up to the researcher to choose suitable simulation-specific functions: variables of AIMD include the Gaussian basis sets for the wavefuntions, exchange-correlation
55 2.4. NEURAL NETWORK POTENTIALS functionals, and pseudopotentials to describe the nuclei. Determination of electrostatic prop- erties such as the multipole moments of a charge distribution are achieved with maximally localized Wannier functions (MLWFs).121 In AIMD, the charge is diffuse and must be assigned to atoms in a chemically reasonable manner.122 I’ll discuss this further in section Chapter 5.
2.4 Neural Network Potentials
The goal of Neural Network Potentials (NNP) is to reduce the electronic structure problem of
DFT/ AIMD approaches with a Machine Learning derived potential. Instead of a complicated analytical Potential Energy Surface (PES), NNP’s develop a neural network structure-energy relationship. This is physically correct, as the Born-Oppenheimer Approximation states45 that the Hamiltonian itself is completely defined by the atomic positions, nuclear charges, and total charge of the system.58
This approach arose out of the technique123 124 125 of representing the PES through a highly general, flexible, and adaptable set of mathematical functions which are optimized to fit ab initio data instead of empirical results.58 These "mathematical potentials" were developed in the 1990’s and use a large number of terms. Machine Learning methods such as artificial neural networks, kernel ridge regression, and support vector machines have been used to efficiently and accurately construct the PES.126 127 128
Neural networks (NN) are theoretically able to produce any multi-dimensional function.129
In brief, neural networks are computational processing systems modeled after the human brain, and are composed of many simple, interconnected processing units, nodes, which make decisions based on some response to external input.130 These nodes are organized into layers of varying size. NNs with many hidden layers are referred to as "Deep" NNs. Here, I use a fully-connected feed-forward NN, meaning each node is connected to all other nodes in the neighboring layers
56 2.4. NEURAL NETWORK POTENTIALS
Figure 2.1: Schematic of a neural network with 2 hidden layers. Input layer values are feed- forward to the output layer. In the case of a NNP, this output layer is total energy E which is the sum of individual atomic energies. Inspired by Ref. 130. with a forward flow of outputs.
j kl Important elements of NNs are the bias weight bi , weight parameters Wij , and activation
j function φi as shown in Figure 2.1 . The learning, or training, process is similar to the traditional sense of the word. Training data is fed to the first layer with a linear or nonlinear transformation
j known as the activation function. The value of node yi is then given by
Nj−1 j j j X j−1,i j−1 yi = φi bi + Wk,i · yk (2.12) k=1
The bias and weight parameters are free parameters to be optimized. After one iteration of the full training set through the NN, called an epoch, the output is optimized through a process of minimization of the loss function (or objective function). The number of iterations in this
Batch Normalization procedure is variable, and weights and biases are only updated after one epoch. The loss function L is usually the mean squared error of the quantity of interest. The
57 2.4. NEURAL NETWORK POTENTIALS errors are then back-propagated through each step starting from the final parameter set at t, similar to a steepest descent algorithm,
∂L W (t + 1) = W (t) − r · (2.13) ∂W (t)
Here r is the learning rate, which affects convergence.
Turning to the specific case of NNPs used in atomic simulations, I’ll restrict my discussion to an overview of the procedure of Wang et al. 63 contained within the open source package
DeePMD-kit. An extremely important element of NNs is processing the input data. The in- put data are the atomic coordinates along with the energies that are used for optimization
(learning), and the output is the inter-atomic potential energy which has permutational, ro- tational, and translational symmetry.131 The training data consists of an ab initio trajectory of coordinates and calculated energies, forces, and virial for the system. Coordinates for each frame of the trajectory are then processed into a symmetry-preserving set of coordinates. For a system of N atoms, with lab coordinates {R1, R2, ..., RN }, neighboring atoms are determined by Rij = Ri − Rj within a certain cutoff around the atom in order to preserve translational symmetry. Atoms are chosen from this list by a user-defined rule to form a local frame. The coordinates are then rotated into this local frame to preserve rotational symmetry and finally are sorted according to the chemical species s(i), and then ascending inverse distance to atom i to preserve permutational symmetry. These descriptors Dij are produced for each coordinate in the local frame, with the option of full-coordinate or only radial coordinate information. The full information is given by:59
1 xij yij zij {Dij} = { , , , } (2.14) Rij Rij Rij Rij
58 2.5. THERMODYNAMICS OF ION SOLVATION
These descriptors are fed through the deep neural network NW (i) that determines Ei, the
α energy of atom i, through it’s coordinate descriptor: Ei = NW (i)({Dij} }j∈N (i)) where N (i) is the neighbor list. In this network, the activation function φ is the nonlinear hyperbolic tangent of the linear combination of weights and biases as described above. The loss function
L(p, pf , pξ) given by:
p p p L(p , p , p ) = ∆E2 + f Σ ∆|F |2 + ξ ∆||Ξ||2 (2.15) f ξ N 3N i i 9N
where ∆E, ∆Fi and ∆Ξ are the root mean squared error in the energy, forces, and virial, respectively. Forces on the atoms and virial are computed by gradients of system energy with respect to position. Optimization is done using the Adam stochastic gradient descent method.132 Training data is divided into batches with loss function and gradient computed at each training step only from data within the current batch. A more detailed discussion of the network can be found in Ref 63. Some network parameters are user defined and can be adjusted to ensure an adequately converged network by analyzing the error in the training and testing data as the batch iterations proceeds. Once this is achieved, the biases and weights for the trained network are frozen, and it can be used on a new system of input coordinates.
2.5 Thermodynamics of Ion Solvation
Reprinted (adapted) with permission from
Doyle, Carrie C., Yu Shi, and Thomas L. Beck. "The importance of the water molecular quadrupole for estimating interfacial potential shifts acting on ions near the liquid–vapor in- terface." The Journal of Physical Chemistry B 123.15 (2019): 3348-3358. Copyright 2020.
American Chemical Society.
59 2.5. THERMODYNAMICS OF ION SOLVATION
For ion solvation science, one of the most important quantites to determine is the chemical potential, or partial molar Gibbs Free Energy
δG ! µi = |T,P,n (2.16) δni
Free energies tell us about the phenomena of binding, equilibrium constants, rate constants activity coefficients, osmotic coefficients, surface tension, and phase transitions. In the following section I will discuss the method we use to calculate free energies, Quasichemical Theory (QCT), which uses a spatial partitioning of the thermodynamic process.
2.5.1 The Potential Distribution Theorem
The chemical potential, µ, of an ion in dilute solution is given by
βµ = ln(ρΛ3) + βµex (2.17)
1 where β = kT , ρ is the density, and λ is the thermal de Broglie wavelength. The excess chemical potential µex or absolute solvation free energy is the difference between the chemical potential of species i and an ideal gas under the same conditions.3 There are several ways to calculate excess chemical potentials from simulation, one method is the Widom Particle Insertion Method133 used in the Potential Distribution Theorem.97 111
The Potential Distribution Theorem expression for the excess chemical potential of a molecule or ion is111 97
ex βµ = lnhexp(β∆U)i = lnhexp(−β∆U)i0 (2.18)
where ∆U = UN+S − UN − US is the interaction energy of the molecule or ion (N) with the solvent (S). The sampling in the first expression, h...i, includes the solute fully coupled to the
60 2.5. THERMODYNAMICS OF ION SOLVATION
solvent and the sampling in the last expression, h...i0, involves the solvent and solute separately with no coupling.97
These expressions in equation 2.18 can be rearranged to give
ex Z βµ β∆Ui βε e i = he i= Pi(ε)e dε (2.19)
for the coupled case and
ex Z (0) −βµi −β∆Ui −βε e = he i0 = Pi (ε)e dε (2.20)
(0) for the uncoupled case where Pi(ε) = hδ(ε − ∆U)i and Pi (ε) = hδ(ε − ∆U)i0 are the prob- ability densities of ion/solvent interaction energies for a coupled and an uncoupled trajectory, respectively. If the distributions overlap, the energy at which these two distributions cross gives the exact excess chemical potential. If they do not overlap, we use an alternate method known as Quasichemical Theory. This method arose out of the fact that the tails of the distributions are poorly sampled in an MD simulation. Spatial partitioning of the thermodynamic process can yield near Gaussian distributions Pi(ε)
2.5.2 Quasichemical Theory
Using this spatial partitioning and a repulsive wall potential M the above excess chemical expression 2.18 can be rewritten as,
ex −βM(r) −βM(r) −β∆U βµ = lnhe i − lnhe i0 − lnhe iM
−β∆U = ln x0(r) − ln p0(r) − lnhe iM (2.21)
ex ex ex =µis (r) + µpk(r) + µos (r)
61 2.5. THERMODYNAMICS OF ION SOLVATION
This equation can be derived from a thoughtful inclusion of an expression equal to unity. These terms are referred to as the inner shell (is), packing(pk), and outershell (os) terms. Inner shell term xo is the probability of finding no solvent in the cavity formed by the repulsive wall with full ion-solvent interaction. The packing term po is the probability of finding no solvent in the cavity of the ion-free solvent. The outershell (or long-ranged) term describes the remaining interaction of ion with solvent beyond the excluded volume.
The steps of this thermodynamic cycle are
1) open a cavity larger than the ion;
2) insert a point charge at the cavity center;
3) convert the point charge into the real ion;
4) remove the cavity constraint and allow the solvent molecules to move into direct contact
with the ion.
The corresponding free energy changes are:
1) the cavity formation free energy (or packing);
2) a purely electrostatic part of the electrochemical potential due to insertion of a test
charge into the cavity;
3) a small quantity that includes a dispersion contribution depending on cavity size;
4) the inner-shell or chemical part of the free energy that includes all chemically specific
effects due to direct contact of the ion with the solvent, such as electrostatics, polarization,
dispersion, and charge transfer.
62 2.5. THERMODYNAMICS OF ION SOLVATION
The textbook expression for the electrochemical potential134,135 for an ion i is
ex ex µi = µb,i + qiφnp, (2.22)
ex where µb,i is the bulk hydration free energy and qi is the ion charge. If we link this expression to the QCT partitioning, it is sensible to define φnp as the purely electrostatic net potential at the center of a cavity into which the ion is inserted [step (2) above]. Interpreting what this net potential means depends on the system being studied. Accurate determination and definition of this net potential is essential to establishing a single-ion solvation free energy scale.
2.5.3 Interfacial Potentials
As discussed above, an interfacial potential is the electrostatic potential produced upon moving a charge across regions of nonuniform charge, such as those at phase boundaries or other interfaces. As shown in equation 2.22, if the charge is nonzero, there will be a contribution to the excess chemical potential. As discussed in section 1.5.2, the measurability of these potentials has been called into question for many years, but recent work has shown that there are measurable physical effects. This is predicated on a careful consideration of experiment and simulation specifics.
It is critically important that the researcher understands the boundary conditions involved and their bearing on the interfaces. If periodic boundary conditions are used, there is no explicit surface interface that separates the bulk liquid from the vapor. The real hydration free energy is then
ex ex µ = µint + qφsp, (2.23)
where µint is the intrinsic free energy evaluated without that surface interface and φsp is the
63 2.5. THERMODYNAMICS OF ION SOLVATION
Liquid
Vapor
Figure 2.2: A schematic representation of the studied system - a spherical liquid water droplet
(blue) containing a spherical cavity (grey). Potentials φsp and φlp denote the potential shift due to crossing the liquid-vapor interface of radius Rdrop, and the cavity-liquid interface of radius
λ. The net potential at the cavity center is then: φnp = φlp + φsp interfacial potential across that interface. Alternatively, it can be defined in terms of the net
ex potential φnp the potential at the center of a neutral cavity and the bulk free energy µb as follows:
ex ex µ = µb + qφnp, (2.24) which is equivalent to 2.22 above. This definition is appropriate for simulations of a finite droplet or a PBC slab, which both contain two interfaces. In those cases, the two “interfaces" that are crossed upon moving from the outer vapor phase into the cavity are: the liquid-vapor interface (yielding the surface potential φsp) and the cavity-liquid interface near the ion (yielding the local potential φlp). A schematic depicting these potentials is provided in Figure 2.2. This definition produces:
φnp = φlp + φsp (2.25)
A second motivation for the physical significance of the net potential is the exact QCT
64 2.6. MACROSCOPIC INTERFACIAL ELECTROSTATICS expression for the ion excess chemical potential derived in Ref. 136:
ex (0) n ex ex µ = −kBT ln Kn ρW + kBT ln p(n) + µXWn − nµW (2.26)
(0) where Kn is the equilibrium constant for binding n solvent molecules to the ion in the gas phase, ρW is the solvent number density in the liquid, p(n) is the probability to observe n solvent
ex molecules bound to the ion in the liquid, µXWn is the free energy to insert the ion/(n-solvent)
ex cluster, and µW is the free energy to insert a single solvent molecule into the bulk liquid. In numerical implementations of QCT, n typically includes the first solvation shell or roughly 4–8 water molecules.111 Thus the cavity size for inserting the cluster will be on the order of 4-6 Å.
ex The net potential can be seen to be located in the µXWn term in the free energy. If a cavity size (for inserting the cluster) is chosen so the inner shell and packing parts of that free
ex energy largely cancel, then the predominant contribution to µXWn will be a long-ranged (largely electrostatic) free energy that is close to the net potential we have defined above. This cavity length scale occurs roughly at the peak of the radial distribution function between the cluster and the next layer of solvating molecules.
2.6 Macroscopic Interfacial Electrostatics
The net potential φnp has physical significance which cannot be overlooked in ion solvation thermodynamics. In the present section I outline the theoretical methods used to investigate it’s origin. In particular, I analyze the molecular multipole distributions at each interface and how they generate the net potential at cavity center. This discussion relies on Refs. 137, 138,
139 and 140. Reprinted (adapted) with permission from
Doyle, Carrie C., Yu Shi, and Thomas L. Beck. "The importance of the water molecular
65 2.6. MACROSCOPIC INTERFACIAL ELECTROSTATICS quadrupole for estimating interfacial potential shifts acting on ions near the liquid–vapor in- terface." The Journal of Physical Chemistry B 123.15 (2019): 3348-3358. Copyright 2020.
American Chemical Society.
2.6.1 Multipole Expansion of a Charge Distribution
Multipole moment expansions can be used to approximate a charge distribution and the result- ing potential. Following Jackson’s derivation138 of the macroscopic electrostatic equations from the microscopic definitions, we consider molecules as collections of bound charge. Performing a spatial average leads to the picture of averaged microscopic charge density as the sum of molecular multipole moment contributions.
For an individual molecule n at position xn with charges j at positions xj = xn + xjn the
first three molecular multipole moments (charge, dipole, and quadrupole) are given by:
X qn = qj j(n)
X pn = qjxjn (2.27) j(n)
X (Qn)αβ =3 qj(xjn)α(xjn)β j(n)
The microscopic average charge density hηi can then be written as the sum of the macro- scopic multipole densities— ρ(x), P(x),Qαβ(x) — which are obtained by summing over molecules.
Additional summation of these multipole moments over the moment order, we obtain the aver- aged microscopic charge density as the sum of the macroscopic multipole densities ρ(x), P(x),
138 and Qαβ(x):
2 X ∂ hη(x, t)i = ρ(x, t) − ∇ · P(x, t) + Qαβ(x, t) + ... (2.28) αβ ∂xα∂xβ
66 2.6. MACROSCOPIC INTERFACIAL ELECTROSTATICS
Substituting eq (2.28) for the charge density ρ and performing the appropriate integrations of
Possion’s Equation,95,141
∇2φ = −∇ · E = −4πρ (2.29) we can determine individual multipole interfacial potential contributions.84,142
2.6.2 Coordinate Systems: Cartesian and Spherical
Wilson, Pohorille, and Pratt142 showed that, for a planar interface, there are contributions to the interfacial potential from both dipole and quadrupole moment densities. In Cartesian coordinates, using (2.28) for ρ(z0) and integrating
Z z δφ(z) = 4π z0ρ(z0)dz0 (2.30) zv from vapor to liquid, the potential across a planar interface is given by:
Z zl 0 0 δφ = 4π Pz(z )dz − 4π[Qzz(zl) − Qzz(zv)] (2.31) zv
Higher order multipoles do not contribute because they involve spatial derivatives of higher- order terms. Such gradients do not exist in the bulk fluid, showing that, for the planar surface, the expansion truncates after the quadrupolar term. This relationship has been demonstrated throughout the literature for different water models.98,142–144
Horvath et al.143 later derived the expression analogous to eq (2.31) for a spherical coordi- nate system. Note that the unprimed quadrupole density Q is in cartesian coordinates, and the
0 D Q primed Qrr is in spherical coordinates. With δφr(r) = δφr (r) + δφr (r) the dipole contribution
67 2.6. MACROSCOPIC INTERFACIAL ELECTROSTATICS is given by Z r D 0 0 δφr (r) = 4π Pr(r )dr (2.32) 0
Q and the quadrupole contribution δφr (r) is given by
Q Q,1 Q,2 δφr (r) = δφr (r) + δφr (r) (2.33)
where
Q,1 0 δφr (r) = −4πQrr(r) (2.34) and
Z r 0 Q,2 dr 0 0 0 δφr (r) = 4π [TrQ(r ) − 3Qrr(r )] (2.35) 0 r0
The planar geometry and spherical geometry both contain multipolar elements, though the contributions from each multipolar element are take a different functional form.143 For the planar and spherical geometries, the dipole contributions to the interfacial potentials are of the same form, reflecting their dependence on water dipole orientation (polarization) alone.
For a spherical system, the quadrupole contribution has two terms (eq 2.33). The first term
Q,1 δφr (r) is a bulk property of the liquid, which has a form analogous to that for the quadrupole part of the planar case above (eq 2.31). In either geometry, it cancels when crossing the two
“interfaces" - passing from vapor, through liquid, and into cavity center - and is equivalent to the Bethe Potential96 described in section 1.5.3
4π φ = ρTrQ (2.36) B 6 0
where ρ is the molecular liquid density and Q0 is the primitive (traced) molecular quadrupole
68 2.6. MACROSCOPIC INTERFACIAL ELECTROSTATICS moment.98,143,145,146 Because typical ions do not penetrate into the solvent molecular interior, this contribution is not important for ions near interfaces.96
Q,2 143 The second term δφr is created by the spherical symmetry breaking of an interface.
For the droplet system, this residual quadrupole contribution is due to the curvature of the separate cavity-liquid and liquid-vapor interfaces. The sign of this term is determined by the quadrupole density profiles. It has the same sign for both interfaces, and for the liquid-vapor interface this second term approaches zero in the limit of infinite droplet radius.
We emphasize that, in the above equations (2.28)-(2.36), the traceless form of the multipole moments cannot be used unless the potential is calculated outside of the charge distribution.
The relationships above follow from Jackson’s expansion of the microscopic charge density into molecular multipole moments. He showed that when the partial charges are viewed as a molecule, the quadrupolar contribution to the microscopic charge density is traceless. This differs from the quadrupolar contribution contained in the macroscopic Maxwell equations, where the trace remains. This can be explained by noticing that the quadrupolar trace terms are not eliminated completely, but move to the macroscopic charge density, or first moment, when outside of the distribution.98,138 When considering the electrostatic potential at a point, the quadrupolar trace term must be considered while within the charge distribution but can be eliminated outside of it where the macroscopic charge density term, and hence the quadrupolar trace is zero.98,140,146 This allows us to use the following direct method for calculating the interfacial contributions to the net potential at the cavity center (or any zero charge density region).
69 2.6. MACROSCOPIC INTERFACIAL ELECTROSTATICS
2.6.3 Molecular Multipole Expansion of Electrostatic Potential
The electrostatic potential at x = 0 due to a molecule n at position xn with charges j at
138 positions xj = xn + xjn is given by:
" # X 1 1 1 1 φ(0) = qj − (xjn)α∇α + (xjn)α(xjn)β∇α∇β − ... (2.37) j xn xn 2 xn
The tensor notation indicates a sum over cartesian coordinates. The origin of our multipole calculation is the water oxygen. The moments are shown in eq (2.27) except the quadrupole
3 P 1 2 moment is replaced by the traceless form, (θn)αβ = 2 j qj[(xjn)α(xjn)β − 3 xjnδαβ]. Outside of a charge distibution, the potential is then,140
" 1 1 1 1 # φ(0) = qn − (pn)α∇α + (θn)αβ∇α∇β − ... (2.38) xn xn 3 xn
Summing eq (2.38) over n molecules and taking an ensemble average produces an estimate of the electrostatic potential in terms of the charges, charge coordinates and molecular coordinates.
With the cavity center located at x = 0, this provides an approximation to the net potential
φnp. (See reference 140 for the octupole term.) Equation (2.38) can be rewritten in more compact notation as 1 φ(0) = Tq − T (p ) + T (θ ) (2.39) n α n α 3 αβ n αβ where T is the molecular coordinate tensor:140
1 T = xn 1 (x ) n α (2.40) Tα =∇α = − 3 xn xn 2 1 3(xn)α(xn)β − xnδαβ Tαβ =∇α∇β = 5 xn xn
70 Chapter 3
Water Liquid-Vapor Interfacial
Potential Shifts
3.1 Preface
Reprinted (adapted) with permission from,
Doyle, Carrie C., Yu Shi, and Thomas L. Beck. "The importance of the water molecular quadrupole for estimating interfacial potential shifts acting on ions near the liquid–vapor in- terface." The Journal of Physical Chemistry B 123.15 (2019): 3348-3358. Copyright 2020.
American Chemical Society.
In this Chapter I present the results from my water droplet study. I attempt to answer the question that I posed in section 1.6: What is the physical origin of φnp? The interfacial potential shifts that act on ions near the water liquid-vapor interface are essential for a proper understanding and determination of the single-ion solvation free energy values. Our work builds on previous related analyses in Refs 97,98,143,145–147. I investigate the radial size-dependence of the potential from both the cavity-liquid and liquid-vapor interface. The question we address
71 3.1. PREFACE is: how do the molecular multipoles contribute to the observed average net potential at the cavity center? We choose droplets large enough to exhibit quasi-bulk behavior in the region between the cavity and the surface. For that bulk region, the average charge density is zero, so there is negligible contribution to the net potential. I simulate droplets consisting of 250–1500 water molecules with cavities of variable size embedded at the droplet center. By modeling droplets, all electrostatic interactions through direct summation of the Coulomb potential are included.
Thus, we avoid periodic boundary finite size effects due to the Ewald treatment of long-ranged electrostatics.113,148,149 Recently, droplet models have been examined in relation to other aspects of bulk behavior.150,151 The range of droplet sizes modeled here was chosen to assess the distance dependence of the multipole contributions from the liquid-vapor surface. Here we utilize only the SPC/E water model to illustrate the basic physics of the origin of the net potential.
The distance dependence of the various multipole net potential contributions can be an- alyzed by a simple scaling expression in the spherical droplet configuration (with the cavity located at the droplet center):
Z R+∆R 1 2 φ(0)l,np ∼ ρMl 4πR dR (3.1) R Rl+1
where the dipole contribution is l = 1, quadrupole l = 2, etc., and ρMl is the multipole density of order l in the inhomogeneous region. The dipole contribution is special because it does not decay in magnitude with distance from the cavity center (see also Ref. 146).
Next, consider the quadrupole contribution. Assuming the quadrupole density is roughly constant over the inhomogeneous domain of size ∆R (and that ∆R << R):
Z R+∆R 1 2 φ(0)l=2,np ∼ ρM2 4πR dR ∼ ρM2 ln(1 + ∆R/R) ∼ ρM2 ∆R/R (3.2) R R3
72 3.2. COMPUTATIONAL METHODS
Thus the quadrupolar contribution decays slowly as 1/R. Higher-order multipole contributions decay as higher powers of 1/R.
First, I calculate the electrostatic potential through construction of the charge density and integrations of Possion’s Equation, the method described in section 2.6.1 and equation 2.29.
Then I use the method of section 2.6.3 and equation 2.39, along with a new plotting method which avoids complications from the Bethe potential by finding the multipolar contributions to the net potential directly. Comparison of the two results demonstrates the simplicity of the new method. My results establish an approximate radial distance at which interfacial contributions die off (charge density averages to zero) providing benchmarks for other studies to
find accurate zeroing of the electrostatic potential in water. Additionally, I find the importance of the quadrupole component due to the local surface around the cavity and the cancelling dipole contributions between the two interfaces. This finding provides an explanation for the wide differences in interfacial potentials for the various water models.
3.2 Computational Methods
The TINKER 7.1.2 code152 was used for all the classical molecular dynamics (MD) simulations.
We study three droplet systems with 250, 512, and 1500 SPC/E water molecules that have radii of approximately 12.1 Å, 15.4 Å, and 22.1 Å, respectively. Finite droplet boundary conditions were applied by implementing a half-harmonic bounding potential on only those water oxygens that traversed a distance of 5 Å outside the approximate radius. All simulations employed a 2 fs timestep and the NVT ensemble using the Berendsen thermostat with 0.1 temperature coupling constant. A temperature of T = 260 K was chosen to reduce the number of evaporating waters, while maintaining adequate diffusion. A 60 ps equilibration was performed on the droplet before cavity formation.
73 3.3. RESULTS AND DISCUSSION
To create the cavity, a neutral point particle was placed at the center of the droplet at x = 0 and restrained to the system’s center of mass. Following the method of Shi and Beck147 the
WCA potential153 was used between the neutral particle and water oxygens to push waters out to several different final cavity radii. The potential is given by M(λ):
c /r12 − c /r6 + c /R12 , r ≤ R 12 6 12 c,λ c,λ M(λ) = (3.3) 0 r > Rc,λ E 2c c = λ , c = 12 12 −6 −6 2 6 6 (λ − Rc,λ) Rc,λ
kB T where λ is the final cavity growth radius, Eλ = 2 , and Rc,λ = 1.05λ (the radius at which the
WCA potential is zero). The cavity was created during 10 consecutive 200 ps growth periods
20 using the functional form fλ(γ) = γ M(λ) with γ = 0.1, 0.2, ..., 1.0. After this 2 ns growth period, the droplet with a cavity was equilibrated for 800 ps before the final 3 ns production run.
3.3 Results and Discussion
To investigate the electrostatic origins of the net potential, we prepared a variety of systems with differing cavity and droplet sizes. This allows us to gauge the size-dependent behavior of the local potential from nearby waters and the surface potential from distant waters97 – termed in other literature as the potential due to the solute-liquid interface and the liquid- vapor interface, respectively.143 Presented here are the results of simulations of a 250 SPC/E water system with cavity radii λ = 2.0, 2.5, 4.0, 4.3, 6.0, 6.6 Å, a 512 SPC/E water system with cavity radii λ = 2.0, 2.5, 4.0, 4.3, 6.0, 6.6, 8.0, 10.0 Å, and a 1500 SPC/E water system with cavity radii λ = 2.0, 2.5, 4.0, 4.3, 6.0, 6.6, 8.0, 10.0 Å. These cavity radii range from small
74 3.3. RESULTS AND DISCUSSION length scales similar to many real ions to nanoscopic sized cavities displaying quasi-macroscopic behavior. Previous work has shown that, for radii larger than about 4 Å, the net potential is relatively stable with increasing size (up to about 16 Å radius).154 For cavities below about 4 Å in radius, the net potential shows a strong size dependence due to local water rearrangements that maximize hydrogen bonding around the small cavities.154
Figure 3.1 shows the electrostatic potential profile obtained using average charge density profiles and eq (2.29) for the droplets of 250, 512, and 1500 waters with cavity radii 4.0 Å and a 1500 water droplet with cavity radii of 2.5 and 6.0 Å centered at x=0. From right to left, the first potential shift of -0.600 V is the surface potential φsp (N = 1500, λ = 6.0 Å case, with nearly equal values for the λ = 2.5 and 4.0 Å cases). The second potential shift (from the bulk region into the cavity) is the local potential φlp with a value of 0.275 V (for λ = 6 Å), and the resulting net potential φnp is -0.325 V. The relatively flat region between the two interfaces is extended in the 1500 water system, but small in the 250 and 512 water systems. This implies that relatively large systems are necessary to obtain adequate bulk-like behavior and interface separation. There is clear charge asymmetry near the cavity, reflected by the large variation in potential. This asymmetry presumably exists locally near the liquid-vapor interface, but is smoothed out due to capillary waves.133
The quantity of interest here is the net potential at the center of the cavity, and perhaps higher-order contributions to the free energy due to asymmetry in the potential distribution.147
One method we can use to determine this potential and its multipole components is to calculate the multipole moment densities and use eqs (2.28)- (2.36) to obtain potential profiles (As in
Fig 3.1). As discussed above, there is added complexity when potentials are measured within the region of charge due to the quadrupole trace term.98 Alternatively, we can avoid these issues by calculating the potential outside of the distribution at the cavity center, x = 0, using
75 3.3. RESULTS AND DISCUSSION
0 0
0
0
0 0
0
0 0 0 Å
Figure 3.1: The electrostatic potential profiles, φ(r), for droplets with cavities centered at r = 0. Displayed are potentials for systems of 1500 SPC/E waters (blue) with a 2.5 Å cavity (dotted), 4.0 Å cavity (line), and 6.0 Å cavity (dashed), a 512 SPC/E waters system (red) with a 4 Å cavity and a 250 SPC/E waters system (green) with a 4 Å cavity. Simulations were performed for 3 ns. Figure S1 in Appendix A.2 shows the charge density profile for the 1500 water system with a 6.0 Å cavity. Notice the net potential for the N = 1500 water droplet case is slightly more positive than for the smaller droplets due to the reduced contribution from the solvent quadrupoles near the more distant liquid-vapor interface.
76 3.3. RESULTS AND DISCUSSION eq (2.38). This allows us to examine the multipole potential components of the net potential, free from extraneous (and cancelling) electrostatic effects. We can also examine the separate contribution from each interface by plotting the multipole potentials at the cavity center in radial bins of molecular position xn. (See Fig. 3.2) The profiles are created by summing the net potential due to water oxygens within 0.5 Å width shells.
Shown in Figure 3.2 are the dipole, quadrupole, and octupole net potential contributions along with the oxygen radial distribution functions (rdfs) for the N = 1500 SPC/E water system with 3 different cavity radii. Note that the first peak of the rdf is largest for the 4 Å cavity size, consistent with Ref. 155 which examines structure near hydrophobic solutes along with solvation thermodynamics. Integration of the potential profiles produces the net potential at the cavity center due to each individual multipole.146 Table 3.1 shows the electrostatic potentials for each interface obtained from these net potential profiles.
In order to relate Figure 3.1 to Figure 3.2, we analyze the data for the 1500 SPC/E water system with cavity radius λ = 6.0 Å. This analysis is summarized in Table 3.2 and extended to cavity radii λ = 2.5 Å and λ = 4.0 Å. Summing the three values of φlp and the two values of φsp in Table 3.1, we find a multipole potential estimate for the local potential contribution
(-0.581 V) and surface potential contribution (0.284 V). As discussed above and shown in Fig.
3.1, the potential profile obtained through multipole densities produces a local potential shift of 0.275 V and a surface potential shift of -0.6 V. These differing values can be reconciled by subtracting the Bethe potential - calculated using eq (2.36) to be 0.885 V - from the local potential and adding it to the surface potential. The resultant local potential of -0.610 V and surface potential of 0.285 V are in good agreement with the data obtained through the multipole estimate. Thus we see that the multipole expansion to quadrupole order is of sufficient accuracy for this relatively large cavity size.
77 3.3. RESULTS AND DISCUSSION
(a ) (b ) (c) 3 .0 2 .5 2 .0 1 .5 g (r) 1 .0 0 .5 0 .0
0 .1
(V) 0 .0 ) ( − 0 .1
φ − 0 .2
0 .1
(V) 0 .0 ) ( − 0 .1
φ − 0 .2
0 .1
(V) 0 .0 ) ( − 0 .1
φ − 0 .2
0 1 0 2 0 0 1 0 2 0 0 1 0 2 0 r (Å) r (Å) r (Å)
Figure 3.2: 1500 SPC/E water system with cavity radii (a) 2.5 Å, (b) 4.0 Å, and (c) 6.0 Å. The upper plots are the radial distribution functions for a neutral cavity particle and water oxygen atoms followed by the φnp radial profiles for the dipole potential, quadrupole potential, and octupole potential.
78 3.3. RESULTS AND DISCUSSION
Table 3.1: Electrostatic Contributions to Net Potential at Different Cavity Radii. Potentials Computed at the Cavity Center with Contributions from Cavity-Liquid and Liquid-Vapor In- terfaces a
potential 2.5 Å 4.0 Å 6.0 Å
D φlp -0.121 -0.243 -0.367
Q φlp -0.150 -0.206 -0.195
O φlp -0.093 -0.049 -0.019
D φsp 0.328 0.329 0.329 . a Q φsp -0.050 -0.050 -0.045
D φnp 0.207 0.086 -0.038
φnp -0.086 -0.219 -0.297
∗ φnp -0.036 -0.169 -0.252 φ -0.209 -0.283 -0.325 Electrostatic potentials for the 1500 SPC/E water system obtained through integration of Figure 3.2. Data reported in V. The dipole (D), quadrupole (Q) and octupole (O) contributions to the net potential at the cavity center from the cavity-liquid (lp) and liquid-vapor (sp) surfaces are in the upper section. The range of integration was chosen to be from the cavity center to about halfway between the surfaces (where charge density is zero) for the cavity-liquid interface and from that halfway point to outside of the droplet surface for the liquid-vapor contribution. In the lower section we combine the upper data to produce the D total dipole φnp, total net potential φnp, total net potential without the liquid-vapor ∗ quadrupole contribution φnp, and total electrostatic potential at the center obtained through a summation of charges φ as in Fig. 3.1. All standard errors were computed (from block averaging) to be less than 0.0014 V in magnitude and thus are not included with each data entry.
79 3.3. RESULTS AND DISCUSSION
Table 3.2: Electrostatic Potential Shifts from the Cavity-Liquid and Liquid-Vapor Interfaces at Different Cavity Radii. a
potential 2.5 Å 4.0 Å 6.0 Å
φ(lp) 0.388 0.322 0.275
φ(lp) − φB -0.497 -0.563 -0.610
φnp(lp) -0.364 -0.498 -0.581
| δlp | 0.133 0.065 0.029
φ(sp) -0.597 -0.605 -0.600
φ(sp) + φB 0.288 0.280 0.285
φnp(sp) 0.278 0.279 0.284
| δsp | 0.010 0.001 0.001 a . Electrostatic potentials for the 1500 SPC/E water system. Values for φ(lp) and φ(sp) are obtained from the potential profiles in Fig. 2, in which (sp) and (lp) values are partitioned at the same distances as Table 3.1. Potentials labeled with np are obtained from a summation of D Q O D Q potentials in Table 3.1, where φnp(lp) = φlp + φlp + φlp and φnp(sp) = φsp + φsp. The extent of agreement of the potentials is shown in | δ |, which is the magnitude of the difference between the two methods of calculation.
The octupole contribution becomes non-negligible for smaller cavities. This is illustrated in
Table 3.2, where we isolate the potential contributions from each interface, and can attribute the increasing disagreement between the multipole estimate and the density profile derived potential shifts at small cavity sizes to the local potential. We note here that the third-order cumulant terms for the separate dipole and quadrupole contributions are of magnitude 7% and 2% relative to the means, respectively. This indicates near-Gaussian behavior for the long-ranged net potential.
Although the potential profiles change with increasing cavity size, the main features are consistent. Small fluctuations within the bulk liquid reflect the structure that is retained within the droplet, but contribute a negligible amount to the net potential. The dipole component of the net potential has a large negative value for the cavity-liquid interface, and a large positive
80 3.3. RESULTS AND DISCUSSION value for the liquid-vapor interface. The cavity-liquid quadrupole term has a large negative value, which varies with cavity size, and the liquid-vapor quadrupole term has a small negative value. The octupole behaves similarly to the quadrupole for both surfaces, though with a smaller magnitude. In addition the octupole contribution from the liquid-vapor interface is negligible because it decays more rapidly than the quadrupole contribution.146
The multipolar net potential contributions from each interface are plotted in Figure 3.3 for all the system and cavity sizes analyzed. The potential values numerically support the visual trends depicted. For all system sizes, the net dipole potential from the cavity-liquid surface stabilizes (after a cavity radius of about 6 Å) and reaches a value that mostly cancels the contribution from the liquid-vapor surface (leaving a total dipole contribution of about -0.1
V). In contrast, the quadrupole potential has a negative contribution from both interfaces (as discussed above) and its magnitude is inversely dependent on the interface’s distance from the cavity center. The cavity-liquid dipole contribution is the slowest to converge with increasing cavity size, presumably due to repulsive dipole-dipole interactions that are strong near the boundary of small cavities (and perhaps other structural, hydrogen-bonding features). In addi- tion, after the local dipole contribution stabilizes around 6 Å, the magnitude of the local dipole contribution likely exceeds that for the distant surface due to the smaller radius of the cavity surface, leading to a slightly higher dipole density. The Laplace pressure within the droplet may play a role also.
Water surface orientations have been previously studied at both interfaces for the SPC/E water model.98,146 The potential shift due to molecular dipole polarization is predominantly positive upon entering the liquid, with a slight negative lobe outside of the large positive peak in Fig. 3.2. These results are consistent with Ref. 98 that discusses the detailed water ordering near the interface. There is a low-density outer layer of waters that tend to order with the
81 3.3. RESULTS AND DISCUSSION hydrogens pointed outward, with a higher density layer of waters with their hydrogens oriented toward the bulk liquid located slightly below the outer layer. Multiple studies have indicated relatively good agreement between ab initio simulations and the SPC/E model for the interface structure.156,157 The average orientation near both the cavity and liquid-vapor surfaces is to some extent model dependent due to the hydrogen donor/acceptor capabilities.98,142,146,154 For the cavity-liquid interface, however, it was found, that (for cavities larger than 6 Å) the struc- tural differences between the SPC/E and TIP5P classical models, and the DFT simulations, diminish substantially (see also Ref. 158).
The dipole potential is directly dependent on orientation as in eq (2.32). Hence an opposite orientation translates into an opposite sign for the potentials due to the two interfaces. In contrast, the quadrupole contribution in eq (2.35) depends on the curvature of the interface and not on the local orientation of molecules.143 For this reason we see the same sign for the quadrupole contribution from both interfaces, and the magnitude decreases uniformly towards
0 as the curvature tends towards planarity. As discussed in the scaling argument (eq (3.1)), we can consider the separate multipole moment potentials φ(0)l,np contained within eq (2.38) and observe directly how they depend on the distance from the cavity center xn. (Note R =| xn |.)
In a much larger N → ∞ system with a distant liquid-vapor surface and a similar-sized cavity, the number of contributing multipole components of the net potential is reduced. The net dipole potential will largely cancel between the two surfaces. The net quadrupole potential then only has a contribution from the local interface, because the quadrupole potential from the distant interface decays as 1/R. (The 1/R dependence is clear from the net potential data in Appendix A.2, Tables S1–S3, by comparing quadrupole contributions from the liquid-vapor interface for the N = 250 and N = 1500 water clusters with a cavity size of 6 Å). This result could have a significant impact on the calculation of net potentials in molecular liquids through
82 3.3. RESULTS AND DISCUSSION
0 .3 0 .2 (a ) 0 .1 0 .0 V − 0 .1 − 0 .2 − 0 .3 − 0 .4
0 .3 0 .2 (b ) 0 .1 0 .0 V − 0 .1 − 0 .2 − 0 .3 − 0 .4
0 .3 0 .2 (c) 0 .1 0 .0 V − 0 .1 − 0 .2 − 0 .3 − 0 .4
2 3 4 5 6 7 8 9 1 0 cavity radius λ (Å)
Figure 3.3: Multipole potential contributions to the Net Potential due to the two interfaces for (a) 250 SPC/E waters, (b) 512 SPC/E waters, and (c) 1500 SPC/E waters for several cavity radii. The Dipole (circle), Quadrupole (square), and Octupole (triangle), Net Potential partitioned by cavity-liquid interface (unfilled) and liquid-vapor interface (filled) contributions. The Liquid-Vapor Octupole contribution is negligible and not shown.
83 3.4. CONCLUSION simulation as it suggests that the bulk value can be extrapolated from a modest-sized system where only potentials within about 5 Å outside of the cavity radius need to be considered. This simplifies the analysis necessary to develop more accurate quantum models of liquid-vapor surface potentials. We note that, if the distant interface is not the liquid-vapor interface, but rather say a liquid/solid interface, then the change in the net potential should occur due only to reorientation of molecular dipoles at that interface (due to interactions with, for example, polar groups or charges on the solid surface).
3.4 Conclusion
The interfacial potential shift between liquid water and vapor has long been an area of research which produces conflicting results between different water models and theory levels.87,146,159
Shown to be an integral part of the calculation of single-ion free energies,76,159–168 an accurate value for the net potential at a cavity center is dependent upon a correct understanding of the physical phenomena that create these potentials. We have examined the definition, molecular origins, and size dependence of effective long-ranged interfacial potentials that can influence ion distributions and chemical interactions near interfaces using the SPC/E water model as a simple probe. We studied finite size water droplets to remove possible artifacts from Ewald summa- tion in periodic boundaries (including non-physical potential shifts due to the mathematical properties of the Ewald potential).111,148,149,169
Due to the requirements that such potentials should be of purely electrostatic origin and independent of ion size and type, it is necessary that the potentials should not involve nearby more chemical, and thus ion specific, interactions of the ions with the surrounding solvent.
A natural physical definition emerges from the exact quasichemical (QCT) formula eq 4 that isolates surface potential effects in the term that involves the free energy for insertion for an
84 3.4. CONCLUSION ion/water cluster into the bulk solvent. By defining a new cavity size appropriate for inserting the ion/water cluster, it can be seen that the QCT hydration free energy for the cluster is close to the cavity net potential defined in this paper.
For cavities of size 4-6 Å or larger, we observe a mild dependence of the net potential on cavity and cluster size. The total electrostatic potential, obtained by direct integration of the charge density, shows oscillatory structure near the cavity-liquid boundary due to molecular ordering there. The liquid-vapor interface displays no such structure due to averaging over
fluctuations of the interface.
An important result of this study is the determination of the length scale at which the total potential stabilizes to a near-constant value in the bulk-like region. This length scale is roughly 5 Å outside of the cavity boundary. These results indicate that simulations should be performed on systems of 500 or more water molecules when attempting to determine absolute hydration free energies.167,168 Otherwise, there could be uncontrolled shifts in the single-ion values.76,159,161–168 Clearly it is important to add the liquid-vapor surface potential value to the results of bulk periodic boundary simulations in order to compare with experimental “real" hydration free energies.160 In addition, the average electrostatic potential at the boundary of the simulation box should be set to zero, leading to another possible shift that could be especially large in periodic boundary condition quantum DFT simulations.
The molecular origins of cavity potentials were then explored through multipole analysis.
It was found that, for cavities of 6 Å or larger, the dipole contributions from the two interfaces largely cancel. This result explains the strong model dependence of previously computed cavity potentials.98,146 This is true even though the molecular dipole moments are relatively consistent across the various models.170 Wide differences in the traceless quadrupole moment values for various models170,171 are thus likely related to the substantially different interfacial potentials.
85 3.4. CONCLUSION
For example, the eigenvalues of the traceless quadrupole matrix for the TIP5P water model show small values, while they are considerably larger for the SPC/E model (and experiment).170
Correspondingly, the computed cavity potential for the TIP5P model is small in magnitude.146
This also helps explain the previous success of the quadrupole model developed by Halliwell and
Nyburg172 and employed by Marcus173 in establishing his single-ion free energy scale. Other continuum-level theories have discussed related electrostatic aspects of solvation.174–176
Two other points are clear from Figs. 3.2 and 3.3. First, the dipole contribution does not decay with droplet size. Rather it approaches a constant at large length scales. On the other hand, the quadrupole contribution does decay roughly as 1/R for large length scales.
− 1 Recently, an n 3 dependence (where n = number of waters) of the droplet surface potential was observed by Houriez et al. 79 after n < 200, which is consistent with out results and can most likely be traced back to the radial dependence of the quadrupolar contribution. The octupole contribution is non-trivial for smaller cavities, but for a cavity of about 6 Å, that contribution is quite small. These results illustrate that the only contribution from a distant liquid-vapor interface in a macroscopic droplet is from nonuniform molecular dipole orientations (and this contribution is largely cancelled by the cavity-liquid dipole contribution). This opens the door to accurate quantum calculations of the effect of alternative surfaces on the water dipole orientations and their impact on the cavity potentials. Our results indicate the inhomogeneous quadrupole density near the cavity-liquid boundary is a major contributor to the effective surface potential experienced by ions near interfaces.
86 Chapter 4
Ethylene Carbonate Liquid-Vapor
Interfacial Potential Shifts
4.1 Preface
Examining the origins of the net potential shift at the water liquid-vapor interface by using it’s multipolar expansion advanced our understanding of the physics of solvation in classical water droplets with the result that the quadrupole was the dominant net potential contribu- tion. This was due to a dipole near cancellation between the cavity-liquid and liquid-vapor interfaces leaving a negative quadrupolar potential137 which is curvature dependent. This in- sight helped to explain why the interfacial potentials of water vary widely with model— as the quadrupole moment is quite different for different models170 171 (but dipole moments are similar). Inspired by this finding, we asked if this picture extends beyond water to other solvent molecules. What insights can be gained from a similar multipolar decomposition? Relating back to section 1.6:Using a complex organic molecule, are there similar general physical prop- erties of the potential shift which emerge? How do water and organic solvents differ in their
87 4.1. PREFACE ion solvation properties? To this end, we simulate droplets of liquid ethylene carbonate (EC) using the recently parametrized OPLS-AA model of You et al177 which shows promising results for dielectric constants, relaxation times, and mobilities.110 Although this model will lack the quantum accuracy we know is necessary for accurate thermodynamic solvation quantities, a simple qualitative picture of multipolar decomposition can be valuable.
Organic Liquids are often used as the solvent in Lithium-Ion (Li-ion) batteries. Li-ion batteries are one of the most widely used energy storage devices today, making up close to 90 percent of those in use.178 This is largely due to the fact that the Li-ion battery cost per cycle is very low178 179 and the continued efforts to safely increase energy densities. Developed in the early 1980s, they consist of an anode, cathode, organic solvent and a separator.180 181 Identifying solvent candidates is one wide area of research where both safety and storage capabilities can be improved upon. Solvent stability is often a cause for battery failure and the solvent structure in bulk and near an interface places limits on ion transport and resultant battery performance.182 180 181 Past efforts have focused mainly on cathode improvements.182
Ethylene carbonate (EC), propylene carbonate (PC), and glycerol carbonate (GC) and var- ious mixtures are commonly used because of their low cost, acceptable temperature window, ability to dissolve ions, and unfavorable reactivity with other battery components. Also, these solvents have broader scientific interest because of the vast differences in liquid properties that arise from very small changes in chemical structure.183 35 34 184 For example, the viscosity of GC is 50 times that of EC or PC with the simple addition of single OH group. For these rea- sons they have become the target of many experimental studies183 35 34 184 36 185 which revealed specific ion effects. There is growing attention to the study of the thermodynamics of ion solvation beyond water, in organic solvents. Both classical14 186 187 188 189 190 191 192 177 193 194 and quantum195 196 14 197 198 199 200 201 simulation studies have offered insight into the observed exper-
88 4.1. PREFACE imental results. Some studies have focused on geometry and coordination numbers193 35 183 202 while others aim to calculate thermodynamic solvation properties.201 193 55 These studies have yet to produce uniformly consistent agreement with experiment.
This is in part due to the shortcomings of classical force field models as shown in our lab’s
first study of EC and PC193 using the AMBER (GAFF)203 model in an attempt to repro- duce experimental thermodynamic data produced in Refs. 183 and 35. The model failed to accurately produce ion-solvent pair interactions found from both experiment and from quan- tum chemical calculations. After a corrected pair interaction was employed, the free energy values had improved agreement, but not the enthalpies and entropies. It was proposed, and later shown in our subsequent quantum study,196 that this discrepancy can be traced to the solvent-solvent interaction (mainly polarization) as the enthalpies and entropies contain a sol- vent reorganization term. Ab initio DFT simulations of Li+ in EC and PC showed the great importance of polarization in the first solvation shell.196
These studies highlighted the distinction between ion solvation in organic solvents and in water. Water is an ideal solvent to compare and contrast with because of the extensive existing studies. One such distinction is the cation-anion assymetric solvation, where it is expected, and later shown that cations are more strongly solvated in EC and PC due to the harsh nature of the carbonyl oxygens and their lack of hydrogen bonding capabilities.193 196 Water, on the other hand, more strongly solvates the anion204 due to the strong hydrogen bonding interaction.159
This chapter contains my analysis of the potentials felt by ions in an organic liquid solvent, ethylene carbonate, and their origins. Comparison to my water study is made throughout, with an identical multipolar decomposition and an additional analysis of the angular orientations of molecules at the interfaces. It is found that there are both solvent-dependent and general properties. Universally, the dipolar contributions to the potential at the cavity center are
89 4.2. COMPUTATIONAL METHODS constant with interfacial distance R, while higher order contributions go to zero in increasing
1 powers of R . Also, there is a bulk length scale at which the charge density goes to zero, hence the potential within the bulk liquid flattens. Solvent specific behavior is apparent from the differing multipole contributions to the net potential at the cavity center in SPC/E water and
EC.
4.2 Computational Methods
Molecular Dynamics Simulations were done using the Gromacs 4.5.5 simulation package.205
Finite boundary conditions (non-periodic) were used for a liquid droplet of 1000 EC molecules.
OPLS-AA110 force field parameters were used following You et al. 177. A half-harmonic bounding potential wall was placed approximately 10 Å from the droplet wall to return evaporating molecules. The cavity growth potential is the WCA potential153 described in Ref. 137 and acts between a point-like particle constrained at the cavity center and EC molecules. For the case of charged species, we use a charge of q = +1.0e or −1.0e. After an initial energy minimization, a cavity is grown to a radius Rc and then equilibrated for 1 ns in the NVT ensemble, using a
Nose-Hoover thermostat with a temperature coupling constant of 2.5 ps and a temperature of
340 K. This temperature was chosen to allow for liquid diffusion, while minimizing the number of evaporating molecules, because the melting point of EC is 309.6 K and the boiling point is
521.2 K.206 Configurations were sampled every 1 ps with a timestep of 1 fs. A long production run of 6 ns was chosen due to the large dielectric relaxation time of EC.177
90 4.3. RESULTS AND DISCUSSION
4.3 Results and Discussion
4.3.1 Multipole moments of EC and Water
Multipole moments describe the charge distribution of a molecule. The first non-vanishing moment of a neutral molecule, the dipole, quantifies the separation of positive and negative charge. The SPC/E model dipole is 2.35 D,170 overestimating the experimental gas phase value of 1.86 D.207 The EC model dipole used here is 5.47 D177 , overestimating the experimental gas phase value of 4.81 D.183
The second non-vanishing moment, the quadrupole, is dependent on the choice of origin.
For EC (see Figure 4.4), we define this origin as the center of mass, and for SPC/E water the origin is the oxygen, as is commonly defined. While these definitions are different, they are consistent with past studies200 98 143 170 100 146and varying the origin position by a small amount only effects the individual multipolar potential magnitudes slightly, while their sum is invari- ant208 138 and general behavior unaffected. Both molecules are oriented with the dipole vector along the positive z-axis and with the planar atoms in the yz-plane. The water molecule is oriented with the angle bisector along the positive z-axis with OH bonds in the yz plane. EC is similarly oriented with the carbonyl axis, and dipole, along the positive z-axis and carbon ring roughly in the yz plane. For the rigid SPC/E model, the orientation described here is exact, leading to the C2V point group symmetry classification140 (invariant with 180◦ rotation and two reflection axes). The consensus of EC ab initio studies show a C2 symmetry (invariant with 180◦ rotation) due to the non-planarity of the carbon ring198,209–211 which prevents the reflection axis symmetry.
Comparing the molecules, the dipole of EC is over twice as large as that of SPC/E water.
However, the quadrupole moments are quite similar. For all molecules (even those without
91 4.3. RESULTS AND DISCUSSION
Table 4.1: Quadrupole Components of ethylene carbonate (EC) and water (SPC/E) a
heightMolecule Traceless Quadrupole Moment Eigenvalues
Qxx Qyy Qzz Qxy Qxz Qyz λ1 λ2 λ3 hQi . EC 2.64 -2.55 -0.09 -0.98 0.00 0.00 2.82 -2.73 -0.09 3.204
SPC/E -2.04 2.04 0.00 0.00 0.00 0.00 -2.04 2.04 0.00 2.348 a Isolated gas phase quadrupole moment components for molecules oriented as described above. 2 1 2 212 213 Quantities in DÅ. Magnitude hQi = ( 3 Q : Q) is invariant with rotation and has been used in bulk measurements200
symmetry) the quadrupole moment is symmetric (Qαβ = Qβα) and there are 5 independent components (due to trace condition, Qxx + Qyy + Qzz = 0). Molecules with C2V symmetry, like water, have the unique properties of being diagonal (Qxy = Qxz = Qyz = 0), with two independent components.140 EC, with C2 symmetry, is nearly diagonal with three independent components due to one off-diagonal term Qxy arising from the slight non-planarity of the EC molecules. Table 4.1 shows the quadrupole moments of these molecules, oriented as described above. The main differences between the two molecules are the nonzero Qxy of EC and that the signs of Qxx and Qyy are opposite for EC and SPC/E, though similar in magnitude.
4.3.2 Electrostatic Potential Analysis: Neutral Cavity
To investigate the behavior of the interfacial electrostatic potential of non-aqueous organic solvent droplets, we follow our previous work in Ref. 137. Figure 4.1 shows the electrostatic potential profiles φ(r) of droplets for a range of cavity sizes with the cavity centered at r = 0.
These profiles were obtained through integration of the charge density profiles, obtained from the radial distribution functions, using equation 2.29. Starting from the vapor phase, the surface potential is seen in the potential rise near approximately r = 32 Å and the local potential is seen in the potential drop around cavity sizes λ = 2.0, 6.0, 10.0Å. For all cavity sizes, the EC surface potential is relatively constant (like water), however, it is both small
92 4.3. RESULTS AND DISCUSSION
0
0
0
0 0
0
0
0
0 0 0 0 Å
Figure 4.1: The electrostatic potential profiles, φ(r), for droplets of 1000 EC molecules. The profiles show data from droplets centered at r = 0 with cavity sizes 2.0 Å, 6.0 Å, 10.0Å and positive. Classical water has a well known large and negative surface potential,98,137,145,214 which is the first distinction between these two molecular liquids. Previous results137 found an SPC/E surface potential of -0.6 V for similar droplets. The local potential for EC is larger and negative, decreasing in magnitude for larger cavity sizes. The characteristic oscillations near the cavity reflect the charge asymmetry (dipole ordering) near the cavity-liquid interface.
These oscillations persist to a small degree deep into the liquid. The high degree of ordering within liquid EC has been noted in other studies of ion solvation.183,193
As discussed in previous work,137 the origins of this potential can be further understood by breaking down the potential into it’s multipole expansion components. In Fig 4.2 we plot the ensemble averaged dipole, quadrupole, and octupole contribution to the net potential at cavity center, φnp, in 0.5 Å width shells, using equation 2.39. This allows us to probe how the molecules located at each interface affect the potential at the cavity center. Here, we see the
93 4.3. RESULTS AND DISCUSSION strong dipolar ordering near the cavity-liquid interface. As the cavity size increases, ordering in neighboring shells penetrates deeper into the bulk. Near the liquid-vapor interface we see a small negative contribution to the net potential. This dipole behavior is similar to that of water, except that EC lacks the larger positive lobe near the liquid-vapor interface which produces the dipolar cancellation.137 We will discuss below how molecular orientations confirm this interpretation. The quadrupole and octupole contributions behave similarly to each other.
The magnitude of their net potential contributions from the cavity-liquid interface are small, and decrease with cavity size. The quadrupole and octupole net potential contribution at the liquid-vapor interface are near zero due to the radial dependence of these terms.137 The quadrupole contribution to net potential is positive nearest to the cavity and slightly negative adjacent to that— a behavior different than the totally negative contributions in water.
94 4.3. RESULTS AND DISCUSSION
(a ) (b ) (c) 3 .0 2 .5 2 .0 1 .5 g (r) 1 .0 0 .5 0 .0 0 .4 0 .2 (V)
) 0 .0 ( − 0 .2
φ − 0 .4 − 0 .6
0 .1
(V) 0 .0 ) ( − 0 .1
φ − 0 .2
0 .1
(V) 0 .0 ) ( − 0 .1
φ − 0 .2
0 5 1 0 1 5 2 0 2 5 3 0 3 5 0 5 1 0 1 5 2 0 2 5 3 0 3 5 0 5 1 0 1 5 2 0 2 5 3 0 3 5 r (Å) r (Å) r (Å)
Figure 4.2: 1000 EC molecule system with cavity radii (a) 2.0 Å(b) 6.0 Å(c) 10.0 Å. The upper plot shows the radial distribution function for a neutral cavity followed by the dipole, quadrupole, and octuple net potential profiles
Integration of these profiles along r produces an estimation of the net potential at the cavity center for each multipole contribution. By partitioning this integration from cavity center to bulk and then from bulk to vapor allows us to estimate these contributions due to the cavity- liquid (local) and liquid-vapor (surface) interface, respectively. These values are shown in Table
3.1 and displayed in Figure 3.3 for a range of cavity sizes. For mesoscopic cavity sizes λ > 4 Å it
95 4.3. RESULTS AND DISCUSSION is notable that only the dipole contribution from both interfaces contributes meaningfully to the net potential. As cavity size increases, the liquid-vapor interface dipole contribution dominates.
This occurs because the local dipole contribution to the net potential from molecules closest to the cavity is large and negative for small cavities (Fig 4.2 (a)), but decreases in magnitude as cavity size increases. The dipole ordering becomes more uniform between neighboring shells which leads to an increasing cancellation between them. The liquid-vapor dipole contribution is mainly constant for different cavity sizes.
0 .1
0 .0
− 0 .1 V − 0 .2
− 0 .3
− 0 .4
2 3 4 5 6 7 8 9 1 0
Figure 4.3: Net potential dependence on cavity size for the contribution due to the local dipole (unfilled circle), local quadrupole (unfilled square), local octupole (unfilled triangle), far dipole (filled circle), and far quadrupole (filled square)
This picture is different from that of water in a few key ways. First, the water cavity-liquid
96 4.3. RESULTS AND DISCUSSION dipolar contribution to the net potential for small cavities is smaller in magnitude and increases to a larger negative value for mesoscopic cavity sizes which largely cancels with positive liquid- vapor dipolar contribution137,146— leading to a near zero dipole net potential. For EC, both dipolar net potential contributions are negative— the cavity-liquid contribution decreases in magnitude with increasing cavity size, while the liquid-vapor contribution is small and constant.
Hence, there is no dipolar cancellation for EC. Second, the water quadrupole is small and negative for the cavity-liquid interface, decreasing in magnitude with cavity size and small and negative for the liquid-vapor interface. The quadrupole for EC follows a similar cavity size dependence, but is positive for both interfaces. Possible reasons for this difference will be discussed below.
4.3.3 Electrostatic Potential and Molecular Orientation
We extend this analysis by also examining the spatial orientations of molecules near the inter- faces. Molecular orientations within the droplet can be defined by two angles. One characterises the dipole orientation, and the other characterises the planar orientation. Previous studies of water molecules have defined these angles to be independent of each other, so that the bivariate distributions define unique orientations.215 216 158 146 Here we define angles θ and ϕ of the vector
~xn from EC center-of-mass to the point particle in the local frame shown in Fig. 4.4. In this frame ~n is normal to the O1-C3-O4 plane, ~rO1O4 is the vector from O1 oxygen to O4 oxygen, and ~b is the bisector of O1-C3-O4 angle. The first angle, ϕ, is the angle between ~n and the
~ projection of ~xn onto the ~rO1O4 ~n plane. The dipole vector, ~p, lies directly on the bisector b in the rigid case. The dipole vector may be slightly misaligned with this vector because the EC model that we use here is not rigid. For this reason we diverge from previous analysis158 which guarantees unique orientations using ~b, by defining θ as the angle between dipole vector ~p and
97 4.3. RESULTS AND DISCUSSION
Table 4.2: Electrostatic Contributions to Net Potential at Different Cavity Radii. Potentials Computed at the Cavity Center with Contributions from Cavity-Liquid and Liquid-Vapor In- terfaces a
potential 2.0 Å 6.0 Å 8.0 Å 10.0 Å
D φlp -0.449 -0.113 -0.066 -0.022 Q φlp 0.102 0.012 0.004 0.002 O φlp 0.093 0.012 0.005 0.004 D φsp -0.160 -0.145 -0.141 -0.160 Q φsp 0.002 0.002 0.002 0.002
φnp -0.412 -0.233 -0.196 -0.176
φ(lp) -0.352 -0.265 -0.241 -0.218
φ(sp) 0.036 -0.042 -0.050 0.046 φ -0.316 -0.223 -0.191 -0.172 . a Electrostatic potentials for the 1000 EC molecules droplet obtained from two methods. The upper table shows data obtained through the first method- integration of Figure 4.2. Data reported in V. The dipole (D), quadrupole (Q) and octupole (O) contributions to the net potential at the cavity center from the cavity-liquid (lp) and liquid-vapor (sp) surfaces are in the upper section. Net potential φnp is a summation of preceding data D Q O D Q φnp = φlp + φlp + φlp + φsp + φsp. The lower table shows data obtained through integration of Figure 4.1. The range of integration to determine cavity-liquid and liquid-vapor boundaries was chosen to be the point at which the charge density is close to zero, which is about halfway between the two surfaces. Electrostatic potential φ is obtained through a simple summation of molecular contributions.
98 4.3. RESULTS AND DISCUSSION
~xn. In this way, we can analyze orientations with the actual dipole vector and it’s dipolar angle
θ, along with a nearly unique measure of the degree to which the carbon plane lies parallel to
(0 < θ < 45) or perpendicular to (45 < θ < 90) the interface.
Figure 4.4: Angles θ and ϕ in a local molecule-defined frame used to define the orientation of
EC molecule. ~n is normal to the O1-C3-O4 plane, ~rO1O4 is the vector from O1 oxygen to O4 oxygen, and ~b is the bisector of O1-C3-O4 angle. These vectors form a local ~x~y~z axis. For a non-rigid model, ~p can be slightly misaligned with ~b in the liquid phase.
Figure 4.5 shows the probability distribution of the projection of ~p onto ~xn for four different radial shells in the 1000 EC droplet with a cavity size of 6 Å. The first shell consists of molecules with their center of mass between 6 Å and 7 Å, the second between 7 Å and 8 Å, third between
8 Å and 9 Å, and the last between 26 Å and 29Å. These shells were chosen after inspection of the full joint probability distributions of the two angles P (θ, φ) which are shown the the
Appendix A.3 Figure S3 and because of the insight they provide into Fig. 4.2. However, the most probable orientations don’t necessarily dictate the sign of the potential contributions.
Rather, the sum of contributions from individual atoms produces the value of the potential
99 4.3. RESULTS AND DISCUSSION from each shell.
The first distinction in the previous section can be explained using the dipolar angle θ. The
first and second shells of the cavity-liquid interface show a preponderance of molecules with the dipole vector pointing slightly away from cavity center (carbonyl oxygens point towards cavity center), in agreement with Arslanargin et al. 193 for an uncharged solute. The third shell shows a preference for the dipole pointing towards the cavity center, consistent with the positive dipolar potential contribution in Fig 4.2 for molecules located in 8 < r < 9, and the distribution becomes increasingly uniform in the bulk (not shown). Again, these orientations support the alternating dipole orientation in the shells nearest to the cavity interface. SPC/E water does not have this strong alternating behavior, perhaps because of the smaller size and dipole moment magnitude of water.
In the last shells of the liquid-vapor interface, 26 < r < 29 Å, the most probable orientation has the dipole vector perpendicular to ~r, meaning the carbonyl axis lies flat along the surface.
The larger tail on the left side of this distribution shows the very slight majority of molecular dipoles pointing away from the cavity center. This confirms the small negative potential seen in Fig 4.2. In contrast, the water liquid-vapor interface has a large density of dipoles pointing towards the cavity center with an outer lower density vapor shell of dipoles pointing away.
Because EC lacks this larger density shell of molecules pointing towards the cavity center (and opposite to the orientation at the cavity), the dipolar net potential contribution does not cancel between the two interfaces.
100 4.3. RESULTS AND DISCUSSION
Figure 4.5: Probability distribution of the projection, uθ, of the molecular dipole vector ~p onto
~xn for 4 shells in the 1000 EC droplet with 6.0 Å cavity. Projection determined by uθ = |p|cos(θ). The plus, diamond, and cross, shows the first, second, and third shells closest to the cavity (black), respectively. The dot shows the shell near the liquid-vapor interface (red).
The behavior of molecules at the interfaces is further clarified by Figure 4.6. The liquid- vapor interface does not have any preferred planar orientation angle as indicated by the uniform distribution for 26 < r < 29 Å. While the dipole is mostly flat along this interface, the molecular plane has no parallel or perpendicular bias. Joint probability distributions in Appendix A.3
Figure S3 show a slight preference for flat orientations in the lowest density shells, r > 31, but this does not result in a nonzero quadrupole net potential, likely because of it’s r dependence.
This differs from propylene carbonate, as seen in the study of You et al. 188, in which a flat planar orientation is strongly preferred in droplets due to the methyl group protruding from
101 4.3. RESULTS AND DISCUSSION the liquid surface. Because EC lacks this methyl group, there is no preferred planar ordering.
The methyl group disrupting the bonding behavior at the surface in PC but not in EC could explain the experimental observations of higher surface tension in EC.184
At the cavity-liquid interface, the carbon plane is more frequently flat, or parallel to the cavity nearest to the point particle. The neighboring two shells have a slight bias for the opposite orientation- in which the carbon ring is perpendicular to the droplet surface. As above, all other shells display a strongly uniform distribution. From this difference we begin to understand the second distinction, made in the previous section, between water and EC. In the
first shell, 6 < r < 7, the quadrupole potential contribution is positive and in the neighboring shell it is negative. From this we posit that the sign of the quadrupole potential contribution can be determined from it’s planar angle orientation and it’s quadrupole moment. For EC, a flat orientation leads to a positive value, and a perpenicular planar orienation leads to a negative value for the quadrupole net potential contribution.
102 4.3. RESULTS AND DISCUSSION
Figure 4.6: Probability distribution of the planar angle, ϕ for 4 shells in the 1000 EC droplet with 6.0 Å cavity. Values of ϕ closest to 0 (or 180) are molecules with the carbon plane mostly flat and ϕ close to 90 are mostly perpendicular. The plus, diamond, and cross shows the first, second, and third shells closest to the cavity (black), respectively. The dot shows the last shell near the liquid-vapor interface (red).
4.3.4 Electrostatic Potential Analysis: Charged Cavities
Next we turn to the behavior of EC molecules with a charged point particle at the cavity center.
Given the results above, if a positive point particle is placed at the cavity center, we would expect the orientation of the cavity-liquid interfacial dipoles pointing away from the cavity center to become more strongly ordered and the value of the dipole net potential contribution to be more negative. In the case of a negatively charged particle, the orientation would reverse— the cavity-liquid dipole net potential contribution would become positive. Figure 4.7 confirms
103 4.3. RESULTS AND DISCUSSION this. In the first shell around the cavity, the magnitude of the contribution to the net dipole potential compared to the neutral case increases by nearly three times for the positive ion and around two times for the negative ion. The total electrostatic potential magnitude in
Figure 4.7(b) increases by about 15 times from a positive ion and 10 times for a negative ion.
This later increase can be attributed to the electrostatic charge effect nearly eliminating the alternating picture that occurs in the neutral case, which cancels much of the cavity-liquid dipole contribution. Table 4.3 shows this drastic dipole net potential magnitude increase. The larger net potential in a positive cavity points to stronger cation solvation compared to anion, which has been shown elsewhere.217 This is also in agreement Arslanargin et al. 193 where the effect of cavity potentials on anion and cation solvation was explored with the result that the cation is more strongly solvated due to the intense electrostatic interactions of the carbonyl end compared to the alkyl end. This is evidenced by larger fluctuations of the electrostatic potential for the cation.
Table 4.3: Electrostatic Contributions to Net Potential for Charged Cavities. a
potential +1.0e -1.0e
D φnp -2.531 1.912
Q φnp -0.038 0.062
O φnp 0.017 0.010
φnp -2.552 1.984
φ -2.528 1.989 . a Electrostatic potentials for the 1000 EC molecules droplet with a 6.0 Å cavity with a charged point particle located at the center. Data obtained from two methods. The upper table shows data obtained through the first method— integration of Figure 4.2. Data reported in V. The dipole (D), quadrupole (Q) and octupole (O) contributions to the net potential at the cavity center. Total net potential φnp obtained from a summation of previous values. Electrostatic potential φ is obtained through a simple summation of molecular contributions.
Notably, the quadrupole net potential contribution to cavity center is mostly positive for an
104 4.3. RESULTS AND DISCUSSION uncharged cavity, becomes more positive for an anion, and is negative for a cation as seen in
Fig 4.3(b). The octupole contribution is largely unchanged and both octupole and quadrupole are very small in magnitude—neither having much of an effect on the net potential. Molecular orientations shown in Figure 4.8 and 4.9 elucidate this change in quadrupolar sign. Close inspection of the planar orientations in the shells where this sign difference is apparent in Fig
4.7 is consistent with what is observed for the neutral case. For both the neutral and anion system, orientations of EC near the cavity are favored to have the carbon plane lying flat on the interface, with increased probability for the anion. In that region, the quadrupole net potential contribution is positive for both systems. For the cation system, orientations where the carbon plane has a slight bias for being perpendicular to the cavity surface show a quadrupole net potential contribution that is negative.
Considering these observations in conjunction with the calculated quadrupole moment of EC and SPC/E molecules in Table 4.1, we begin to understand the sign difference for quadrupole net potential contributions. In the cases where the EC molecular plane is mostly flat on the interface, the out-of-plane Qxx term dominates, leading to a positive net potential contribu- tion. Regions where the molecules lies mostly perpendicular to the surface produce a negative quadrupole net potential contribution as the in-plane Qyy term dominates. Furthermore, in
SPC/E water, molecules prefer a largely flat H-O-H plane along cavity surface (with dipole pointing slightly away). Unlike EC, the out-of-plane Qxx term in SPC/E water is negative.
At the cavity-liquid interface this out-of-plane Qxx term dominates resulting in a negative quadrupole potential.
105 4.3. RESULTS AND DISCUSSION
0 - 0 0 0 0 0 0 0 0
0 0 0 0 ϕ 0 0 0 0 0 0 0 0 0 0 0 0
0 00 0 0 0 ϕ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 00 0 0 ϕ 0 0 0 0 0 0 0 0 0 0 0 0 0
Figure 4.7: 1000 EC molecule system with cavity radii 6.0 Å and a point particle with a neutral (solid black), positive (dashed red), or negative (dashed blue) point particle. (a) The upper plot show the radial distribution function of particle-EC molecules followed by the dipole, quadrupole, and octupole net potential profiles Φnp. (b) The electrostatic potential profiles, φ(r). The profiles show data from droplets centered at r = 0.
106 4.3. RESULTS AND DISCUSSION
Figure 4.8: Probability distribution of the planar angle, ϕ for 4 shells in the 1000 EC droplet with 6.0 Å cavity. A positive point particle with charge +1.0e is placed at the cavity center. Values of ϕ closest to 0 (or 180) are molecules with the carbon plane mostly flat and ϕ close to 90 are mostly perpendicular. The plus, diamond, and cross shows the first, second, and third shells closest to the cavity (black), respectively. The dot shows the last shell near the liquid-vapor interface (red).
107 4.4. CONCLUSION
Figure 4.9: Probability distribution of the planar angle, ϕ for 4 shells in the 1000 EC droplet with 6.0 Å cavity. A negative point particle with charge -1.0e is placed at the cavity center. Values of ϕ closest to 0 (or 180) are molecules with the carbon plane mostly flat and ϕ close to 90 are mostly perpendicular. The plus, diamond and cross shows the first and second shells closest to the cavity (black), respectively. The dot shows the last shell near the liquid-vapor interface (red).
4.4 Conclusion
In this chapter I presented a detailed study of the origin of the interfacial potential shift going from vapor phase in through the liquid ethylene carbonate (EC) and then to a cavity using a multipolar analysis and an examination of orientations at the interface. This study was compared to a congruent analysis of SPC/E water. Specifically, we examined the definition, molecular origin, and size dependence of the net potential at the cavity center. While simple
108 4.4. CONCLUSION classical models cannot accurately model complex interactions of quantum polarization and thermodynamics, they can be used as a simple probe into solvent behavior.
Our previous study of liquid SPC/E water droplets provided insights into the strong model dependence146 98 of cavity potentials and the length scale at which the total electrostatic poten- tial stabilizes in the bulk. In addition, we explained how the dipole net potential contribution does not decay with interfacial distance r, while the quadrupole and higher order moments
1 decay as powers of r .
In EC, we observe that a large droplet of 1000 molecules is necessary for a stabilized electro- static potential in the bulk. As expected, the dipolar net potential contributions do not decay with interfacial distance. Instead, the distant liquid-vapor dipole net potential contribution is small due to the predominately perpendicular dipole orientations at the droplet surface. Con- tributions from the cavity-liquid interface decrease with cavity size as the alternating dipole ordering of EC progresses in the liquid. Higher order contributions such as the quadrupole presumably do decay with distance and are very small for cavities larger than a few Å, with negligible contributions from the liquid-vapor surface. Hence, the r dependant behavior is difficult to definitively determine here.
It is clear from this analysis that the dipole net potential contribution— a constant negative value from the liquid-vapor interface and a negative size-dependant value from the cavity-liquid interface— are the dominant multipolar contributions to the net potential. This result agrees with Barnes et al. 200, which found that EC interaction energies are dominated by dipolar effects.
For cavities larger than 4-5 Å they are the only non-negligible components. This is significantly different to what we observed with SPC/E water, where orientations at the liquid-vapor inter- face are very similar to those at the cavity-liquid interface, leading to a dipolar cancellation.
EC, on the other hand, has quite different orientational behavior at these two interfaces. Con-
109 4.4. CONCLUSION sequently, in a large macroscopic EC droplet the only net potential contribution will be from a constant, small, and negative distant liquid-vapor dipole, a negative size-dependent cavity- liquid dipole and smaller positive cavity-liquid quadrupole.
In the context of anion and cation solvation, the expected result of stronger cation elec- trostatic interaction was confirmed. Previous analysis193 217 196 attributed this to the favorable cation interaction with the harsh carbonyl end of EC. Here, we can trace this phenomena to the strong EC dipole polarization and subsequent dipole net potential with a cation placed at the center of a cavity.
Finally, we connected the planar orientation of each molecule with the quadrupole net poten- tial contribution at the cavity center. We found that the quadrupole net potential is dependent on surface curvature, the detailed orientation of molecules at the interface, and the quadrupole moment of the molecule. Water and EC, oriented the same way, have similar quadrupole mo- ments except for the opposite sign of the square Qxx and Qyy terms. Furthermore, in the first solvation shell, both molecules are largely flat on the cavity surface. Because of the opposite sign of the quadrupole moment terms, the net quadrupole potential takes on opposite signs in these shells. Detailed analysis of the molecular orientations in charged and uncharged EC cavities shows how the quadrupole potential sign varies depending on whether the molecule lies more planar or perpendicular to the cavity surface.
In closing, the physical effects that establish interfacial potentials have universal properties as well as solvent dependant ones. Accurate calculation of physical properties is contingent upon modeling these effects correctly. Concretely, the ratio of dipole to quadrupole moment determines the ability of classical force field models to produce an accurate phase diagram of water.218 219 The delicate interaction of dipoles, quadrupoles, and higher order terms are an element of determining molecular behavior (multipole densities, orientations, etc.) within a
110 4.4. CONCLUSION bulk and at interfaces.143 This behavior creates the charge asymmetry that produces the net potential relevant to ion solvation. In addition to multipolar interactions, molecular geome- try, packing, dispersion, polarisation, and hydrogen-bonding effect the molecular arrangement around a cavity and the resulting interfacial potentials. The high degree of hydrogen-bonding in water creates a network around the cavity which is maintained as the cavity grows, leading to the relatively stable local potential.154 Ethylene carbonate and related organic liquids lack this strong network, which perhaps leads to ordering according to electrostatic forces near the cavity and deeper into the liquid that results in alternating dipoles and a near cancelling local dipolar potential. The small, near-spherical size and charge density of water compared to the larger elongated shape of EC may also be a factor in this observed behavior. In general, there will be a contribution from each molecular multipole at each interface, and a length scale at which these contributions go to zero in the bulk. For a given molecular orientation, distant dipole contributions will be constant as the distance to the liquid-vapor interface grows, while higher order terms will vanish as they are dependent on r. Cavity-liquid interface dipole net potential contributions are dependent on cavity radii. The values of the multipole potential contributions, however, are solvent dependent, in that they are generated from the detailed orientation of molecules at the interfaces and their intrinsic molecular moments.
111 Chapter 5
Electrostatic Properties from Neural
Network Potentials
5.1 Preface
In the final chapter I present an initial study directed at quantum-designed models for molecular liquids. My previous studies proceeded with the knowledge of the limitations of classical MD.
Understanding the origin of the electrostatic potential felt by an ion in simple model liquids was enhanced, general behaviors as well as solvent specific ones were explored. In the current chapter the focus is instead on novel ways of achieving quantum-level accuracy simulations at the time scales and length scales only possible with less-accurate classical MD. This ambitious goal would make predictive studies of molecular liquids possible and resolve many outstanding questions in solvation science such as phase equilibrium in mixtures, ion channel biophysics, and specific ion effects.
One developing area that has shown promise in resolving the cost-accuracy dilemma is that of neural network potentials (NNPs). This category of machine-learning techniques attempts
112 5.1. PREFACE to reproduce the interatomic potential energy surface (PES) of a system as discussed in section
2.4. This high-dimensional many-body function is approximated by a force field in classical
MD, but in NNPs it is replaced by a complex neural network. Inspired by the recent successes of the DeePMD NNP of Wang and coworkers65 we attempt to validate the ability of neural networks to produce accurate MD simulations of liquid-vapor interfaces and determine some electrostatic properties thereof. To our knowledge, this is the first study of the electrostatic properties of liquid interfaces using NNPs, as well as multipole moments higher than dipolar order (Thermodynamic properties of solid-liquid interfaces were studied in Ref. 220 and the electrostatics of dipole order in bulk liquids were studied in Ref. 65). Verifying the accurate modeling of these properties could open the door to a wide range of new simulation methods.
First, I will explain the mechanics of the open source simulation package DeepMD-kit of
Wang et al. along with our revisions of existing networks in the package, in order to produce a neural network that outputs electrostatic properties. AIMD reference data is produced for a short simulation of a bulk (neat) water system and a similar interfacial system with a 4 Å cavity constrained to the center. After ensuring that the DeePMD network for the separate systems is adequately trained, we train a combined model which incorporates input data from both systems. That way, both the waters near the cavity center and those farther away are accurately modeled. Finally, we train our newly developed NN model using DeePMD-kit for the dipole, traceless quadrupole, and primitive quadrupole moments using CP2K data. When the new NNs are fed a long-time scale DeepMD simulation, these networks produce long-time scale, quantum accuracy electrostatic properties for larger systems.
113 5.2. COMPUTATIONAL METHODS
5.2 Computational Methods
In the following sections we discuss the methodology used to determine the dipole moment and quadrupole moment (primitive and traceless) in bulk and interfacial liquid water of quantum accuracy using DeePMD-kit,63 the basic theory of which was described in section 2.4. Inspired by the recent successes in the prediction of Wannier Centers (WC)65 of Charge we combine
DeepMD-kit generated trajectories and a modified version of the DNN referred to as Deep
Wannier (DW)65 to determine molecular multipole moments.
Simulation details for the AIMD training set for both the dynamic properties and the calculation of the multipole moments is presented first. This is followed by a description of the DeePMD network used for the reconstruction of the PES, as well as an explanation of the interface with available MD software (LAMMPS) which uses this network for long-time simulations. Then, I describe the DNNs used to determine the multipole moments and how we used them with the newly generated simulations to determine the multipole moments.
5.2.1 Ab Initio Simulation
Generating the training data is done using CP2K 2.6.1 Quickstep module AIMD software.118
This corresponds to the leftmost boxes in Fig. 5.1 for a bulk simulation and a cavity (interfacial) simulation. A neat water system of 64 water molecules was used for the bulk system with a box size of L = 12.4295 Å in PBC. For the interfacial system, 64 water molecules are placed in a L
= 13.0236 Å box. A cavity of 4.1 Å was created using a harmonic potential that interacts with a classical particle at the center of the simulation cell and the oxygen atoms of water molecules, given by,
V (r) = k(r − 4.305)2 (5.1)
114 5.2. COMPUTATIONAL METHODS
Figure 5.1: Schematic of the workflow for generating molecular multipole moments from DeePMD generated trajectories. DeePMD network is unchanged from its original form in Reference Zhang et al. 59. Networks DeePD, DeePM, DeePPM are trained using the framework of DeePMD but with slightly altered code in order to train for the dipole moment, traceless quadrupole moment, and primitive quadrupole moment respectively.
115 5.2. COMPUTATIONAL METHODS with k = 40 kcal/mol/Å2
The DFT simulations were performed with the Gaussian-type orbital basis sets DZVP-
MOLOPT-SR-GTH and the Goedecker-Teter-Hutter (GTH) pseudopotentials. The functional used is the revised Perdew, Burke, Ernzerhof (revPBE) with the dispersion correction Grimme
D3 with a plane wave cutoff of 400 Ry. A temperature of 330K in the NVT ensemble was maintained with the Nose-Hoover thermostat chain of length 3. Electrostatics are treated with the Ewald method. The trajectory consists of a 0.5 fs timestep with a total length of 30 ps.
Some frames are used for NN training, while others are used for testing/validation.
Coordinates, Energy, and Forces The atomic positions in each frame are recorded along with the total potential energy E and force on each atom Fi.
Dipole and Quadrupole Moments Both the first and second molecular multipole moments where calculated in a local molecular frame identical to the local frame used to generate input descriptors in the DeepMD method.
The dipole and quadrupole moments were calculated using the AIMD snapshots described above. The quantum mechnical traceless molecular quadrupole moment can be obtained using the maximally localized Wannier functions (MLWFs).121 Each frame has a molecular dipole moment Pi and quadrupole moment Qµν on each water molecule. The moleular dipole and traceless quadrupole moment can be calcualted from the AIMD simulation frames using Refs.
221 ad 122
Z X P = drρer + ZiRi (5.2) cell i
Z 1 2 1 X 2 Qµν = drρe(3rµrν − δµνr ) + (ZiRi,µRi,ν − δµνRi ) (5.3) 2 cell 2 i
th where Zi and Ri are the atomic charge and position of the i molecule and Ri and r are
116 5.2. COMPUTATIONAL METHODS refereced to the oxygen atom. The Maximally Localized Wannier Function (MLWF)121 ap- proach is used in the CP2K CRAZY method to calculate the electron dipole moments as
P4 P3 n=1 2e µ=1 ri,µ=1 where e is the electron charge, φi is the nth MLWF and rn is the center of it,
L i 2π r hr i = F(lnhφ |e L µ |φ i) (5.4) n,µ 2π n n
The electric quadrupole moment element is given by
2 L −i 2π r −i 2π r 2 −i 2π r 2 −i 2π r 2 hr r i = hr i hr i + {ln|hφ |e L µ e L ν |φ i| −ln|hφ |e L µ |φ i| −ln|hφ |e L ν |φ i| } µ ν n µ n ν n 16π2 n n n n n n (5.5)
5.2.2 DeepMD reconstruction of the PES
The methodology of Wang et al. 63 was followed closely to reconstruct the PES for the systems using ab initio energies, forces, and coordinates. As done in Ref. 59 and discussed in section 2.4 the potential energy E of each configuration is produced by a sum of atomic energies Ei which are determined by the local coordinates inside of a cutoff radius. Descriptors are assembled for each atom which are fed into the DeePMD. This method is symmetry preserving due to the descriptors being constructed in a local coordinate frame. Permutational symmetry is preserved though ordering neighboring atoms by their species first and then in order of decreasing inverse distance from the atom of interest. The local coordinate frame {ex, ey, ez} is setup following
63 Wang et al. : ex is the vector along the O-H bond, where atom H is closest to the oxygen atom, ez is perpendicular to the plane of the water molecule, and ez is the cross product of ezand ex.
The simulation data is divided into 4,000,000 training batches where the loss function and gradient are computed at each training step from data within the batch. The loss function
117 5.2. COMPUTATIONAL METHODS is shown in equation 2.15. We use 5 hidden layers with decreasing number of neurons with
(M1,M2,M3,M4,M5) = (240, 120, 60, 30, 15). After the model is sufficiently trained, the model parameters can be frozen for use in MD simulations.
We use these network parameters to train a DeePMD network model for the 3 different systems using the data generated in section 5.2.1. These systems are a bulk system trained on just the bulk data, a cavity (interfacial) system trained on just the cavity data, and a combined system which is trained on both cavity and bulk data as shown in Fig. 5.1. The three center squares correspond to three network models for a pure bulk, pure interfacial, and combined interfacial system. This is made possible by the DeePMD-kit framework, which can include multiple systems in training, and has recently been used to successfully calculate the free energy of proton transfer for a solid Titanium Dioxide-liquid water interface.220
Classical trajectories DeepMD-kit provides LAMMPS (a common classical MD software package) support through a third-party package to produce serial MD simulations using the frozen DeePMD model to compute the atomic interactions. In this way, simulations of much longer time scales, and larger length scales than AIMD allows for are now accessible at nearly quantum accuracy. This corresponds to the rightmost boxes in Fig. 5.1.
Different types of systems can be modeled by using different initial positions and parameters than the AIMD training simulation. For example, the 3 different models we trained can each be used for either a bulk or cavity simulation using definitions in LAMMPS. Here I’ll narrow the scope to just the Combined Interfacial model used for DeePMD trajectories of a (1) bulk system and (2) 4.0 Å cavity system.
First, the bulk system was generated with LAMMPs and DeePMD and is composed of
64 waters. The trajectory has a timestep of 0.0005 ps with data recorded every 20 steps.
A temperature of 330K is used in the NVT ensemble. The simulation cell is a 12.429531 x
118 5.2. COMPUTATIONAL METHODS
12.429531 x 12.429531 Å box. The first 50 ps of equilibration is discarded and simulations of
30ps and 250ps are completed.
Second, for the cavity system, we use a 128 water system. The cavity is maintained in
LAMMPS through use of harmonic cavity as in the CP2K simulation. The system size is given
N 4π 1 3 3 −3 3 by L = ( ρ + 3 rc ) where N is the number of waters, ρ = 33.3285(nm) (or 0.997g/cm ) is the number density, and rc is the cavity radius at the center of the cubic simulation box. All interfacial DeePMD simulations are 1500 ps where the first 500 ps is discarded for equilibration.
The timestep is 0.5 fs and trajectory is recorded every 0.01 ps (20 steps). Simulation parameters are the same as above.
5.2.3 DNN for determination of Multipole Moments: DeePD, DeePM,
DeePPM
We build upon previous work here by adding an additional DNN to the procedure. In this DNN we input both coordinate information through the same local descriptors as done previously, and output the quantum mechanical multipole moments calculated in a local frame. For each molecule, only those atoms within a cutoff radius Rc are included in the descriptors. The traceless molecular quadrupole has 5 independent components due to the trace condition Qxx +
Qyy + Qzz = 0 and symmetric property Qxy = Qyx, Qxz = Qzx, and Qyz = Qzy. Training is done in 1,000,000 batches.
DeepMD-kit provides a module which accomplishes this task, but for learning the 4 Wannier center (WC) locations (assigned to each oxygen). This module63 inputs the atom coordinates and outputs predictions for the 12 cartesian WC coordinates of each molecule— learning via a loss function that computes the mean squared error of the computed quantum WC. The computed molecular information that is used in the loss function we will refer to as the tensor
119 5.3. RESULTS AND DISCUSSION data.
Therefore, accessing Dipole and Quadrupole predictions requires editing the source code to train for a different number of components, a different size of tensor data. We train three separate models for the dipole, traceless quadrupole, and primitive quadrupole moment which we name DeePD, DeePM, and DeePPM as shown in Fig. 5.1. In the dipole case, DeePD, we train the model for 3 components— µx, µy, µz in the local coordinate frame. In the traceless quadrupole case, DeePM, we train on 5 components— Qxx, Qyy, Qxy, Qxz, and Qyz in the local coordinate frame. Finally, we train DeePPM for all six components Qxx, Qyy, Qzz, Qxy, Qxz, and Qyz. Additionally, it is necessary to edit the source code to disable the step of rotating the tensor data from lab frame to local frame.
5.2.4 DNN for determination of DeePMD multipole moments
Combining the DeepMD trajectories with the DNN for multipole moments allows us to ac- cess long-time quantum accuracy multipole moments. After generating the trajectories with
LAMMPs and DeePMD, we can input them into our trained DeePD, DeePM, and DeePPM networks through a python script which uses the "model inference" method of DeePMD-kit.
The multipole DNN predicts multipole moments for atoms in each frame of the trajectory.
Provided that both our frozen DeepMD model and multipole DNN models are trained to suf-
ficient accuracy, this allows us to calculate multipole moments and the resulting electrostatic potentials at timescales that were previously inaccessible.
5.3 Results and Discussion
The loss curves for the dynamic DeePMD model, dipole DeePD model, traceless quadrupole
DeePM model, and primitive quadrupole DeePPM are shown in Figure 5.2. For each figure, the
120 5.3. RESULTS AND DISCUSSION training and testing error curves show well-trained models. The final accuracy of DeePMD is
0.3 meV for energy and 56 meV/Å for forces. DeePD, DeePM, and DeePPM achieve accuracy of 0.0447 D, 0.0337 DÅ, and 0.0112 DÅ, respectively.
Figure 5.2: Loss functions for neural network training for each network, DeePMD, DeePD, DeePM, DeePPM.
To ensure that the interfacial system is modeled accurately, we check that the DeePMD
Combined Interfacial Model works for a pure bulk system. First, we analyze the structural properties through the RDFs of atoms in the system ((1) above). Figure 5.3 shows the radial distribution function for the bulk AIMD training data from CP2K and the bulk 64 water system DeePMD trajectory of 30 ps. There is excellent agreement between the AIMD data and
DeePMD data, showing that the structural properties of the bulk system are correctly modeled
121 5.3. RESULTS AND DISCUSSION
Å2 by the Combined Interfacial DNN. Also, the diffusion constant, D, was found to be 0.539 ps
Å2 in the AIMD simulation and 0.521 ps for the DeePMD trajectory, showing the dynamics is sufficiently modeled.
Figure 5.3: Radial Distribution Function for oxygen-oxygen (solid), oxygen-hydrogen (dashed) and hydrogen-hydrogen (dotted) for the AIMD training data trajectory (red) and DeePMD trajectory.
Next, we apply our network models—DeePD, DeePM, DeePPM— for the multipole mo- ments to the bulk system. The bulk system is valuable, as the Bethe potential described in section 2.36 is a bulk property which can be calculated from the primitive quadrupole moment,
Q. Figure 5.4 shows the dipole distribution for system (1) above. Both the DeePMD generated
30 ps and 250 ps simulation are shown along with the AIMD dipole data. Excellent agreement is seen between AIMD and DeePMD trajectories. The average dipole moment of the AIMD data is 2.741 ± 0.068. The 30 ps DeePMD trajectory has an average dipole moment of 2.733 ± 0.067, and 2.736 ± 0.067 for the 250 ps trajectory.
122 5.3. RESULTS AND DISCUSSION
Figure 5.4: Ensemble averaged dipole moment distributions for the AIMD training data tra- jectory (red) and DeePMD trajectory of 30 ps (black) and 250 ps (blue) using DeePD.
123 5.3. RESULTS AND DISCUSSION
Figure 5.5: Ensemble average primitive quadrupole moment eigenvalue distributions for the AIMD training data trajectory (red) and DeePMD trajectory of 30 ps (black) and 250 ps (blue) using DeePPM.
124 5.3. RESULTS AND DISCUSSION
Figure 5.6: Ensemble average distribution of the trace of the primitive quadrupole moment for the AIMD training data trajectory (red) and DeePMD trajectory of 30 ps (black) and 250 ps (blue) using DeePPM.
125 5.3. RESULTS AND DISCUSSION
Figure 5.7: Ensemble averaged double dot product of the primitive quadrupole moment distri- butions for the AIMD training data trajectory (red) and DeePMD trajectory of 30 ps (black) and 250 ps (blue) using DeePPM.
Figures 5.5, 5.6, and 5.7 show the ensemble average eigenvalues, trace, and double dot prod- uct of the primitive quadrupole moment, respectively for the AIMD and DeePMD systems. The trace is calculated through T r(Q) = Qxx + Qyy + Qzz and the double dot product is given by
2 1 2 hQi = ( 3 Q : Q) . The AIMD trace is −8.647 ± 0.002 and both the 30 ps and 250 ps DeePMD trajectories are −8.646 ± 0.002. This value results in a Bethe potential estimate of 3.619V for both both methods. Again, there is excellent agreement for primitve quadrupole eigenvalues, trace and double product. This shows how the DeePMD, DeePD, DeePM, and DeePM net- works, trained on the Combined Interfacial System can both reproduce and determine multipole electrostatics of high accuracy, even for longer trajectories of a bulk system.
Finally, we look at the system of most importance, the interfacial cavity system. As de-
126 5.3. RESULTS AND DISCUSSION scribed above, we use the Combined Interfacial model for DeePMD trajectories of a 4.0 Å, 128 water system. The box size for this system is approximately 16.04 Å. A simulation of 1000ps
(1ns) is analyzed. The radial distribution function for this system is shown in Figure 5.8
Figure 5.8: RDFs of the interfacial cavity system of the DeePMD trajectory of 1 ns and the AIMD 30 ps trajectory.
To assess the agreement of the moments between AIMD training data and NN generated data, we plot the average dipole moment for molecules located in 1 Å width shells around the cavity boundary. Figure 5.9 shows the average dipole for the 1 ns DeePMD trajectory and the 30 ps AIMD training data. Dipole moments in shells around the cavity are in excellent agreement. Both data sets show a lack of polarization around the neutral cavity compared to shells further into the bulk-like area. The double dot product iQh of the traceless quadrupole moment, trained in the DeePM network is shown in figure 5.10 for spherical shells around the cavity. This is a simple way to gauge the magnitude of the quadrupole moment, and to confirm
127 5.3. RESULTS AND DISCUSSION that the network correctly models the quadrupole moment.
Figure 5.9: Ensemble average dipole moment for spherical shells of width 1Å around the cavity. The DeePMD trajectory is 1 ns and the AIMD consists of a 30 ps trajectory. Dipole obtained with DeePD.
128 5.4. CONCLUSION
Figure 5.10: Ensemble average traceless quadrupole double dot product for spherical shells of width 1Å around the cavity. The DeePMD trajectory is 1 ns and the AIMD consists of a 30 ps trajectory. Traceless quadrupole moment values obtained with DeePM.
5.4 Conclusion
The ability to generate long time scale and length scale trajectories at ab initio accuracy is a great achievement by the field of Neural Network Potentials. Ensuring that electrostatic properties can first be extracted from the NN data is an additional step forward and verifying their accuracy is also important. Electrostatic properties of ab initio accuracy at time scales beyond a few hundred picoseconds was previously not possible without incorporating various approximations.
In the pursuit of a predictive theory of ionic and molecular liquids, this result provides a few options for continued research. The strength of ab initio models is their ability to
129 5.4. CONCLUSION incorporate the quantum physics and chemical reactivity that allow for models to adapt to external parameters such as temperatures, pressures, and interfaces. By increasing the time scales and length scales possible with these high-level models through NNPs, we are able to tackle a large number of solvation science problems either directly or indirectly. Directly, Neural
Network Potentials can be used to generate trajectories and properties of interest. Alternatively, we could use NNPs to generate electrostatic properties from which more traditional MD models can be parametrized.
In the indirect method we perhaps sacrifice some accuracy for the time-tested reliability of classical MD. For example, we could use the long-timescale multipoles to optimize the polar- ization and dispersion response using a Drude Oscillator particle.222 223 The model should, in principle, contain the correct electrostatic physical response which allows for it to be trans- ferable, and the optimization process could provide insights into which parameters are most important for that correct physical response.
130 Chapter 6
Conclusions
This thesis I discussed the pervasiveness of solvation science in chemistry, biology, and physics.
I first discussed the use of computer simulations in the field and their importance as a bridge between theory and experiment. It is apparent that current models are hitting a wall in terms of their predictive power for solving the problems of solvation science. Properties of mixtures, mechanisms of ion transport in biological membranes, and ion specific solvation were highlighted. One of the essential elements for advancing our understanding of these problems is having a proper interpretation of the interfacial potentials.
In chapter 3 I advanced our understanding of the liquid water/vapor interface through the simplest model of SPC/E water and classical MD. Existing studies provided evidence along multiple theoretical and experimental routes for an effective net potential of -0.4 V. The question remained of how that number was generated. Using a novel plotting method, I demonstrated how the molecular quadrupole potential is the dominant element of the net potential at a cavity center. Nearly equal and opposite dipolar potentials at the cavity interface and liquid-vapor interface cancel and octupole potential contributions are very small. This result explains why different water models with similar dipole moments but widely varying quadrupole moments
131 produce such different interfacial potential values. Additionally, this technique is free of the often misunderstood Bethe potential contribution that is seen in most traditional interfacial potential calculations, but which is a bulk property that cancels in the presence of two interfaces.
I showed how (1) in all solvents, there will be a length scale at which interfacial potentials attain any rational meaning: where the charge density goes to zero representing a bulk-like area. Furthermore, I suggested that (2) multipolar contributions to the net potential should follow a simple radial dependence.
This result motivated my study of ethylene carbonate using the OPLS-AA model in Chapter
4. Although these classical MD models do not accurately incorporate polarization, my interest was in the general behavior of the molecule and it’s comparison to water. Ethylene carbonate belongs to a class of organic molecules of academic interest for their widely varying physical properties from small changes in structure. They are often used as solvents in Lithium-ion batteries and there are immense efforts to improve performance. I found that properties (1) and (2) above do hold for this organic solvent. Specific values of the multipolar potentials, however, do depend on the detailed molecular orientation of the solvent and the value of the multipole moments. Ethylene carbonate unlike water has a strong dipolar structure which arises from the cavity symmetry breaking and persists deep into the liquid. This, and the fact that the higher order potential contributions are very small results in the liquid-vapor dipole potential as the dominant contribution to the net potential at a cavity center.
These studies aimed to improve physical understanding with knowledge of the shortcomings of classical MD models. These results provided a basis for predicting estimates of electrostatic potentials felt by ions and molecules given the molecular multipole and average orientations at the interface(s). Interfacial potential behavior was clarified for two important solvents. Testing this hypothesis on other solvents such as PC or DMC would strengthen these conclusions.
132 For PC, which has a strong orientational preference for the carbon plane lying flat at the droplet surface, there should be a substantial negative quadrupolar contribution to the net potential from the liquid-vapor interface. DMC, which has a very small dipole moment and large quadrupole moment would be expected to show far less dipolar ordering around the cavity.
Larger quadrupole net potential contributions would be the logically conclusion, though the detailed orientations of molecules at the interfaces is needed.
In my final chapter I turned to the pursuit of calculating more accurate electrostatic po- tentials. Progress in the field of Neural Network Potentials has created a promising new way of modeling liquids. Predictive models of solvation science depend on the model’s ability to generate physical phenomena of polarization, dispersion, chemical reactivity and other forces contained in the quantum description. Towards this end, I showed that an existing NNP,
DeePMD of Wang et al. 63 can be accurately trained to produce the electrostatic properties of the molecular dipole and quadrupole in both the bulk and interfacial water systems. This is an exciting validation of the correct modeling of interfacial electrostatic properties, and moments higher than quadrupolar order.
I suggest future work could consist of using the long timescale NNP trained dipole and quadrupole moments to calculate the net potential at cavity center using the method of my
first two chapters. This would provide a final check of the accuracy of NNP’s in electrostatic calculations. If consistency is achieved with previous calculations, free energy calculations could proceed by using the resulting internal interaction energies in the NNP simulations. If modeling of water is successful, the potential shifts at the interface of other solvents could be studied with these techniques. Most directly, our results can be used to investigate the multipolar electrostatic polarization of different molecular species composing the interfaces between liquids, vapors, or solids. Validation of electrostatic properties in mixtures and at
133 different temperatures or phases would be an additional achievement. Bulk and interfacial multipole moments, obtained for long timescales, could be useful tools to parametrize simple classical models, as the electostatic response to a local environment is produced at the ab initio theory level. Model development using these NNPs directly or as a tool are a promising step forward in the creation of predictive models of ion solvation.
134 Bibliography
[1] Freitas, R. A. Nanomedicine, volume I: basic capabilities; Landes Bioscience Georgetown,
TX, 1999; Vol. 1.
[2] Gleick, P. H. Pacific Institute for Studies in Dev., Environment & Security. Stockholm
Env. Institute, Oxford Univ. Press. 473p 1993, 9 .
[3] Frenkel, D.; Smit, B. Understanding molecular simulation: from algorithms to applica-
tions; Elsevier, 2001; Vol. 1.
[4] Gora, P. F. The theory of Brownian MotionL A Hundred Years’ Anniversary; d: ds,
Spring 2006.
[5] Morrell, W.; Hildebrand, J. The Journal of Chemical Physics 1936, 4, 224–227.
[6] Metropolis, N.; Rosenbluth, A. W.; Rosenbluth, M. N.; Teller, A. H.; Teller, E. The
journal of chemical physics 1953, 21, 1087–1092.
[7] Tildesley, M. A. D. Computers Simulation of Liquids; Oxford University Press: New
York, 1987.
[8] Sangster, J. Octanol-water partition coefficients: fundamentals and physical chemistry;
John Wiley & Sons, 1997; Vol. 1.
135 BIBLIOGRAPHY
[9] Chen, B.; Siepmann, J. I. Journal of the American Chemical Society 2000, 122, 6464–
6467.
[10] Economou, I. G. Industrial & engineering chemistry research 2002, 41, 953–962.
[11] Ferrari, E.; Burton, R.; Davey, R. J.; Gavezzotti, A. Journal of computational chemistry
2006, 27, 1211–1219.
[12] Bai, P.; Siepmann, J. I. Fluid phase equilibria 2011, 310, 11–18.
[13] Chen, B.; Xing, J.; Siepmann, J. I. The Journal of Physical Chemistry B 2000, 104,
2391–2401.
[14] Borodin, O.; Smith, G. D. The Journal of Physical Chemistry B 2009, 113, 1763–1776.
[15] Ashcroft, F.; Gadsby, D.; Miller, C. Introduction. The blurred boundary between channels
and transporters: We dedicate this volume to the memory of Peter Läuger, a pioneer of
the link between channels and pumps. 2009.
[16] Ashcroft, F. M. Ion channels and disease; Academic press, 1999.
[17] Albers, R. W.; Siegel, G. J.; Xie, Z.-J. Basic neurochemistry; Elsevier, 2012; pp 40–62.
[18] Song, H. D.; Beck, T. L. The Journal of Physical Chemistry C 2013, 117, 3701–3712.
[19] Peng, X.; Zhang, Y.; Chu, H.; Li, Y.; Zhang, D.; Cao, L.; Li, G. Journal of chemical
theory and computation 2016, 12, 2973–2982.
[20] Chen, Z.; Beck, T. L. The Journal of Physical Chemistry B 2016, 120, 3129–3139.
[21] Lim, H.-H.; Stockbridge, R. B.; Miller, C. Nature chemical biology 2013, 9, 721.
[22] Abernethy, J. L. Franz Hofmeister-The impact of his life and research on chemistry. 1967.
136 BIBLIOGRAPHY
[23] Kunz, W. Specific ion effects; World Scientific, 2010.
[24] Laughlin, R. G. The aqueous phase behavior of surfactants; Academic Pr, 1994; Vol. 6.
[25] Kronberg, B.; Lindman, B. Surfactants and polymers in aqueous solution; John Wiley &
Sons Ltd., Chichester, 2003.
[26] Jungwirth, P.; Tobias, D. J. Chemical reviews 2006, 106, 1259–1281.
[27] Craig, V. S.; Henry, C. L. Specific Ion Effects; World Scientific, 2010; pp 191–214.
[28] Zhong, J.; Kumar, M.; Francisco, J. S.; Zeng, X. C. Accounts of chemical research 2018,
51, 1229–1237.
[29] Ninham, B. W.; Nostro, P. L. Molecular forces and self assembly: in colloid, nano sciences
and biology; Cambridge University Press, 2010.
[30] Baldwin, R. L. Biophysical journal 1996, 71, 2056–2063.
[31] Heydweiller, A. Ann. Phys 1910, 33, 145–185.
[32] Wagner, C. Phys. Z 1924, 25, 474–477.
[33] Onsager, L.; Samaras, N. N. The Journal of Chemical Physics 1934, 2, 528–536.
[34] Sarri, F.; Tatini, D.; Ambrosi, M.; Carretti, E.; Ninham, B. W.; Dei, L.; Nostro, P. L.
Journal of Molecular Liquids 2018, 255, 397–405.
[35] Peruzzi, N.; Nostro, P. L.; Ninham, B. W.; Baglioni, P. Journal of Solution Chemistry
2015, 44, 1224–1239.
[36] Mazzini, V.; Craig, V. S. Current opinion in colloid & interface science 2016, 23, 82–93.
137 BIBLIOGRAPHY
[37] Tavares, F.; Bratko, D.; Prausnitz, J. Current opinion in colloid & interface science 2004,
9, 81–86.
[38] Gradzielski, M. Current opinion in colloid & interface science 2004, 9, 256–263.
[39] Dill, K.; Bromberg, S. Molecular driving forces: statistical thermodynamics in biology,
chemistry, physics, and nanoscience; Garland Science, 2012.
[40] Lo Nostro, P.; Ninham, B. W. Chemical reviews 2012, 112, 2286–2322.
[41] Milo, R.; Jorgensen, P.; Moran, U.; Weber, G.; Springer, M. Nucleic acids research 2010,
38, D750–D753.
[42] Smith, A. M.; Lee, A. A.; Perkin, S. The journal of physical chemistry letters 2016, 7,
2157–2163.
[43] Parsons, D. F.; Boström, M.; Nostro, P. L.; Ninham, B. W. Physical Chemistry Chemical
Physics 2011, 13, 12352–12367.
[44] Parsons, D. F.; Walsh, R. B.; Craig, V. S. The Journal of chemical physics 2014, 140,
164701.
[45] Born, M. Zeitschrift für Physik 1920, 1, 45–48.
[46] Atkins, P. W.; De Paula, J.; Keeler, J. Atkins’ physical chemistry; Oxford university
press, 2018.
[47] Hospital, A.; Goñi, J. R.; Orozco, M.; Gelpí, J. L. Advances and applications in bioinfor-
matics and chemistry: AABC 2015, 8, 37.
[48] French, R. H.; Parsegian, V. A.; Podgornik, R.; Rajter, R. F.; Jagota, A.; Luo, J.;
138 BIBLIOGRAPHY
Asthagiri, D.; Chaudhury, M. K.; Chiang, Y.-m.; Granick, S. Reviews of Modern Physics
2010, 82, 1887.
[49] Woods, L.; Dalvit, D. A. R.; Tkatchenko, A.; Rodriguez-Lopez, P.; Rodriguez, A. W.;
Podgornik, R. Reviews of Modern Physics 2016, 88, 045003.
[50] Rick, S. W. Journal of computational chemistry 2016, 37, 2060–2066.
[51] Caleman, C.; Hub, J. S.; van Maaren, P. J.; van der Spoel, D. Proceedings of the National
Academy of Sciences 2011, 108, 6838–6842.
[52] Carloni, P.; Rothlisberger, U.; Parrinello, M. Accounts of Chemical Research 2002, 35,
455–464.
[53] Lipfert, J.; Doniach, S.; Das, R.; Herschlag, D. Annual review of biochemistry 2014, 83,
813–841.
[54] Horinek, D.; Mamatkulov, S. I.; Netz, R. R. The Journal of chemical physics 2009, 130,
124507.
[55] Jones, J.; Anouti, M.; Caillon-Caravanier, M.; Willmann, P.; Lemordant, D. Fluid phase
equilibria 2009, 285, 62–68.
[56] Iftimie, R.; Minary, P.; Tuckerman, M. E. Proceedings of the National Academy of Sciences
2005, 102, 6654–6659.
[57] Paquet, E.; Viktor, H. L. Advances in Chemistry 2018, 2018 .
[58] Behler, J. Angewandte Chemie International Edition 2017, 56, 12828–12840.
[59] Zhang, L.; Han, J.; Wang, H.; Car, R.; Weinan, E. Physical review letters 2018, 120,
143001.
139 BIBLIOGRAPHY
[60] Chmiela, S.; Tkatchenko, A.; Sauceda, H. E.; Poltavsky, I.; Schütt, K. T.; Müller, K.-R.
Science advances 2017, 3, e1603015.
[61] Bonati, L.; Parrinello, M. Physical review letters 2018, 121, 265701.
[62] Cheng, B.; Engel, E. A.; Behler, J.; Dellago, C.; Ceriotti, M. Proceedings of the National
Academy of Sciences 2019, 116, 1110–1115.
[63] Wang, H.; Zhang, L.; Han, J.; Weinan, E. Computer Physics Communications 2018, 228,
178–184.
[64] Zhang, L.; Lin, D.-Y.; Wang, H.; Car, R.; Weinan, E. Physical Review Materials 2019,
3, 023804.
[65] others„ et al. arXiv preprint arXiv:1906.11434 2019,
[66] LeCun, Y.; Bengio, Y.; Hinton, G. nature 2015, 521, 436–444.
[67] Conway, B. Journal of Solution Chemistry 1978, 7, 721–770.
[68] Marcus, Y. Ion Solvation John Wiley. 1985.
[69] Grossfield, A.; Ren, P.; Ponder, J. W. Journal of the American Chemical Society 2003,
125, 15671–15682.
[70] Lamoureux, G.; Roux, B. The journal of physical chemistry B 2006, 110, 3308–3322.
[71] Marcus, Y. Journal of the Chemical Society, Faraday Transactions 1: Physical Chemistry
in Condensed Phases 1987, 83, 2985–2992.
[72] Ashbaugh, H. S.; Asthagiri, D. The Journal of chemical physics 2008, 129, 204501.
140 BIBLIOGRAPHY
[73] Latimer, W. M.; Pitzer, K. S.; Slansky, C. M. Molecular Structure And Statistical Ther-
modynamics: Selected Papers of Kenneth S Pitzer; World Scientific, 1993; pp 485–489.
[74] Tissandier, M. D.; Cowen, K. A.; Feng, W. Y.; Gundlach, E.; Cohen, M. H.;
Earhart, A. D.; Coe, J. V.; Tuttle, T. R. The Journal of Physical Chemistry A 1998,
102, 7787–7794.
[75] Pollard, T.; Beck, T. L. The Journal of chemical physics 2014, 140, 224507.
[76] Vlcek, L.; Chialvo, A. A. Fluid Phase Equilibria 2016, 407, 58–75.
[77] Donald, W. A.; Williams, E. R. The Journal of Physical Chemistry B 2010, 114, 13189–
13200.
[78] Vlcek, L.; Chialvo, A. A.; Simonson, J. M. The Journal of Physical Chemistry A 2013,
117, 11328–11338.
[79] Houriez, C.; Réal, F.; Vallet, V.; Mautner, M.; Masella, M. The Journal of chemical
physics 2019, 151, 174504.
[80] Pollard, T. P.; Beck, T. L. The Journal of chemical physics 2018, 148, 222830.
[81] Shi, Y.; Beck, T. L. to be submitted,
[82] Guggenheim, E. The Journal of Physical Chemistry 2002, 33, 842–849.
[83] Pethica, B. A. Physical Chemistry Chemical Physics 2007, 9, 6253–6262.
[84] Pratt, L. R. J. Phys. Chem. 1992, 96, 25–33.
[85] Hünenberger, P.; Reif, M. Single-ion solvation: experimental and theoretical approaches
to elusive thermodynamic quantities; Royal Society of Chemistry, 2011.
141 BIBLIOGRAPHY
[86] You, X.; Chaudhari, M.; Pratt, L. Aqua incognita: why ice floats on water and galileo
400 years on. 2014.
[87] Lyklema, J. Substantia 2017, 1 .
[88] Conboy, J. C.; Richmond, G. L. The Journal of Physical Chemistry B 1997, 101, 983–990.
[89] Scheu, R.; Rankin, B. M.; Chen, Y.; Jena, K. C.; Ben-Amotz, D.; Roke, S. Angewandte
Chemie International Edition 2014, 53, 9560–9563.
[90] Bastos-González, D.; Pérez-Fuentes, L.; Drummond, C.; Faraudo, J. Current Opinion in
Colloid & Interface Science 2016, 23, 19–28.
[91] Creux, P.; Lachaise, J.; Graciaa, A.; Beattie, J. K.; Djerdjev, A. M. The Journal of
Physical Chemistry B 2009, 113, 14146–14150.
[92] Mishra, H.; Enami, S.; Nielsen, R. J.; Stewart, L. A.; Hoffmann, M. R.; Goddard, W. A.;
Colussi, A. J. Proceedings of the National Academy of Sciences 2012, 109, 18679–18683.
[93] Eugene, A. J.; Pillar, E. A.; Colussi, A. J.; Guzman, M. I. Langmuir 2018, 34, 9307–9313.
[94] Lee, J. K.; Walker, K. L.; Han, H. S.; Kang, J.; Prinz, F. B.; Waymouth, R. M.;
Nam, H. G.; Zare, R. N. Proceedings of the National Academy of Sciences 2019, 116,
19294–19298.
[95] Landau, L. D.; Pitaevskii, L.; Lifshitz, E. Electrodynamics of Continuous Media, 2nd ed.;
Pergamon Press: Oxford, U.K., 1984; Vol. 8.
[96] Kathmann, S. M.; Kuo, I. W.; Mundy, C. J.; Schenter, G. K. J. Phys. Chem. B 2011,
115, 4369–4377.
[97] Beck, T. L. Chem. Phys. Lett. 2013, 561, 1–13.
142 BIBLIOGRAPHY
[98] Cendagorta, J. R.; Ichiye, T. J. Phys. Chem. B 2015, 119, 9114–9122.
[99] Leung, K. The Journal of Physical Chemistry Letters 2010, 1, 496–499.
[100] Kathmann, S. M.; Kuo, I.-F. W.; Mundy, C. J.; Schenter, G. K. The Journal of Physical
Chemistry B 2011, 115, 4369–4377.
[101] Fawcett, W. R. Langmuir 2008, 24, 9868–9875.
[102] Farrell, J. R.; McTigue, P. Journal of Electroanalytical Chemistry and Interfacial Elec-
trochemistry 1982, 139, 37–56.
[103] Randles, J. Transactions of the Faraday Society 1956, 52, 1573–1581.
[104] Verlet, L. Physical review 1967, 159, 98.
[105] Berendsen, H.; Grigera, J.; Straatsma, T. Journal of Physical Chemistry 1987, 91, 6269–
6271.
[106] Kusalik, P. G.; Svishchev, I. M. Science 1994, 265, 1219–1221.
[107] Mark, P.; Nilsson, L. The Journal of Physical Chemistry A 2001, 105, 9954–9960.
[108] Reddy, M. R.; Berkowitz, M. Chemical physics letters 1989, 155, 173–176.
[109] Smith, P. E.; van Gunsteren, W. F. Chemical physics letters 1993, 215, 315–318.
[110] Jorgensen, W. L.; Maxwell, D. S.; Tirado-Rives, J. Journal of the American Chemical
Society 1996, 118, 11225–11236.
[111] Beck, T. L.; Paulaitis, M. E.; Pratt, L. R. The Potential Distribution Theorem and Models
of Molecular Solutions; Cambridge University Press, Cambridge, 2006.
[112] Reif, M. M.; Hünenberger, P. H. The Journal of chemical physics 2011, 134, 144103.
143 BIBLIOGRAPHY
[113] Hub, J. S.; de Groot, B. L.; Grubmuller, H.; Groenhof, G. Journal of chemical theory and
computation 2014, 10, 381–390.
[114] Born, M.; Oppenheimer, R. Annalen der physik 1927, 389, 457–484.
[115] Kohn, W.; Becke, A. D.; Parr, R. G. The Journal of Physical Chemistry 1996, 100,
12974–12980.
[116] LIPPERT, B. G.; PARRINELLO, J. H.; MICHELE, Molecular Physics 1997, 92, 477–
488.
[117] Lippert, G.; Hutter, J.; Parrinello, M. Theoretical Chemistry Accounts 1999, 103, 124–
140.
[118] VandeVondele, J.; Krack, M.; Mohamed, F.; Parrinello, M.; Chassaing, T.; Hutter, J.
Computer Physics Communications 2005, 167, 103–128.
[119] Hohenberg, P.; Kohn, W. B 1964, 136, 864–871.
[120] Kühne, T. D.; Krack, M.; Mohamed, F. R.; Parrinello, M. Physical review letters 2007,
98, 066401.
[121] Marzari, N.; Vanderbilt, D. Physical review B 1997, 56, 12847.
[122] Sagui, C.; Pomorski, P.; Darden, T. A.; Roland, C. The Journal of chemical physics 2004,
120, 4530–4544.
[123] Thompson, K. C.; Jordan, M. J.; Collins, M. A. The Journal of chemical physics 1998,
108, 8302–8316.
[124] Braams, B. J.; Bowman, J. M. International Reviews in Physical Chemistry 2009, 28,
577–606.
144 BIBLIOGRAPHY
[125] Handley, C. M.; Behler, J. The European Physical Journal B 2014, 87, 152.
[126] Blank, T. B.; Brown, S. D.; Calhoun, A. W.; Doren, D. J. The Journal of chemical physics
1995, 103, 4129–4137.
[127] Behler, J. Physical Chemistry Chemical Physics 2011, 13, 17930–17955.
[128] Handley, C. M.; Popelier, P. L. The Journal of Physical Chemistry A 2010, 114, 3371–
3383.
[129] Cybenko, G. Mathematics of control, signals and systems 1989, 2, 303–314.
[130] Behler, J. International Journal of Quantum Chemistry 2015, 115, 1032–1050.
[131] Han, J.; Zhang, L.; Car, R.; Weinan, E. arXiv preprint arXiv:1707.01478 2017,
[132] Kingma, D. P.; Ba, J. arXiv preprint arXiv:1412.6980 2014,
[133] Rowlinson, J. S.; Widom, B. Molecular Theory of Capillarity; Oxford, Oxford, 1982.
[134] Schmickler, W. Interfacial Electrochemistry; Oxford University Press, Oxford, 1996.
[135] Fawcett, W. R. Liquids, Solutions, and Interfaces: from Classical Macroscopic Descrip-
tions to Modern Microscopic Details; Oxford, New York, 2004.
[136] Asthagiri, D.; Dixit, P. D.; Merchant, S.; Paulaitis, M. E.; Pratt, L. R.; Rempe, S. B.;
Varma, S. Chem. Phys. Lett. 2010, 485, 1–7.
[137] Doyle, C. C.; Shi, Y.; Beck, T. L. The Journal of Physical Chemistry B 2019, 123,
3348–3358.
[138] Jackson, J. D. Classical Electrodynamics, 3rd ed.; Wiley: New York, 1975.
145 BIBLIOGRAPHY
[139] Horváth, L.; Manghi, M.; Palmeri, J. University of Montpellier, Montpellier, France,
2018; Personal communication.
[140] Stone, A. J. The Theory of Intermolecular Forces, 2nd ed.; Oxford University Press:
Oxford, U.K., 2013.
[141] Wilson, M. A.; Pohorille, A.; Pratt, L. R. J. Phys. Chem. 1987, 91, 4873–4878.
[142] Wilson, M. A.; Pohorille, A.; Pratt, L. R. J. Chem. Phys. 1989, 90, 5211–5213.
[143] Horváth, L.; Beu, T.; Manghi, M.; Palmeri, J. J. Chem. Phys. 2013, 138, 154702.
[144] Lee Warren, G.; Patel, S. J. Chem. Phys. 2007, 127, 064509.
[145] Harder, E.; Roux, B. J. Chem. Phys. 2008, 129, 12B613.
[146] Remsing, R. C.; Baer, M. D.; Schenter, G. K.; Mundy, C. J.; Weeks, J. D. J. Phys. Chem.
Lett. 2014, 5, 2767–2774.
[147] Shi, Y.; Beck, T. L. J. Chem. Phys. 2013, 139, 044504.
[148] Hummer, G.; Pratt, L. R.; Garcia, A. E. J. Phys. Chem. 1996, 100, 1206–1215.
[149] Reif, M. M.; Hünenberger, P. H. J. Chem. Phys. 2011, 134, 144103.
[150] S. M. A. Malek,; Sciortino, F.; Poole, P. H.; Saika-Volvod, I. J. Phys.: Condens. Matt.
2018, 30, 144005.
[151] Nandi, P. K.; Burnham, C. J.; Futera, Z.; English, N. J. ACS Earth Space Chem. 2017,
1, 187–196.
[152] Ponder, J. W. TINKER: Software Tools for Molecular Design, version 7.1. Washington
University, School of Medicine: Saint Louis, MO, 2015.
146 BIBLIOGRAPHY
[153] Weeks, J. D.; Chandler, D.; Andersen, H. C. J. Chem. Phys. 1971, 54, 5237–5247.
[154] Ashbaugh, H. S. J. Phys. Chem. B 2000, 104, 7235–7238.
[155] Ashbaugh, H. S.; Pratt, L. R. Rev. Mod. Phys. 2006, 78, 159–178.
[156] Kuo, I. W.; Mundy, C. J.; Eggimann, B. L.; McGrath, M. J.; Siepmann, J. I.; Chen, B.;
Vieceli, J.; Tobias, D. J. J. Phys. Chem. B. 2006, 110, 3738–3746.
[157] Serva, A.; Pezzotti, S.; Bougueroua, S.; Galimberti, D. R.; Gaigeot, M.-P. J. Mol. Struct.
2018, 1165, 71–78.
[158] Jedlovszky, P.; Předota, M.; Nezbeda, I. Mol. Phys. 2006, 104, 2465–2476.
[159] Pollard, T. P.; Beck, T. L. Current opinion in colloid & interface science 2016, 23,
110–118.
[160] Tissandier, M. D.; Cowen, K. A.; Feng, W. Y.; Gundlach, E.; Cohen, M. H.;
Earhart, A. D.; Coe, J. V.; Tuttle, T. R. J. Phys. Chem. A 1998, 102, 7787–7794.
[161] Grossfield, A.; Ren, P.; Ponder, J. W. J. Am. Chem. Soc. 2003, 125, 15671–15682.
[162] Lamoureux, G.; Roux, B. J. Phys. Chem. B 2006, 110, 3308–3322.
[163] Hünenberger, P.; Reif, M. Single-Ion Solvation: Experimental and Theoretical Ap-
proaches to Elusive Thermodynamics Quantities.; RSC Publishing, Cambridge, 2011.
[164] Leung, K.; Rempe, S. B.; von Lilienfeld, O. A. The Journal of chemical physics 2009,
130, 204507.
[165] Pollard, T. P.; Beck, T. L. J. Chem. Phys. 2014, 141, 18C512.
[166] Pollard, T.; Beck, T. L. J. Chem. Phys. 2014, 140, 224507.
147 BIBLIOGRAPHY
[167] Duignan, T. T.; Baer, M. D.; Schenter, G. K.; Mundy, C. J. J. Chem. Phys. 2017, 147,
161716.
[168] Duignan, T. T.; Baer, M. D.; Mundy, C. J. J. Chem. Phys. 2018, 148, 222819.
[169] Hub, J. S.; de Groot, B. L.; Grubmüller, H.; Groenhof, G. J. Chem. Theory Comput.
2014, 10, 381–390.
[170] Niu, S.; Tan, M.-L.; Ichiye, T. J. Chem. Phys. 2011, 134, 134501.
[171] Dharmawardhana, C. C.; Ichiye, T. The Journal of chemical physics 2017, 147, 194103.
[172] Halliwell, H. F.; Nyburg, S. C. Trans. Faraday Soc. 1963, 59, 1126–1140.
[173] Marcus, Y. Ion Solvation; John Wiley, New York, 1985.
[174] Dinpajooh, M.; Matyushov, D. V. J. Chem. Phys. 2015, 143, 044511.
[175] Slavchov, R. I.; Dimitrova, I. M.; Ivanov, T. J. Chem. Phys. 2015, 143, 154707.
[176] Slavchov, R. I.; Ivanov, T. I. J. Chem. Phys. 2014, 140, 074503.
[177] You, X.; Chaudhari, M. I.; Rempe, S. B.; Pratt, L. R. The Journal of Physical Chemistry
B 2015, 120, 1849–1853.
[178] Amrouche, S. O.; Rekioua, D.; Rekioua, T.; Bacha, S. International Journal of Hydrogen
Energy 2016, 41, 20914–20927.
[179] Chawla, N.; Bharti, N.; Singh, S. Batteries 2019, 5, 19.
[180] Goodenough, J. B.; Park, K.-S. Journal of the American Chemical Society 2013, 135,
1167–1176.
[181] Melot, B. C.; Tarascon, J.-M. Accounts of chemical research 2013, 46, 1226–1238.
148 BIBLIOGRAPHY
[182] Xu, K. Chemical reviews 2014, 114, 11503–11618.
[183] Peruzzi, N.; Ninham, B. W.; Lo Nostro, P.; Baglioni, P. The Journal of Physical Chemistry
B 2012, 116, 14398–14405.
[184] Sarri, F.; Tatini, D.; Tanini, D.; Simonelli, M.; Ambrosi, M.; Ninham, B. W.; Cappe-
rucci, A.; Dei, L.; Nostro, P. L. Journal of Molecular Liquids 2018, 266, 711–717.
[185] Mazzini, V.; Craig, V. S. ACS central science 2018, 4, 1056–1064.
[186] Borodin, O.; Smith, G. D. The Journal of Physical Chemistry B 2006, 110, 4971–4977.
[187] Borodin, O. The Journal of Physical Chemistry B 2009, 113, 11463–11478.
[188] You, X.; Chaudhari, M. I.; Pratt, L. R.; Pesika, N.; Aritakula, K. M.; Rick, S. W. The
Journal of chemical physics 2013, 138, 114708.
[189] Soetens, J.-C.; Millot, C.; Maigret, B.; Bakó, I. Journal of Molecular Liquids 2001, 92,
201–216.
[190] Chaudhari, M. I.; Nair, J. R.; Pratt, L. R.; Soto, F. A.; Balbuena, P. B.; Rempe, S. B.
Journal of chemical theory and computation 2016, 12, 5709–5718.
[191] Chaudhari, M. I.; Muralidharan, A.; Pratt, L. R.; Rempe, S. B. Modeling Electrochemical
Energy Storage at the Atomic Scale; Springer, Cham, 2018; pp 53–77.
[192] Muralidharan, A.; Chaudhari, M. I.; Pratt, L. R.; Rempe, S. B. Scientific reports 2018,
8, 1–8.
[193] Arslanargin, A.; Powers, A.; Beck, T. L.; Rick, S. W. The Journal of Physical Chemistry
B 2015, 120, 1497–1508.
[194] Lee, S.; Park, S. S. Journal of Molecular Liquids 2012, 175, 97–102.
149 BIBLIOGRAPHY
[195] Kumar, N.; Seminario, J. M. The Journal of Physical Chemistry C 2016, 120, 16322–
16332.
[196] Pollard, T. P.; Beck, T. L. The Journal of chemical physics 2017, 147, 161710.
[197] Skarmoutsos, I.; Ponnuchamy, V.; Vetere, V.; Mossa, S. The Journal of Physical Chem-
istry C 2015, 119, 4502–4515.
[198] Masia, M.; Probst, M.; Rey, R. The Journal of Physical Chemistry B 2004, 108, 2016–
2027.
[199] Bhatt, M. D.; Cho, M.; Cho, K. Modelling and Simulation in Materials Science and
Engineering 2012, 20, 065004.
[200] Barnes, T. A.; Kaminski, J. W.; Borodin, O.; Miller III, T. F. The Journal of Physical
Chemistry C 2015, 119, 3865–3880.
[201] Leung, K.; Tenney, C. M. The Journal of Physical Chemistry C 2013, 117, 24224–24235.
[202] Smith, J. W.; Lam, R. K.; Sheardy, A. T.; Shih, O.; Rizzuto, A. M.; Borodin, O.; Har-
ris, S. J.; Prendergast, D.; Saykally, R. J. Physical Chemistry Chemical Physics 2014,
16, 23568–23575.
[203] Wang, J.; Wolf, R. M.; Caldwell, J. W.; Kollman, P. A.; Case, D. A. Journal of compu-
tational chemistry 2004, 25, 1157–1174.
[204] Hummer, G.; Pratt, L. R.; Garcia, A. E. The Journal of Physical Chemistry 1996, 100,
1206–1215.
[205] Hess, B.; Kutzner, C.; Van Der Spoel, D.; Lindahl, E. Journal of chemical theory and
computation 2008, 4, 435–447.
150 BIBLIOGRAPHY
[206] Lide, D. R. CRC handbook of chemistry and physics; CRC press, 2004; Vol. 85.
[207] Clough, S. A.; Beers, Y.; Klein, G. P.; Rothman, L. S. The Journal of Chemical Physics
1973, 59, 2254–2259.
[208] De Visschere, P. Journal of Physics D: Applied Physics 2006, 39, 4278.
[209] Fortunato, B.; Mirone, P.; Fini, G. Spectrochimica Acta Part A: Molecular Spectroscopy
1971, 27, 1917–1927.
[210] Alonso, J. L.; Cervellati, R.; Degli Esposti, A.; Lister, D. G.; Palmieri, P. Journal of the
Chemical Society, Faraday Transactions 2: Molecular and Chemical Physics 1986, 82,
337–356.
[211] Matias, P. M.; Jeffrey, G.; Wingert, L. M.; Ruble, J. R. Journal of Molecular Structure:
THEOCHEM 1989, 184, 247–260.
[212] Gray, C.; Gubbins, K. Molecular Physics 1981, 42, 843–860.
[213] Gubbins, K.; Gray, C.; Machado, J. Molecular Physics 1981, 42, 817–841.
[214] Wilson, M. A.; Pohorille, A.; Pratt, L. R. J. Chem. Phys. 1988, 88, 3281–3285.
[215] Jedlovszky, P.; Vincze, A.; Horvai, G. The Journal of chemical physics 2002, 117, 2271–
2280.
[216] Jedlovszky, P.; Vincze, Á.; Horvai, G. Physical Chemistry Chemical Physics 2004, 6,
1874–1879.
[217] Wood, R. H.; Craft, Q. D. Journal of Solution Chemistry 1978, 7, 799–812.
[218] Abascal, J. L.; Vega, C. Physical review letters 2007, 98, 237801.
151 BIBLIOGRAPHY
[219] Abascal, J. L.; Vega, C. The Journal of Physical Chemistry C 2007, 111, 15811–15822.
[220] Andrade, M. F. C.; Ko, H.-Y.; Zhang, L.; Car, R.; Selloni, A. Chemical Science 2020,
11, 2335–2341.
[221] Silvestrelli, P. L.; Parrinello, M. The Journal of chemical physics 1999, 111, 3572–3580.
[222] Lopes, P. E.; Roux, B.; MacKerell, A. D. Theoretical Chemistry Accounts 2009, 124,
11–28.
[223] Cipcigan, F. S.; Sokhan, V. P.; Crain, J.; Martyna, G. J. Journal of Computational
Physics 2016, 326, 222–233.
152 Appendix A
Appendix
A.1 Molecular Multipole Definition
This section follows section 6.6 of Jackson 138. Words are otherwise original. The definition of molecular multipole appear during the derivation of the Macroscopic Maxwell Equations for the Microscopic version. This provides the physical proof for considering a molecule as a set of point multipoles originating at a fixed location. For simplification I set all magnetic quantities
B and H equal to zero. The macroscopic Maxwell Equations are given by
∇ · D = ρ (A.1)
Microscopically, the system should be considered quantum. The discussion here is analogous except that the quantites can be replaced by expectation values. Electrons and Nuclei can be considered point charges when dimensions are much greater than 0.0001 Å. Interactions are determined by the microscopic Maxwell Equations
∇ · e = 4πη (A.2)
153 A.1. MOLECULAR MULTIPOLE DEFINITION
Averaging the microscopic equations, leads to the familiar homogeneous equations and the following inhomogeneous equations
∇ · E = 4πhη(x, t)i (A.3)
In a medium composed of atoms and molecules, the microscopic charge density η is the sum of free charge and bound charge where xj is the position of charge j at time t and the charge
th P density of the n molecule is ηn = j(n) qjδ(x − xj)
X ηfree = qjδ(x − xj) (A.4) j(free)
X ηbound = ηn(x, t) (A.5) n(molecules)
Time dependence is ignored because averaging takes place at an instant of time. Define the coordinate of a fixed point within the molecule be the origin xn(t) and the location of charges
th relative to that origin be xjn(t). The average charge of n molecule is then:
Z 3 0 0 hηn(x, t)i = d xf(x )ηn(x − x , t)
Z X 3 0 0 = qj d xf(x )δ(x − x − xjn − xn, t) (A.6) j(n)
X = qjf(x − xn − xjn) j(n)
Because length of xjn is always on the scale of atomic dimensions, a Taylor expansion can be
154 A.2. SUPPORTING INFORMATION: CHAPTER 3
made about (x − xn):
2 X 1 X ∂ hηn(x, t)i = qj f(x − xn) − xjn · ∇f(x − xn) + f(x − xn) + ... (A.7) j(n) 2 αβ ∂xα∂xβ
The sums of charge and distance are the molecular multipole moments given in Eq. 2.27 of the main text. The first, second and third terms here are averages of point charge density, divergence of the average point dipole density, etc. at x = xn:
2 1 X ∂ 0 hηn(x, t)i = hqnδ(x − xn)i − ∇ · hpn(x − xn)i + h(Qn)αβδ(x − xn)i + ... (A.8) 6 αβ ∂xα∂xβ
Summing over molecules leads to Eq. 2.28 in the main text
A.2 Supporting Information: Chapter 3
Figure S1 and S2 show plots of the ensemble averaged charge density profile and the charge density profile multiplied by 4πr2 for the largest droplet system studied with a 6 Å cavity centered at r=0. These were used to obtain the electrostatic potential profiles in Fig. 1 of the main text.
155 A.2. SUPPORTING INFORMATION: CHAPTER 3
0 0
0 0
0 0
0 0 3 Å 0 00 ) ( ρ 0 0
0 0
0 0
0 0 0 0 0 0 Å
The ensemble averaged charge density profile ρ(r) for the N = 1500 SPC/E water system with a 6 Å cavity size. This includes 3ns of data.
0
0
0
0 Å
2 0 )4
( 0 ρ
0
0
0 0 0 0 0 Å
The ensemble averaged charge density profile multiplied by 4πr2 for the N = 1500 SPC/E water system with a 6 Å cavity size. This includes 3ns of data. Integration of this plot produced total charge q(r) which is used to find the electric field E(r) and potential profile Φ(r).
Complete data for each system size and cavity size analogous to the Table 1 in the main text. All data in V. The upper section of each table shows the dipole (D), quadrupole (Q), and octupole (O) contribution to the net potential at cavity center separated into the cavity-liquid
(lp) and liquid-vapor (sp) interface. The lower section combines the upper data to show the
D total dipole φnp, total net potential φnp, total net potential without the liquid-vapor quadrupole
∗ contribution φnp and total electrostatic potential at the center of obtained through summation
156 A.2. SUPPORTING INFORMATION: CHAPTER 3 of charges φ.
250 SPC/E waters
potential 2.0 Å 2.5Å 4.0 Å 4.3 Å 6.0 Å 6.6 Å
D φlp -0.018 -0.081 -0.266 -0.283 -0.379 -0.365 Q φlp -0.147 -0.168 -0.217 -0.222 -0.197 -0.197 O φlp -0.100 -0.097 -0.053 -0.046 -0.025 -0.019 D φsp 0.345 0.345 0.365 0.370 0.371 0.347 Q φsp -0.098 -0.098 -0.088 -0.088 -0.089 -0.084
D φnp 0.327 0.264 0.099 0.087 -0.008 -0.018
φnp -0.018 -0.099 -0.259 -0.269 -0.319 -0.318 ∗ φnp 0.080 -0.001 -0.171 -0.181 -0.230 -0.234 φ -0.189 -0.233 -0.311 -0.315 -0.339 -0.316
512 SPC/E waters
potential 2.0 Å 2.5 Å 4.0 Å 4.3 Å 6.0 Å 6.6 Å 8.0 Å 10.0 Å
D φlp -0.022 -0.095 -0.272 -0.289 -0.351 -0.387 -0.445 -0.431 Q φlp -0.173 -0.156 -0.206 -0.199 -0.194 -0.182 -0.167 -0.144 O φlp -0.118 -0.093 -0.046 -0.046 -0.021 -0.017 -0.012 -0.006 D φsp 0.346 0.343 0.346 0.348 0.347 0.340 0.360 0.327 Q φsp -0.074 -0.075 -0.072 -0.073 -0.071 -0.073 -0.064 -0.051
D φnp 0.324 0.248 0.074 0.059 -0.024 -0.047 -0.085 -0.104
φnp -0.041 -0.076 -0.250 -0.259 -0.310 -0.319 -0.328 -0.305 ∗ φnp 0.033 -0.001 -0.178 -0.186 -0.239 -0.246 -0.264 -0.254 φ -0.176 -0.214 -0.304 -0.313 -0.338 -0.341 -0.339 -0.321 .
157 A.2. SUPPORTING INFORMATION: CHAPTER 3
1500 SPC/E waters
potential 2.0 Å 2.5 Å 4.0 Å 4.3 Å 6.0 Å 6.6 Å 8.0 Å 10.0 Å
D φlp -0.055 -0.121 -0.243 -0.275 -0.367 -0.381 -0.407 -0.415 Q φlp -0.114 -0.150 -0.206 -0.211 -0.195 -0.184 -0.160 -0.135 O φlp -0.102 -0.093 -0.049 -0.039 -0.019 -0.017 -0.010 -0.006 D φsp 0.331 0.328 0.329 0.331 0.329 0.332 0.321 0.309 Q φsp -0.050 -0.050 -0.050 -0.049 -0.045 -0.049 -0.050 -0.048
D φnp 0.276 0.207 0.086 0.056 -0.038 -0.049 -0.086 -0.106
φnp 0.010 -0.086 -0.219 -0.243 -0.297 -0.299 -0.306 -0.295 ∗ φnp 0.060 -0.036 -0.169 -0.194 -0.252 -0.250 -0.256 -0.247 φ -0.162 -0.209 -0.283 -0.294 -0.325 -0.323 -0.318 -0.302 .
158 A.3. SUPPORTING INFORMATION: CHAPTER 4
A.3 Supporting Information: Chapter 4
Joint probability distributions P (uθ, ϕ) for molecules in shells near the interfaces. Darker re- gions indicate higher probability while lighter indicate lower probability. The neutral, positive, and negative point particle cases are shown in Figure S1, S2, and S3 respectively. Shells are defined as shown in Figure S1. Shells not shown display a largely uniform distribution.
(a) 2.0 1.5 1.0 g(r) 0.5 0.0
1.5 1.0 0.5
(r) (V) 0.0 D
np
¡ ¢ 0.5
¡ 1.0
0.08 0.06 0.04 0.02
(r) (V) 0.00 Q np
¡ 0.02 ¢
¡ 0.04
¡ 0.06 0.08 0.06 0.04 0.02
(r) (V) 0.00 O np
¡ 0.02 ¢
¡ 0.04
¡ 0.06 5 10 15 20 25 30 r (Å)
159 A.3. SUPPORTING INFORMATION: CHAPTER 4
Shells through EC droplet
Shell Name A B C D E F G H I Boundary 6,7 7,8 8,9 26,27 27,28 28,29 31,32 32,33 33,34 Definition of shells in droplet. Values for r given in Å with r = i, j for i < r ≤ j
160 A.3. SUPPORTING INFORMATION: CHAPTER 4
(a) A (b) B (c) C
(d) D (e) E (f) F
(g) G (h) H (i) I
[ ] Figures S1, S2, and S3 display joint probability distributions P (uθ, ϕ) for molecules with center of mass located in 1 Å width shells near the cavity-liquid and liquid-vapor interface.
161 A.3. SUPPORTING INFORMATION: CHAPTER 4
(a) A (b) B (c) C
(d) D (e) E (f) F
(g) G (h) H (i) I
[ ] Figures S1, S2, and S3 display joint probability distributions P (uθ, ϕ) for molecules with center of mass located in 1 Å width shells near the cavity-liquid and liquid-vapor interface. Positive point particle with charge q = +1.0e
162 A.3. SUPPORTING INFORMATION: CHAPTER 4
(a) A (b) B (c) C
(d) D (e) E (f) F
(g) G (h) H (i) I
[ ] Figures S1, S2, and S3 display joint probability distributions P (uθ, ϕ) for molecules with center of mass located in 1 Å width shells near the cavity-liquid and liquid-vapor interface. Negative point particle with charge q = −1.0e
163