1
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Sitlapersad RS, Thornton AR, den Otter WK. A simple efficient algorithm for molecular simulations of constant potential electrodes. J Chem Phys 2024; 160:034107. [PMID: 38235800 DOI: 10.1063/5.0171502] [Citation(s) in RCA: 1] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/09/2023] [Accepted: 12/21/2023] [Indexed: 01/19/2024] Open
Abstract
Increasingly, society requires high power, high energy storage devices for applications ranging from electric vehicles to buffers on the electric grid. Supercapacitors are a promising contribution to meeting these demands, though there still remain unsolved practical problems. Molecular dynamics simulations can shed light on the relevant molecular level processes in electric double layer capacitors, but these simulations are computationally very demanding. Our focus here is on the algorithmic complexity of the constant potential method (CPM), which uses dedicated electrostatics solvers to maintain a fixed potential difference between two conducting electrodes. We show how any standard electrostatics solver-capable of calculating the energies and forces on all atoms-can be used to implement CPM with a minimum of coding. As an example, we compare our generalized implementation of CPM, based on invocations of the particle-particle-particle-mesh routine of the Large-scale Atomic/Molecular Massively Parallel Simulator, with a traditional implementation based on a dedicated re-implementation of Ewald summation. Both methods yield comparable results on four test systems, with the former achieving a substantial gain in speed and improved scalability. The step from dedicated electrostatic solvers to generic routines is made possible by noting that CPM's traditional narrow Gaussian point-spread of atomic charges on the electrodes effectively endows point-like atoms with chemical hardness, i.e., an intra-atomic energy quadratic in the charge.
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Affiliation(s)
- Ranisha S Sitlapersad
- Department of Fluid and Thermal Engineering and MESA+ Institute for Nanotechnology, University of Twente, Enschede, The Netherlands
| | - Anthony R Thornton
- Department of Fluid and Thermal Engineering and MESA+ Institute for Nanotechnology, University of Twente, Enschede, The Netherlands
| | - Wouter K den Otter
- Department of Fluid and Thermal Engineering and MESA+ Institute for Nanotechnology, University of Twente, Enschede, The Netherlands
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2
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Kondrat S, Feng G, Bresme F, Urbakh M, Kornyshev AA. Theory and Simulations of Ionic Liquids in Nanoconfinement. Chem Rev 2023; 123:6668-6715. [PMID: 37163447 DOI: 10.1021/acs.chemrev.2c00728] [Citation(s) in RCA: 19] [Impact Index Per Article: 19.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 05/12/2023]
Abstract
Room-temperature ionic liquids (RTILs) have exciting properties such as nonvolatility, large electrochemical windows, and remarkable variety, drawing much interest in energy storage, gating, electrocatalysis, tunable lubrication, and other applications. Confined RTILs appear in various situations, for instance, in pores of nanostructured electrodes of supercapacitors and batteries, as such electrodes increase the contact area with RTILs and enhance the total capacitance and stored energy, between crossed cylinders in surface force balance experiments, between a tip and a sample in atomic force microscopy, and between sliding surfaces in tribology experiments, where RTILs act as lubricants. The properties and functioning of RTILs in confinement, especially nanoconfinement, result in fascinating structural and dynamic phenomena, including layering, overscreening and crowding, nanoscale capillary freezing, quantized and electrotunable friction, and superionic state. This review offers a comprehensive analysis of the fundamental physical phenomena controlling the properties of such systems and the current state-of-the-art theoretical and simulation approaches developed for their description. We discuss these approaches sequentially by increasing atomistic complexity, paying particular attention to new physical phenomena emerging in nanoscale confinement. This review covers theoretical models, most of which are based on mapping the problems on pertinent statistical mechanics models with exact analytical solutions, allowing systematic analysis and new physical insights to develop more easily. We also describe a classical density functional theory, which offers a reliable and computationally inexpensive tool to account for some microscopic details and correlations that simplified models often fail to consider. Molecular simulations play a vital role in studying confined ionic liquids, enabling deep microscopic insights otherwise unavailable to researchers. We describe the basics of various simulation approaches and discuss their challenges and applicability to specific problems, focusing on RTIL structure in cylindrical and slit confinement and how it relates to friction and capacitive and dynamic properties of confined ions.
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Affiliation(s)
- Svyatoslav Kondrat
- Institute of Physical Chemistry, Polish Academy of Sciences, 01-224 Warsaw, Poland
- Institute for Computational Physics, University of Stuttgart, Stuttgart 70569, Germany
| | - Guang Feng
- State Key Laboratory of Coal Combustion, School of Energy and Power Engineering, Huazhong University of Science and Technology (HUST), Wuhan 430074, China
- Nano Interface Centre for Energy, School of Energy and Power Engineering, Huazhong University of Science and Technology, Wuhan 430074, China
| | - Fernando Bresme
- Department of Chemistry, Molecular Sciences Research Hub, White City Campus, London W12 0BZ,United Kingdom
- Thomas Young Centre for Theory and Simulation of Materials, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
- London Centre for Nanotechnology, Imperial College London, London SW7 2AZ, United Kingdom
| | - Michael Urbakh
- School of Chemistry and the Sackler Center for Computational Molecular and Materials Science, Tel Aviv University, Tel Aviv 6997801, Israel
| | - Alexei A Kornyshev
- Department of Chemistry, Molecular Sciences Research Hub, White City Campus, London W12 0BZ,United Kingdom
- Thomas Young Centre for Theory and Simulation of Materials, Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
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3
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Hoang Ngoc Minh T, Stoltz G, Rotenberg B. Frequency and field-dependent response of confined electrolytes from Brownian dynamics simulations. J Chem Phys 2023; 158:104103. [PMID: 36922117 DOI: 10.1063/5.0139258] [Citation(s) in RCA: 3] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 02/17/2023] Open
Abstract
Using Brownian dynamics simulations, we investigate the effects of confinement, adsorption on surfaces, and ion-ion interactions on the response of confined electrolyte solutions to oscillating electric fields in the direction perpendicular to the confining walls. Nonequilibrium simulations allows to characterize the transitions between linear and nonlinear regimes when varying the magnitude and frequency of the applied field, but the linear response, characterized by the frequency-dependent conductivity, is more efficiently predicted from the equilibrium current fluctuations. To that end, we (rederive and) use the Green-Kubo relation appropriate for overdamped dynamics, which differs from the standard one for Newtonian or underdamped Langevin dynamics. This expression highlights the contributions of the underlying Brownian fluctuations and of the interactions of the particles between them and with external potentials. Although already known in the literature, this relation has rarely been used to date, beyond the static limit to determine the effective diffusion coefficient or the DC conductivity. The frequency-dependent conductivity always decays from a bulk-like behavior at high frequency to a vanishing conductivity at low frequency due to the confinement of the charge carriers by the walls. We discuss the characteristic features of the crossover between the two regimes, most importantly how the crossover frequency depends on the confining distance and the salt concentration, and the fact that adsorption on the walls may lead to significant changes both at high and low frequencies. Conversely, our results illustrate the possibility to obtain information on diffusion between walls, charge relaxation, and adsorption by analyzing the frequency-dependent conductivity.
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Affiliation(s)
- Thê Hoang Ngoc Minh
- CNRS, Physicochimie des Électrolytes et Nanosystèmes Interfaciaux, Sorbonne Université, F-75005 Paris, France
| | | | - Benjamin Rotenberg
- CNRS, Physicochimie des Électrolytes et Nanosystèmes Interfaciaux, Sorbonne Université, F-75005 Paris, France
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4
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Jeanmairet G, Rotenberg B, Salanne M. Microscopic Simulations of Electrochemical Double-Layer Capacitors. Chem Rev 2022; 122:10860-10898. [PMID: 35389636 PMCID: PMC9227719 DOI: 10.1021/acs.chemrev.1c00925] [Citation(s) in RCA: 35] [Impact Index Per Article: 17.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/19/2022]
Abstract
![]()
Electrochemical double-layer
capacitors (EDLCs) are devices allowing
the storage or production of electricity. They function through the
adsorption of ions from an electrolyte on high-surface-area electrodes
and are characterized by short charging/discharging times and long
cycle-life compared to batteries. Microscopic simulations are now
widely used to characterize the structural, dynamical, and adsorption
properties of these devices, complementing electrochemical experiments
and in situ spectroscopic analyses. In this review,
we discuss the main families of simulation methods that have been
developed and their application to the main family of EDLCs, which
include nanoporous carbon electrodes. We focus on the adsorption of
organic ions for electricity storage applications as well as aqueous
systems in the context of blue energy harvesting and desalination.
We finally provide perspectives for further improvement of the predictive
power of simulations, in particular for future devices with complex
electrode compositions.
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Affiliation(s)
- Guillaume Jeanmairet
- Sorbonne Université, CNRS, Physico-chimie des Électrolytes et Nanosystèmes Interfaciaux, PHENIX, F-75005 Paris, France.,Réseau sur le Stockage Electrochimique de l'Energie (RS2E), FR CNRS 3459, 80039 Amiens, France
| | - Benjamin Rotenberg
- Sorbonne Université, CNRS, Physico-chimie des Electrolytes et Nanosystèmes Interfaciaux, PHENIX, F-75005 Paris, France.,Réseau sur le Stockage Électrochimique de l'Énergie (RS2E), FR CNRS 3459, 80039 Amiens, France
| | - Mathieu Salanne
- Réseau sur le Stockage Electrochimique de l'Energie (RS2E), FR CNRS 3459, 80039 Amiens, France.,Sorbonne Université, CNRS, Physico-chimie des Electrolytes et Nanosystèmes Interfaciaux, PHENIX, F-75005 Paris, France.,Institut Universitaire de France (IUF), 75231 Paris Cedex 05, France
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5
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Schlaich A, Jin D, Bocquet L, Coasne B. Electronic screening using a virtual Thomas-Fermi fluid for predicting wetting and phase transitions of ionic liquids at metal surfaces. NATURE MATERIALS 2022; 21:237-245. [PMID: 34764431 DOI: 10.1038/s41563-021-01121-0] [Citation(s) in RCA: 13] [Impact Index Per Article: 6.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/24/2020] [Accepted: 09/03/2021] [Indexed: 06/13/2023]
Abstract
Of relevance to energy storage, electrochemistry and catalysis, ionic and dipolar liquids display unexpected behaviours-especially in confinement. Beyond adsorption, over-screening and crowding effects, experiments have highlighted novel phenomena, such as unconventional screening and the impact of the electronic nature-metallic versus insulating-of the confining surface. Such behaviours, which challenge existing frameworks, highlight the need for tools to fully embrace the properties of confined liquids. Here we introduce a novel approach that involves electronic screening while capturing molecular aspects of interfacial fluids. Although available strategies consider perfect metal or insulator surfaces, we build on the Thomas-Fermi formalism to develop an effective approach that deals with any imperfect metal between these asymptotes. Our approach describes electrostatic interactions within the metal through a 'virtual' Thomas-Fermi fluid of charged particles, whose Debye length sets the screening length λ. We show that this method captures the electrostatic interaction decay and electrochemical behaviour on varying λ. By applying this strategy to an ionic liquid, we unveil a wetting transition on switching from insulating to metallic conditions.
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Affiliation(s)
- Alexander Schlaich
- Université Grenoble Alpes, CNRS, LIPhy, Grenoble, France
- Institute for Computational Physics, University of Stuttgart, Stuttgart, Germany
| | - Dongliang Jin
- Université Grenoble Alpes, CNRS, LIPhy, Grenoble, France
| | - Lyderic Bocquet
- Laboratoire de Physique de l'Ecole Normale Supérieure, CNRS, Université PSL, Sorbonne Université, Paris, Paris, France
| | - Benoit Coasne
- Université Grenoble Alpes, CNRS, LIPhy, Grenoble, France.
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6
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Microscopic origin of the effect of substrate metallicity on interfacial free energies. Proc Natl Acad Sci U S A 2021; 118:2108769118. [PMID: 34876519 DOI: 10.1073/pnas.2108769118] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Accepted: 10/19/2021] [Indexed: 11/18/2022] Open
Abstract
We investigate the effect of the metallic character of solid substrates on solid-liquid interfacial thermodynamics using molecular simulations. Building on the recent development of a semiclassical Thomas-Fermi model to tune the metallicity in classical molecular dynamics simulations, we introduce a thermodynamic integration framework to compute the evolution of the interfacial free energy as a function of the Thomas-Fermi screening length. We validate this approach against analytical results for empty capacitors and by comparing the predictions in the presence of an electrolyte with values determined from the contact angle of droplets on the surface. The general expression derived in this work highlights the role of the charge distribution within the metal. We further propose a simple model to interpret the evolution of the interfacial free energy with voltage and Thomas-Fermi length, which allows us to identify the charge correlations within the metal as the microscopic origin of the evolution of the interfacial free energy with the metallic character of the substrate. This methodology opens the door to the molecular-scale study of the effect of the metallic character of the substrate on confinement-induced transitions in ionic systems, as reported in recent atomic force microscopy and surface force apparatus experiments.
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7
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Yuan J, Wang Y. Conformation and Ionization Behavior of Charge-Regulating Polyelectrolyte Brushes in a Poor Solvent. J Phys Chem B 2021; 125:10589-10596. [PMID: 34494845 DOI: 10.1021/acs.jpcb.1c04451] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/20/2022]
Abstract
Understanding the structural response of weak polyelectrolyte brushes upon external stimuli is crucial for their applications ranging from modifying surface properties to the development of smart and intelligent materials. In this work, coarse-grained molecular dynamics simulations were carried out to investigate the conformation and ionization behavior of charge-regulating polyelectrolyte brushes under poor solvent conditions, using an implicit solvent model. The results show that, while the thickness of a sparse polyelectrolyte brush shows a similar behavior to that of a single chain, namely, a monotonic change as a function of solvent quality (modeled by an effective segment-segment attraction strength parameter) and solution pH, a dense polyelectrolyte brush exhibits more complex behavior. An unexpected reexpansion is observed when the effective segment-segment attraction strength is further increased, especially in the case of a high pH. In the latter case, strong attraction in polymer segments promotes the formation of large, interchain, cylindrical aggregates, leading to an increase in brush thickness.
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Affiliation(s)
- Jiaxing Yuan
- School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Yanwei Wang
- Department of Chemical and Materials Engineering, School of Engineering and Digital Sciences, Nazarbayev University, 53 Kabanbay Batyr Avenue, Nur-Sultan 010000, Kazakhstan.,Laboratory of Computational Materials Science for Energy Applications, Center for Energy and Advanced Materials Science, National Laboratory Astana, 53 Kabanbay Batyr Avenue, Nur-Sultan 010000, Kazakhstan
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8
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Solis FJ, Olvera de la Cruz M. Pimples reduce and dimples enhance flat dielectric surface image repulsion. J Chem Phys 2021; 155:104703. [PMID: 34525828 DOI: 10.1063/5.0058810] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
In solid-liquid, or liquid-liquid, interfaces with dielectric contrast, charged particles interact with the induced polarization charge of the interface. These interactions contribute to an effective self-energy of the bulk ions and mediate ion-ion interactions. For flat interfaces, the self-energy and the mediated interactions are neatly constructed by the image charge method. For other geometries, explicit results are scarce and the problem must be treated via approximations or direct computation. The case of interfaces with roughness is of great practical importance. This article provides analytical results, valid to first-order in perturbation theory, for the self-energy of particles near rough substrates. Explicit formulas are provided for the case of a sinusoidal deformation of a flat surface. Generic deformations can be treated by superposition. In addition to results for the self-energy, the surface polarization charge is presented as a quadrature. The interaction between an ion and the deformed surface is modified by the change in relative distance as well as by the local curvature of the surface. Solid walls, with a lower dielectric constant than the liquid, repel all ions. We show that the repulsion is reduced by local convexity and enhanced by concavity; dimples are more repulsive than pimples.
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Affiliation(s)
- Francisco J Solis
- School of Mathematical and Natural Sciences, Arizona State University, Glendale, Arizona 85306, USA
| | - Monica Olvera de la Cruz
- Department of Chemical and Biological Engineering, Northwestern University, Evanston, Illinois 60208, USA
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9
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Maxian O, Peláez RP, Greengard L, Donev A. A fast spectral method for electrostatics in doubly periodic slit channels. J Chem Phys 2021; 154:204107. [PMID: 34241178 DOI: 10.1063/5.0044677] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/15/2022] Open
Abstract
We develop a fast method for computing the electrostatic energy and forces for a collection of charges in doubly periodic slabs with jumps in the dielectric permittivity at the slab boundaries. Our method achieves spectral accuracy by using Ewald splitting to replace the original Poisson equation for nearly singular sources with a smooth far-field Poisson equation, combined with a localized near-field correction. Unlike existing spectral Ewald methods, which make use of the Fourier transform in the aperiodic direction, we recast the problem as a two-point boundary value problem in the aperiodic direction for each transverse Fourier mode for which exact analytic boundary conditions are available. We solve each of these boundary value problems using a fast, well-conditioned Chebyshev method. In the presence of dielectric jumps, combining Ewald splitting with the classical method of images results in smoothed charge distributions, which overlap the dielectric boundaries themselves. We show how to preserve the spectral accuracy in this case through the use of a harmonic correction, which involves solving a simple Laplace equation with smooth boundary data. We implement our method on graphical processing units and combine our doubly periodic Poisson solver with Brownian dynamics to study the equilibrium structure of double layers in binary electrolytes confined by dielectric boundaries. Consistent with prior studies, we find strong charge depletion near the interfaces due to repulsive interactions with image charges, which points to the need for incorporating polarization effects in understanding confined electrolytes, both theoretically and computationally.
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Affiliation(s)
- Ondrej Maxian
- Courant Institute, New York University, New York, New York 10012, USA
| | - Raúl P Peláez
- Department of Theoretical Condensed Matter Physics, Universidad Autónoma de Madrid, 28049 Madrid, Spain
| | - Leslie Greengard
- Courant Institute, New York University, New York, New York 10012, USA
| | - Aleksandar Donev
- Courant Institute, New York University, New York, New York 10012, USA
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10
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Abstract
Many key industrial processes, from electricity production, conversion, and storage to electrocatalysis or electrochemistry in general, rely on physical mechanisms occurring at the interface between a metallic electrode and an electrolyte solution, summarized by the concept of an electric double layer, with the accumulation/depletion of electrons on the metal side and of ions on the liquid side. While electrostatic interactions play an essential role in the structure, thermodynamics, dynamics, and reactivity of electrode-electrolyte interfaces, these properties also crucially depend on the nature of the ions and solvent, as well as that of the metal itself. Such interfaces pose many challenges for modeling because they are a place where quantum chemistry meets statistical physics. In the present review, we explore the recent advances in the description and understanding of electrode-electrolyte interfaces with classical molecular simulations, with a focus on planar interfaces and solvent-based liquids, from pure solvent to water-in-salt electrolytes.
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Affiliation(s)
- Laura Scalfi
- Physicochimie des Électrolytes et Nanosystèmes Interfaciaux, CNRS 8234, Sorbonne Université, F-75005 Paris, France
| | - Mathieu Salanne
- Physicochimie des Électrolytes et Nanosystèmes Interfaciaux, CNRS 8234, Sorbonne Université, F-75005 Paris, France
- Réseau sur le Stockage Electrochimique de l’Energie (RS2E), FR CNRS 3459, 80039 Amiens Cedex, France
| | - Benjamin Rotenberg
- Physicochimie des Électrolytes et Nanosystèmes Interfaciaux, CNRS 8234, Sorbonne Université, F-75005 Paris, France
- Réseau sur le Stockage Electrochimique de l’Energie (RS2E), FR CNRS 3459, 80039 Amiens Cedex, France
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11
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Abstract
Electrolyte-filled subnanometre pores exhibit exciting physics and play an increasingly important role in science and technology. In supercapacitors, for instance, ultranarrow pores provide excellent capacitive characteristics. However, ions experience difficulties in entering and leaving such pores, which slows down charging and discharging processes. In an earlier work we showed for a simple model that a slow voltage sweep charges ultranarrow pores quicker than an abrupt voltage step. A slowly applied voltage avoids ionic clogging and co-ion trapping—a problem known to occur when the applied potential is varied too quickly—causing sluggish dynamics. Herein, we verify this finding experimentally. Guided by theoretical considerations, we also develop a non-linear voltage sweep and demonstrate, with molecular dynamics simulations, that it can charge a nanopore even faster than the corresponding optimized linear sweep. For discharging we find, with simulations and in experiments, that if we reverse the applied potential and then sweep it to zero, the pores lose their charge much quicker than they do for a short-circuited discharge over their internal resistance. Our findings open up opportunities to greatly accelerate charging and discharging of subnanometre pores without compromising the capacitive characteristics, improving their importance for energy storage, capacitive deionization, and electrochemical heat harvesting. Narrowing pores filled with an electrolyte usually slows down their charge-discharge dynamics. Here the authors demonstrate through molecular dynamics simulations and experiments with novolac-derived carbon electrodes how non-linear voltage sweeps can accelerate charging and discharging of subnanometer pores.
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12
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Malossi RM, Girotto M, dos Santos AP, Levin Y. Simulations of electrolyte between charged metal surfaces. J Chem Phys 2020; 153:044121. [DOI: 10.1063/5.0012073] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Rodrigo Mór Malossi
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970 Porto Alegre, RS, Brazil
| | - Matheus Girotto
- Instituto de Fisica, Universidade de Sao Paulo, Rua do Matao, 1371, 05508-090 Sao Paulo, SP, Brazil
| | - Alexandre P. dos Santos
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970 Porto Alegre, RS, Brazil
| | - Yan Levin
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, 91501-970 Porto Alegre, RS, Brazil
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13
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Abstract
We employ a three-dimensional molecular dynamics to simulate a driven polymer translocation through a nanopore by applying an external force, for four pore diameters and two external forces. To see the polymer and pore interaction effects on translocation time, we studied nine interaction energies. Moreover, to better understand the simulation results, we investigate polymer center of mass, shape factor and the monomer spatial distribution through the translocation process. Our results reveal that increasing the polymer-pore interaction energy is accompanied by an increase in the translocation time and decrease in the process rate. Furthermore, for pores with greater diameter, the translocation becomes faster. The shape analysis of the polymer indicates that the polymer shape is highly sensitive to the interaction energy. In great interactions, the monomers come close to the pore from both sides. As a result, the translocation becomes fast at first and slows down at last. Overall, it can be concluded that the external force does not play a major role in the shape and distribution of translocated monomers. However, the interaction energy between monomer and nanopore has a major effect especially on the distribution of translocated monomers on the trans side.
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14
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Haege C, Jagiella S, Giesselmann F. Molecular Electron Density Distribution and X-Ray Diffraction Patterns of Smectic A Liquid Crystals - A Simulation Study. Chemphyschem 2019; 20:2466-2472. [PMID: 31361389 DOI: 10.1002/cphc.201900538] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2019] [Indexed: 11/07/2022]
Abstract
X-ray diffraction (XRD) is one of the most important methods to assess the long-range translational order in smectic A (SmA) liquid crystals. Nevertheless, the knowledge about the influence of the molecular electron density distribution (MEDD) on the XRD pattern is rather limited because it is not possible to vary the orientational order, the translational order and the MEDD independently in an experiment. We here present a systematic simulation study in which we examine this effect and show that the MEDD indeed has a major impact on the general appearance of the XRD pattern. More specifically, we find that the smectic layer peaks and the intensity ratios thereof strongly depend on the width of the MEDD. The classic approach by Leadbetter et al. to determine the smectic translational order parameter ∑ from XRD intensities works if the MEDD is quite narrow. In all other cases the influence of the MEDD has to be taken into account.
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Affiliation(s)
- Christian Haege
- Institute of Physical Chemistry, University of Stuttgart, Pfaffenwaldring 55, 70569, Stuttgart, Germany
| | - Stefan Jagiella
- Institute of Physical Chemistry, University of Stuttgart, Pfaffenwaldring 55, 70569, Stuttgart, Germany
| | - Frank Giesselmann
- Institute of Physical Chemistry, University of Stuttgart, Pfaffenwaldring 55, 70569, Stuttgart, Germany
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15
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Kondrat S, Vasilyev OA, Kornyshev AA. Feeling Your Neighbors across the Walls: How Interpore Ionic Interactions Affect Capacitive Energy Storage. J Phys Chem Lett 2019; 10:4523-4527. [PMID: 31318564 DOI: 10.1021/acs.jpclett.9b01623] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Progress in low-dimensional carbon materials has intensified research on supercapacitors with nanostructured/nanoporous electrodes. The theoretical and simulation work so far has focused on charging single nanopores or nanoporous networks and the effects due to ionic interactions inside the pores, while the effect of interpore ion-ion correlations has received less attention. Herein, we study how the interactions between the ions in the neighboring pores across the pore walls affect capacitive energy storage. We develop a simple lattice model for the ions in a stack of parallel-aligned nanotubes, solve it by using the perturbation and "semi-mean-field" theories, and test the results by Monte Carlo simulations. We demonstrate that the interpore ionic interactions can have a profound effect on charge storage; in particular, such interactions can enhance or diminish the stored energy density, depending on the sign of like-charge interactions. We also find that charging can proceed either continuously or via a phase transition. Our results call for more detailed investigations of the properties of carbon pore walls and suggest that tuning their electrostatic response may be promising for the rational design of an optimal supercapacitor.
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Affiliation(s)
- Svyatoslav Kondrat
- Department of Complex Systems , Institute of Physical Chemistry, PAS , Kasprzaka 44/52 , 01-224 Warsaw , Poland
| | - Oleg A Vasilyev
- Max Planck Institute for Intelligent Systems , Heisenbergstrasse 3 , 70569 Stuttgart , Germany
- 4th Institute for Theoretical Physics , University of Stuttgart , Pfaffenwaldring 57 , 70569 Stuttgart , Germany
| | - Alexei A Kornyshev
- Department of Chemistry , Imperial College London , Molecular Sciences Research Hub, White City Campus , London W12 0BZ , United Kingdom
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16
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Bedrov D, Piquemal JP, Borodin O, MacKerell AD, Roux B, Schröder C. Molecular Dynamics Simulations of Ionic Liquids and Electrolytes Using Polarizable Force Fields. Chem Rev 2019; 119:7940-7995. [PMID: 31141351 PMCID: PMC6620131 DOI: 10.1021/acs.chemrev.8b00763] [Citation(s) in RCA: 274] [Impact Index Per Article: 54.8] [Reference Citation Analysis] [Abstract] [MESH Headings] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/11/2018] [Indexed: 11/30/2022]
Abstract
Many applications in chemistry, biology, and energy storage/conversion research rely on molecular simulations to provide fundamental insight into structural and transport properties of materials with high ionic concentrations. Whether the system is comprised entirely of ions, like ionic liquids, or is a mixture of a polar solvent with a salt, e.g., liquid electrolytes for battery applications, the presence of ions in these materials results in strong local electric fields polarizing solvent molecules and large ions. To predict properties of such systems from molecular simulations often requires either explicit or mean-field inclusion of the influence of polarization on electrostatic interactions. In this manuscript, we review the pros and cons of different treatments of polarization ranging from the mean-field approaches to the most popular explicit polarization models in molecular dynamics simulations of ionic materials. For each method, we discuss their advantages and disadvantages and emphasize key assumptions as well as their adjustable parameters. Strategies for the development of polarizable models are presented with a specific focus on extracting atomic polarizabilities. Finally, we compare simulations using polarizable and nonpolarizable models for several classes of ionic systems, discussing the underlying physics that each approach includes or ignores, implications for implementation and computational efficiency, and the accuracy of properties predicted by these methods compared to experiments.
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Affiliation(s)
- Dmitry Bedrov
- Department
of Materials Science & Engineering, University of Utah, 122 South Central Campus Drive, Room 304, Salt Lake City, Utah 84112, United States
| | - Jean-Philip Piquemal
- Laboratoire
de Chimie Théorique, Sorbonne Université,
UMR 7616 CNRS, CC137, 4 Place Jussieu, Tour 12-13, 4ème étage, 75252 Paris Cedex 05, France
- Institut
Universitaire de France, 75005, Paris Cedex 05, France
- Department
of Biomedical Engineering, The University
of Texas at Austin, Austin, Texas 78712, United States
| | - Oleg Borodin
- Electrochemistry
Branch, Sensors and Electron Devices Directorate, Army Research Laboratory, 2800 Powder Mill Road, Adelphi, Maryland 20703, United
States
| | - Alexander D. MacKerell
- Department
of Pharmaceutical Sciences, School of Pharmacy, University of Maryland, 20 Penn Street, Baltimore, Maryland 21201, United
States
| | - Benoît Roux
- Department
of Biochemistry and Molecular Biology, Gordon Center for Integrative
Science, University of Chicago, 929 57th Street, Chicago, Illinois 60637, United States
| | - Christian Schröder
- Department
of Computational Biological Chemistry, University
of Vienna, Währinger Strasse 17, A-1090 Vienna, Austria
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17
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Jochum C, AdŽić N, Stiakakis E, Derrien TL, Luo D, Kahl G, Likos CN. Structure and stimuli-responsiveness of all-DNA dendrimers: theory and experiment. NANOSCALE 2019; 11:1604-1617. [PMID: 30311616 DOI: 10.1039/c8nr05814h] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
We present a comprehensive theoretical and experimental study of the solution phase properties of a DNA-based family of nanoparticles - dendrimer-like DNA molecules (DL-DNA). These charged DNA dendrimers are novel macromolecular aggregates, which hold high promise in targeted self-assembly of soft matter systems in the bulk and at interfaces. To describe the behaviour of this family of dendrimers (with generations ranging from G1 to G7), we use a theoretical model in which base-pairs of a single DL-DNA molecule are modeled by charged monomers, whose interactions are chosen to mimic the equilibrium properties of DNA correctly. Experimental results on the sizes and conformations of DL-DNA are based on static and dynamic light scattering; and molecular dynamics simulations are employed to model the equilibrium properties of DL-DNA, which compare favorably to the findings from experiments while at the same time providing a host of additional information and insight into the molecular structure of the nanostructures. We also examine the salt-responsiveness of these macromolecules, finding that despite the strong screening of electrostatic interactions brought about by the added salt, the macromolecules shrink only slightly, their size robustness stemming from the high bending rigidity of the DNA-segments. The study of these charged dendrimer systems is an important field of research in the area of soft matter due to their potential role for various interdisciplinary applications, ranging from molecular cages and carriers for drug delivery in a living organism to the development of dendrimer- and dendron-based ultra-thin films in the area of nanotechnology. These findings are essential to determine if DL-DNA is a viable candidate for the experimental realization of cluster crystals in the bulk, a novel form of solid with multiple site occupancy.
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Affiliation(s)
- Clemens Jochum
- Institute for Theoretical Physics, TU Wien, Wiedner Hauptstraße 8-10, A-1040 Vienna, Austria.
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18
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Breitsprecher K, Holm C, Kondrat S. Charge Me Slowly, I Am in a Hurry: Optimizing Charge-Discharge Cycles in Nanoporous Supercapacitors. ACS NANO 2018; 12:9733-9741. [PMID: 30088913 DOI: 10.1021/acsnano.8b04785] [Citation(s) in RCA: 44] [Impact Index Per Article: 7.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
Nanoporous supercapacitors attract much attention as green energy storage devices with remarkable cyclability and high power and energy densities. However, their use in high-frequency applications is limited by relatively slow charging processes, while accelerating charging without compromising the energy storage still remains a challenging task. Here, we study in detail the charging and discharging behavior of nanoporous supercapacitors with narrow pores, which provide exceptionally high capacitances and stored energy densities. We scrutinize the dynamic modes of charging, revealing, in particular, a transient formation of crowded and dilute ionic-liquid phases inside the pores, which leads to co-ion trapping and correspondingly slow charging. We show how trapping can be circumvented by applying a slow voltage sweep, and we demonstrate that it can accelerate the overall charging process considerably if the sweep rate is chosen appropriately. While one might be tempted to apply a similar strategy to discharging, we find that the best discharge rates are obtained when the voltage is switched off in a step-like fashion, whereby the optimal charge and discharge times differ a few-fold. We unveil the scaling laws for such optimal quantities, which allow one to predict quantitatively the charging behavior for realistically long pores. On the basis of our findings, we propose an optimal charge-discharge cycle and elaborate on optimization strategies.
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Affiliation(s)
- Konrad Breitsprecher
- Institute for Computational Physics , Universität Stuttgart , Allmandring 3 , D-70569 Stuttgart , Germany
| | - Christian Holm
- Institute for Computational Physics , Universität Stuttgart , Allmandring 3 , D-70569 Stuttgart , Germany
| | - Svyatoslav Kondrat
- Institute for Computational Physics , Universität Stuttgart , Allmandring 3 , D-70569 Stuttgart , Germany
- Department of Complex Systems , Institute of Physical Chemistry, Polish Academy of Sciences , Kasprzaka 44/52 , 01-224 Warsaw , Poland
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19
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Girotto M, Malossi RM, Dos Santos AP, Levin Y. Lattice model of ionic liquid confined by metal electrodes. J Chem Phys 2018; 148:193829. [PMID: 30307233 DOI: 10.1063/1.5013337] [Citation(s) in RCA: 17] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We study, using Monte Carlo simulations, the density profiles and differential capacitance of ionic liquids confined by metal electrodes. To compute the electrostatic energy, we use the recently developed approach based on periodic Green's functions. The method also allows us to easily calculate the induced charge on the electrodes permitting an efficient implementation of simulations in a constant electrostatic potential ensemble. To speed up the simulations further, we model the ionic liquid as a lattice Coulomb gas and precalculate the interaction potential between the ions. We show that the lattice model captures the transition between camel-shaped and bell-shaped capacitance curves-the latter characteristic of ionic liquids (strong coupling limit) and the former of electrolytes (weak coupling). We observe the appearance of a second peak in the differential capacitance at ≈0.5 V for 2:1 ionic liquids, as the packing fraction is increased. Finally, we show that ionic size asymmetry decreases substantially the capacitance maximum, when all other parameters are kept fixed.
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Affiliation(s)
- Matheus Girotto
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - Rodrigo M Malossi
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - Alexandre P Dos Santos
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
| | - Yan Levin
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970, Porto Alegre, RS, Brazil and Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany
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20
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Naji A, Hejazi K, Mahgerefteh E, Podgornik R. Charged nanorods at heterogeneously charged surfaces. J Chem Phys 2018; 149:134702. [PMID: 30292214 DOI: 10.1063/1.5044391] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We study the spatial and orientational distribution of charged nanorods (rodlike counterions) as well as the effective interaction mediated by them between two plane-parallel surfaces that carry fixed (quenched) heterogeneous charge distributions. The nanorods are assumed to have an internal charge distribution, specified by a multivalent monopolar moment and a finite quadrupolar moment, and the quenched surface charge is assumed to be randomly distributed with equal mean and variance on the two surfaces. While equally charged surfaces are known to repel within the traditional mean-field theories, the presence of multivalent counterions has been shown to cause attractive interactions between uniformly charged surfaces due to the prevalence of strong electrostatic couplings that grow rapidly with the counterion valency. We show that the combined effects due to electrostatic correlations (caused by the coupling between the mean surface field and the multivalent, monopolar, charge valency of counterions) as well as the disorder-induced interactions (caused by the coupling between the surface disorder field and the quadrupolar moment of counterions) lead to much stronger attractive interactions between two randomly charged surfaces. The interaction profile turns out to be a nonmonotonic function of the intersurface separation, displaying an attractive minimum at relatively small separations, where the ensuing attraction can exceed the maximum strong-coupling attraction (produced by multivalent monopolar counterions between uniformly charged surfaces) by more than an order of magnitude.
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Affiliation(s)
- Ali Naji
- School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
| | - Kasra Hejazi
- School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
| | - Elnaz Mahgerefteh
- School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran
| | - Rudolf Podgornik
- School of Physical Sciences and Kavli Institute for Theoretical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
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21
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Wu H, Luijten E. Accurate and efficient numerical simulation of dielectrically anisotropic particles. J Chem Phys 2018; 149:134105. [DOI: 10.1063/1.5048203] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Huanxin Wu
- Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
| | - Erik Luijten
- Department of Physics and Astronomy, Northwestern University, Evanston, Illinois 60208, USA
- Department of Materials Science and Engineering, Northwestern University, Evanston, Illinois 60208, USA
- Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA
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22
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Breitsprecher K, Abele M, Kondrat S, Holm C. The effect of finite pore length on ion structure and charging. J Chem Phys 2018; 147:104708. [PMID: 28915735 DOI: 10.1063/1.4986346] [Citation(s) in RCA: 25] [Impact Index Per Article: 4.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/13/2022] Open
Abstract
Nanoporous supercapacitors play an important role in modern energy storage systems, and their modeling is essential to predict and optimize the charging behaviour. Two classes of models have been developed that consist of finite and infinitely long pores. Here, we show that although both types of models predict qualitatively consistent results, there are important differences emerging due to the finite pore length. In particular, we find that the ion density inside a finite pore is not constant, but increases linearly from the pore entrance to the pore end, where the ions form a strongly layered structure. This hinders a direct quantitative comparison between the two models. In addition, we show that although the ion density between the electrodes changes appreciably with the applied potential, this change has a minor effect on charging. Our simulations also reveal a complex charging behaviour, which is adsorption-driven at high voltages, but it is dominated either by co-ion desorption or by adsorption of both types of ions at low voltages, depending on the ion concentration.
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Affiliation(s)
- Konrad Breitsprecher
- Institute for Computational Physics, Universität Stuttgart, Allmandring 3, D-70569 Stuttgart, Germany
| | - Manuel Abele
- Institute for Computational Physics, Universität Stuttgart, Allmandring 3, D-70569 Stuttgart, Germany
| | - Svyatoslav Kondrat
- Institute for Computational Physics, Universität Stuttgart, Allmandring 3, D-70569 Stuttgart, Germany
| | - Christian Holm
- Institute for Computational Physics, Universität Stuttgart, Allmandring 3, D-70569 Stuttgart, Germany
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23
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Dos Santos AP, Girotto M, Levin Y. Simulations of Coulomb systems confined by polarizable surfaces using periodic Green functions. J Chem Phys 2018; 147:184105. [PMID: 29141417 DOI: 10.1063/1.4997420] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Abstract
We present an efficient approach for simulating Coulomb systems confined by planar polarizable surfaces. The method is based on the solution of the Poisson equation using periodic Green functions. It is shown that the electrostatic energy arising from the surface polarization can be decoupled from the energy due to the direct Coulomb interaction between the ions. This allows us to combine an efficient Ewald summation method, or any other fast method for summing over the replicas, with the polarization contribution calculated using Green function techniques. We apply the method to calculate density profiles of ions confined between the charged dielectric and metal surfaces.
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Affiliation(s)
- Alexandre P Dos Santos
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970 Porto Alegre, RS, Brazil
| | - Matheus Girotto
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970 Porto Alegre, RS, Brazil
| | - Yan Levin
- Instituto de Física, Universidade Federal do Rio Grande do Sul, Caixa Postal 15051, CEP 91501-970 Porto Alegre, RS, Brazil
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24
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Fahrenberger F, Hickey OA, Smiatek J, Holm C. The influence of charged-induced variations in the local permittivity on the static and dynamic properties of polyelectrolyte solutions. J Chem Phys 2015; 143:243140. [DOI: 10.1063/1.4936666] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Florian Fahrenberger
- Institute for Computational Physics, University of Stuttgart, Stuttgart 70569, Germany
| | - Owen A. Hickey
- Institute for Computational Physics, University of Stuttgart, Stuttgart 70569, Germany
| | - Jens Smiatek
- Institute for Computational Physics, University of Stuttgart, Stuttgart 70569, Germany
| | - Christian Holm
- Institute for Computational Physics, University of Stuttgart, Stuttgart 70569, Germany
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25
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Zhou J, Schmid F. Computer simulations of single particles in external electric fields. SOFT MATTER 2015; 11:6728-6739. [PMID: 26238433 DOI: 10.1039/c5sm01485a] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
Applying electric fields is an attractive way to control and manipulate single particles or molecules, e.g., in lab-on-a-chip devices. However, the response of nanosize objects in electrolyte solution to external fields is far from trivial. It is the result of a variety of dynamical processes taking place in the ion cloud surrounding charged particles and in the bulk electrolyte, and it is governed by an intricate interplay of electrostatic and hydrodynamic interactions. Already systems composed of one single particle in electrolyte solution exhibit a complex dynamical behaviour. In this review, we discuss recent coarse-grained simulations that have been performed to obtain a molecular-level understanding of the dynamic and dielectric response of single particles and single macromolecules to external electric fields. We address both the response of charged particles to constant fields (DC fields), which can be characterized by an electrophoretic mobility, and the dielectric response of both uncharged and charged particles to alternating fields (AC fields), which is described by a complex polarizability. Furthermore, we give a brief survey of simulation algorithms and highlight some recent developments.
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Affiliation(s)
- Jiajia Zhou
- School of Chemistry & Enviroment, Center of Soft Matter Physics and its Application, Beihang University, Xueyuan Road 37, Beijing 100191, China.
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26
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Pujos JS, Maggs AC. Convexity and Stiffness in Energy Functions for Electrostatic Simulations. J Chem Theory Comput 2015; 11:1419-27. [PMID: 26574353 DOI: 10.1021/acs.jctc.5b00023] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
We study the properties of convex functionals which have been proposed for the simulation of charged molecular systems within the Poisson-Boltzmann approximation. We consider the extent to which the functionals reproduce the true fluctuations of electrolytes and thus the one-loop correction to mean field theory-including the Debye-Hückel correction to the free energy of ionic solutions. We also compare the functionals for use in numerical optimization of a mean field model of a charged polymer and show that different functionals have very different stiffnesses leading to substantial differences in accuracy and speed.
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Affiliation(s)
- Justine S Pujos
- Laboratoire PCT, Gulliver CNRS-ESPCI UMR 7083 , 10 rue Vauquelin, 75231 Paris Cedex 05, France
| | - A C Maggs
- Laboratoire PCT, Gulliver CNRS-ESPCI UMR 7083 , 10 rue Vauquelin, 75231 Paris Cedex 05, France
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27
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Fahrenberger F, Holm C. Computing the Coulomb interaction in inhomogeneous dielectric media via a local electrostatics lattice algorithm. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2014; 90:063304. [PMID: 25615219 DOI: 10.1103/physreve.90.063304] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/18/2013] [Indexed: 06/04/2023]
Abstract
The local approach to computing electrostatic interactions proposed by Maggs and adapted by Rottler and Pasichnyk for molecular-dynamics simulations is extended to situations where the dielectric background medium is inhomogeneous. We furthermore correct a problem of the original algorithm related to the correct treatment of the global dipole moment, provide an error estimate for the accuracy of the algorithm, and suggest a different form of the treatment of the self-energy problem. Our implementation is highly scalable on many cores, and we have validated and compared its performance against theoretical predictions and simulation data obtained by other algorithmic approaches.
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Affiliation(s)
- F Fahrenberger
- Institut für Computerphysik, Universität Stuttgart, Allmandring 3, 70569 Stuttgart, Germany
| | - C Holm
- Institut für Computerphysik, Universität Stuttgart, Allmandring 3, 70569 Stuttgart, Germany
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