We present a new theoretical method for rapid calculation of the solvation free energy in water b... more We present a new theoretical method for rapid calculation of the solvation free energy in water by combining molecular simulation and the classical density functional theory (DFT). The DFT calculation is based on an accurate free-energy functional for water that incorporates the simulation results for long-range correlations and the fundamental measure theory for the molecular excluded-volume effects. The numerical performance of the theoretical method has been validated with simulation results and experimental data for the solvation free energies of halide (F À , Cl À , Br À , and I À ) and alkali (Li þ , Na þ , K þ , Rb þ , and Cs þ ) ions in water at ambient conditions. Because simulation is applied only to the particular thermodynamic condition of interest, the hybrid method is computationally much more efficient than conventional ways of solvation free energy calculations.
Most theoretical studies of an electrical double layer, which is formed by an electrolyte in cont... more Most theoretical studies of an electrical double layer, which is formed by an electrolyte in contact with a charged electrode, employ a primitive model in which the solvent is represented by a dielectric continuum. This implicit-solvent model is convenient because computations are comparatively simple. However, it suppresses oscillations in the density profiles of ionic species that result from the discreteness of the solvent molecules. Furthermore, the implicit-solvent model yields poor results for the capacitance. In comparison with experiment at fixed electrode charge density, it predicts a too small electrode potential, and the resultant capacitance is too large. This latter discrepancy can be compensated in part by postulating the existence of an often fictitious inner layer whose properties are parametrized to agree best with experiment. The use of an implicit solvent model and an inner layer helps in correlating experimental results but rests on a faulty microscopic picture. Unfortunately, explicit consideration of solvent molecules poses both theoretical and numerical difficulties and, as a result, studies using an explicit solvent model have been few and far between. In this study, we consider a simple nonprimitive or explicit solvent model in which each solvent molecule is represented by a dimer composed of touching positive and negative hard spheres, with a resulting dipole moment that is equal to that of a water molecule, and the ions are represented by charged hard spheres. The density profiles and charge−potential relationship of this model are examined using the classical density functional theory. We find that the introduction of an explicit solvent increases the electrode potential, at fixed electrode charge, without the need to postulate a parametrized inner layer. Because of the solvent polarity, the ion profiles become strong oscillatory and show local charge inversion near a highly charged electrode surface at all ion concentrations.
Entropic force has been a focus of many recent theoretical studies because of its fundamental imp... more Entropic force has been a focus of many recent theoretical studies because of its fundamental importance in solution thermodynamics and its close relevance to a broad range of practical applications. Whereas previous investigations are mostly concerned with the potential energy as a onedimensional function of the separation, here we propose a hybrid method for studying multidimensional systems by combining Monte Carlo simulation for the microscopic configurations of the solvent and the density functional theory for the free energy. We demonstrate that the hybrid method predicts the potential of mean force between a test particle and various concave objects in a hard-sphere solvent in excellent agreement with the results from alternative but more expensive computational methods. In particular, the hybrid method captures the entropic force between asymmetric particles and its dependence on the particle size and shape that underlies the "lock and key" interactions. Because the same molecular model is used for the theory and simulation, we expect that the hybrid method provides a new avenue to efficient computation of entropic forces in complex molecular systems.
Ionic liquids have attracted much recent theoretical interest for broad applications as environme... more Ionic liquids have attracted much recent theoretical interest for broad applications as environmentallyfriendly solvents in separation and electrochemical processes. Because of the intrinsic complexity of organic ions and strong electrostatic correlations, the electrochemical properties of ionic liquids often defy the descriptions of conventional mean-field methods including the venerable, and over-used, Gouy-Chapman-Stern (GCS) theory. Classical density functional theory (DFT) has proven to be useful in previous studies of the electrostatic properties of aqueous electrolytes but until recently it has not been applied to ionic liquids. Here we report predictions from the DFT on the interfacial properties of ionic liquids near neutral or charged surfaces. By considering the molecular size, topology, and electrostatic correlations, we have examined major factors responsible for the unique features of electric-double layers of ionic-liquid including formation of long-range and alternating structures of cations and anions at charged surfaces.
Thermodynamic properties of a single chain in a confined space have been studied before with the ... more Thermodynamic properties of a single chain in a confined space have been studied before with the polymer scaling theory and computer simulations. However, a comprehensive understanding of the entropic effects due to the molecular excluded volume and chain connectivity is emerging only recently, especially in the limit of large polymer packing densities as often encountered in biological systems. In this work, we propose a polymer density functional theory ͑DFT͒ to study the entropic forces for the confinement of single polymer chains in spherical cavities. At conditions accessible to Monte Carlo simulations, we show that the DFT predictions are in excellent agreement with the simulation results for the distributions of polymer segments as well as the free energy of confinement. The numerical efficiency of the DFT allows us to unify key conclusions from various theoretical analyses and experimental observations.
b S Supporting Information E lectric energy storage becomes increasingly imperative as more elect... more b S Supporting Information E lectric energy storage becomes increasingly imperative as more electricity is generated from renewable sources such as solar and wind at remote areas. Supercapacitors (also known as electric double layer capacitors or ultracapacitors) have the advantages of high power density, fast charging/discharging cycles (within seconds), and extremely long cycle life. 1,2 These advantages make supercapacitors attractive for broad applications such as in memory backup power and in the gasoline/electric hybrid bus. 3 Carbon materials are promising as cost-effective electrode materials for supercapacitors. 4 Beyond the traditional activated carbons, 5 new forms such as carbide-derived carbons, 6,7 graphene, 2,8 and carbon onions 9 have been used for supercapacitors that display distinct characteristics such as ultrathin form 8 and ultrahigh power. 9 An interesting observation that has attracted great attention recently is the "anomalous" increase of the capacitance as the pore size approaches the electrolyteÀion dimensions. 6 This anomalous increase is observed not only in the traditional electrolyte such as tetraethylammonium tetrafluoroborate in acetonitrile (TEA-BF 4 /ACN) 6 but also in ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl) imide (EMIM-TFSI). 10 The pore-size dependence of capacitance has been explained in terms of desolvation in the case of the traditional electrolyte 11À13 and image forces that exponentially screen the electrostatic interaction among ions within the narrow pore. 14À16 Atomistic molecular dynamics simulation of a supercapacitor made of an ionic liquid confined inside a nanotube also showed the increase in capacitance as the diameter of the tube decreases to the ion size. 17 However, it remains unclear how the capacitance changes with the pore size in the intermediate poresize range (that is, from one to multiple times the electrolyteÀ ion diameters). Understanding the dependence of the capacitance on the pore size for these mesosclaes would facilitate developing mesoporous and hierarchical porous carbons for supercapacitors. 18À20 Recently, Centeno et al. 21 measured the capacitances of 28 carbon electrodes of very different average pore sizes, ranging from 0.7 to 15 nm. They found that the surface-area normalized capacitance randomly fluctuates around a relatively constant value; they did not observe the anomalous increase of capacitance. The disagreement with the previous experiments 6,10 has profound implication for practical applications of supercapacitors and thus calls for a deeper understanding of the size dependence of the capacitance in the micropore (<2 nm) to mesopore (2À50 nm) ranges where one expects overlapping of electric double layers.
Conventional wisdom for controlling the nanoparticle size and shape during synthesis is that part... more Conventional wisdom for controlling the nanoparticle size and shape during synthesis is that particle growth favors the direction of a facet with the highest surface energy. However, the particle solvation free energy, which dictates the particle stability and growth, depends not only on the surface area and surface free energy but also on other geometric measures such as the solvent excluded volume and the surface curvature and their affiliated thermodynamic properties. In this work, we study the geometrical effects on the solvation free energies of nonspherical nanoparticles using morphometric thermodynamics and density functional theories. For idealized systems that account for only molecular excluded-volume interactions, morphometric thermodynamics yields a reliable solvation free energy when the particle size is significantly larger than the solvent correlation length. However, noticeable deviations can be identified in comparison to the microscopic theories for predicting the solvation free energies of small nanoparticles. This conclusion also holds for predicting the potential of mean force underlying the colloidal "key-and-lock" interactions. Complementary to the microscopic theories, morphometric thermodynamics requires negligible computational cost, therefore making it very appealing for a broad range of practical applications.
Fourier space approach to the classical density functional theory for multi-Yukawa and square-wel... more Fourier space approach to the classical density functional theory for multi-Yukawa and square-well fluids A new lattice density functional theory for polymer adsorption at solid-liquid interface Density and chain conformation profiles of square-well chains confined in a slit by density-functional theory
Preferential interaction between DNA and small ions in mixed-size counterion systems: Monte Carlo... more Preferential interaction between DNA and small ions in mixed-size counterion systems: Monte Carlo simulation and density functional study Genome packaging inside viral capsids is strongly influenced by the molecular size and the backbone structure of RNA/DNA chains and their electrostatic affinity with the capsid proteins. Coarsegrained models are able to capture the generic features of non-specific interactions and provide a useful testing ground for theoretical developments. In this work, we use the classical density functional theory (DFT) within the framework of an extended primitive model for electrolyte solutions to investigate the self-organization of flexible and semi-flexible linear polyelectrolytes in spherical capsids that are permeable to small ions but not polymer segments. We compare the DFT predictions with Monte Carlo (MC) simulation for the density distributions of polymer segments and small ions at different backbone flexibilities and several solution conditions. In general, the agreement between DFT and MC is near quantitative except when the simulation results are noticeably influenced by the boundary effects. The numerical efficiency of the DFT calculations makes it promising as a useful tool for quantification of the structural and thermodynamic properties of viral nucleocapsids in vivo and at conditions pertinent to experiments.
Organic electrolytes such as tetraethylammonium tetrafluoroborate dissolved in acetonitrile (TEA-... more Organic electrolytes such as tetraethylammonium tetrafluoroborate dissolved in acetonitrile (TEA-BF 4 /ACN) are widely used in commercial supercapacitors and academic research, but conflicting experimental results have been reported regarding the dependence of surface-area-normalized capacitance on the pore size. Here we show from a classical density functional theory the dependence of capacitance on the pore size from 0.5 to 3.0 nm for a model TEA-BF 4 /ACN electrolyte. We find that the capacitance−pore size curve becomes roughly flat after the first peak around the ion diameter, and the peak capacitance is not significantly higher than the large-pore average. We attribute the invariance of capacitance with the pore size to the formation of an electric double-layer structure that consists of counterions and highly organized solvent molecules. This work highlights the role of the solvent molecules in modulating the capacitance and reconciles apparently conflicting experimental reports. SECTION: Energy Conversion and Storage; Energy and Charge Transport
We present a new theoretical method for rapid calculation of the solvation free energy in water b... more We present a new theoretical method for rapid calculation of the solvation free energy in water by combining molecular simulation and the classical density functional theory (DFT). The DFT calculation is based on an accurate free-energy functional for water that incorporates the simulation results for long-range correlations and the fundamental measure theory for the molecular excluded-volume effects. The numerical performance of the theoretical method has been validated with simulation results and experimental data for the solvation free energies of halide (F À , Cl À , Br À , and I À ) and alkali (Li þ , Na þ , K þ , Rb þ , and Cs þ ) ions in water at ambient conditions. Because simulation is applied only to the particular thermodynamic condition of interest, the hybrid method is computationally much more efficient than conventional ways of solvation free energy calculations.
Most theoretical studies of an electrical double layer, which is formed by an electrolyte in cont... more Most theoretical studies of an electrical double layer, which is formed by an electrolyte in contact with a charged electrode, employ a primitive model in which the solvent is represented by a dielectric continuum. This implicit-solvent model is convenient because computations are comparatively simple. However, it suppresses oscillations in the density profiles of ionic species that result from the discreteness of the solvent molecules. Furthermore, the implicit-solvent model yields poor results for the capacitance. In comparison with experiment at fixed electrode charge density, it predicts a too small electrode potential, and the resultant capacitance is too large. This latter discrepancy can be compensated in part by postulating the existence of an often fictitious inner layer whose properties are parametrized to agree best with experiment. The use of an implicit solvent model and an inner layer helps in correlating experimental results but rests on a faulty microscopic picture. Unfortunately, explicit consideration of solvent molecules poses both theoretical and numerical difficulties and, as a result, studies using an explicit solvent model have been few and far between. In this study, we consider a simple nonprimitive or explicit solvent model in which each solvent molecule is represented by a dimer composed of touching positive and negative hard spheres, with a resulting dipole moment that is equal to that of a water molecule, and the ions are represented by charged hard spheres. The density profiles and charge−potential relationship of this model are examined using the classical density functional theory. We find that the introduction of an explicit solvent increases the electrode potential, at fixed electrode charge, without the need to postulate a parametrized inner layer. Because of the solvent polarity, the ion profiles become strong oscillatory and show local charge inversion near a highly charged electrode surface at all ion concentrations.
Entropic force has been a focus of many recent theoretical studies because of its fundamental imp... more Entropic force has been a focus of many recent theoretical studies because of its fundamental importance in solution thermodynamics and its close relevance to a broad range of practical applications. Whereas previous investigations are mostly concerned with the potential energy as a onedimensional function of the separation, here we propose a hybrid method for studying multidimensional systems by combining Monte Carlo simulation for the microscopic configurations of the solvent and the density functional theory for the free energy. We demonstrate that the hybrid method predicts the potential of mean force between a test particle and various concave objects in a hard-sphere solvent in excellent agreement with the results from alternative but more expensive computational methods. In particular, the hybrid method captures the entropic force between asymmetric particles and its dependence on the particle size and shape that underlies the "lock and key" interactions. Because the same molecular model is used for the theory and simulation, we expect that the hybrid method provides a new avenue to efficient computation of entropic forces in complex molecular systems.
Ionic liquids have attracted much recent theoretical interest for broad applications as environme... more Ionic liquids have attracted much recent theoretical interest for broad applications as environmentallyfriendly solvents in separation and electrochemical processes. Because of the intrinsic complexity of organic ions and strong electrostatic correlations, the electrochemical properties of ionic liquids often defy the descriptions of conventional mean-field methods including the venerable, and over-used, Gouy-Chapman-Stern (GCS) theory. Classical density functional theory (DFT) has proven to be useful in previous studies of the electrostatic properties of aqueous electrolytes but until recently it has not been applied to ionic liquids. Here we report predictions from the DFT on the interfacial properties of ionic liquids near neutral or charged surfaces. By considering the molecular size, topology, and electrostatic correlations, we have examined major factors responsible for the unique features of electric-double layers of ionic-liquid including formation of long-range and alternating structures of cations and anions at charged surfaces.
Thermodynamic properties of a single chain in a confined space have been studied before with the ... more Thermodynamic properties of a single chain in a confined space have been studied before with the polymer scaling theory and computer simulations. However, a comprehensive understanding of the entropic effects due to the molecular excluded volume and chain connectivity is emerging only recently, especially in the limit of large polymer packing densities as often encountered in biological systems. In this work, we propose a polymer density functional theory ͑DFT͒ to study the entropic forces for the confinement of single polymer chains in spherical cavities. At conditions accessible to Monte Carlo simulations, we show that the DFT predictions are in excellent agreement with the simulation results for the distributions of polymer segments as well as the free energy of confinement. The numerical efficiency of the DFT allows us to unify key conclusions from various theoretical analyses and experimental observations.
b S Supporting Information E lectric energy storage becomes increasingly imperative as more elect... more b S Supporting Information E lectric energy storage becomes increasingly imperative as more electricity is generated from renewable sources such as solar and wind at remote areas. Supercapacitors (also known as electric double layer capacitors or ultracapacitors) have the advantages of high power density, fast charging/discharging cycles (within seconds), and extremely long cycle life. 1,2 These advantages make supercapacitors attractive for broad applications such as in memory backup power and in the gasoline/electric hybrid bus. 3 Carbon materials are promising as cost-effective electrode materials for supercapacitors. 4 Beyond the traditional activated carbons, 5 new forms such as carbide-derived carbons, 6,7 graphene, 2,8 and carbon onions 9 have been used for supercapacitors that display distinct characteristics such as ultrathin form 8 and ultrahigh power. 9 An interesting observation that has attracted great attention recently is the "anomalous" increase of the capacitance as the pore size approaches the electrolyteÀion dimensions. 6 This anomalous increase is observed not only in the traditional electrolyte such as tetraethylammonium tetrafluoroborate in acetonitrile (TEA-BF 4 /ACN) 6 but also in ionic liquid 1-ethyl-3-methylimidazolium bis(trifluoromethylsulfonyl) imide (EMIM-TFSI). 10 The pore-size dependence of capacitance has been explained in terms of desolvation in the case of the traditional electrolyte 11À13 and image forces that exponentially screen the electrostatic interaction among ions within the narrow pore. 14À16 Atomistic molecular dynamics simulation of a supercapacitor made of an ionic liquid confined inside a nanotube also showed the increase in capacitance as the diameter of the tube decreases to the ion size. 17 However, it remains unclear how the capacitance changes with the pore size in the intermediate poresize range (that is, from one to multiple times the electrolyteÀ ion diameters). Understanding the dependence of the capacitance on the pore size for these mesosclaes would facilitate developing mesoporous and hierarchical porous carbons for supercapacitors. 18À20 Recently, Centeno et al. 21 measured the capacitances of 28 carbon electrodes of very different average pore sizes, ranging from 0.7 to 15 nm. They found that the surface-area normalized capacitance randomly fluctuates around a relatively constant value; they did not observe the anomalous increase of capacitance. The disagreement with the previous experiments 6,10 has profound implication for practical applications of supercapacitors and thus calls for a deeper understanding of the size dependence of the capacitance in the micropore (<2 nm) to mesopore (2À50 nm) ranges where one expects overlapping of electric double layers.
Conventional wisdom for controlling the nanoparticle size and shape during synthesis is that part... more Conventional wisdom for controlling the nanoparticle size and shape during synthesis is that particle growth favors the direction of a facet with the highest surface energy. However, the particle solvation free energy, which dictates the particle stability and growth, depends not only on the surface area and surface free energy but also on other geometric measures such as the solvent excluded volume and the surface curvature and their affiliated thermodynamic properties. In this work, we study the geometrical effects on the solvation free energies of nonspherical nanoparticles using morphometric thermodynamics and density functional theories. For idealized systems that account for only molecular excluded-volume interactions, morphometric thermodynamics yields a reliable solvation free energy when the particle size is significantly larger than the solvent correlation length. However, noticeable deviations can be identified in comparison to the microscopic theories for predicting the solvation free energies of small nanoparticles. This conclusion also holds for predicting the potential of mean force underlying the colloidal "key-and-lock" interactions. Complementary to the microscopic theories, morphometric thermodynamics requires negligible computational cost, therefore making it very appealing for a broad range of practical applications.
Fourier space approach to the classical density functional theory for multi-Yukawa and square-wel... more Fourier space approach to the classical density functional theory for multi-Yukawa and square-well fluids A new lattice density functional theory for polymer adsorption at solid-liquid interface Density and chain conformation profiles of square-well chains confined in a slit by density-functional theory
Preferential interaction between DNA and small ions in mixed-size counterion systems: Monte Carlo... more Preferential interaction between DNA and small ions in mixed-size counterion systems: Monte Carlo simulation and density functional study Genome packaging inside viral capsids is strongly influenced by the molecular size and the backbone structure of RNA/DNA chains and their electrostatic affinity with the capsid proteins. Coarsegrained models are able to capture the generic features of non-specific interactions and provide a useful testing ground for theoretical developments. In this work, we use the classical density functional theory (DFT) within the framework of an extended primitive model for electrolyte solutions to investigate the self-organization of flexible and semi-flexible linear polyelectrolytes in spherical capsids that are permeable to small ions but not polymer segments. We compare the DFT predictions with Monte Carlo (MC) simulation for the density distributions of polymer segments and small ions at different backbone flexibilities and several solution conditions. In general, the agreement between DFT and MC is near quantitative except when the simulation results are noticeably influenced by the boundary effects. The numerical efficiency of the DFT calculations makes it promising as a useful tool for quantification of the structural and thermodynamic properties of viral nucleocapsids in vivo and at conditions pertinent to experiments.
Organic electrolytes such as tetraethylammonium tetrafluoroborate dissolved in acetonitrile (TEA-... more Organic electrolytes such as tetraethylammonium tetrafluoroborate dissolved in acetonitrile (TEA-BF 4 /ACN) are widely used in commercial supercapacitors and academic research, but conflicting experimental results have been reported regarding the dependence of surface-area-normalized capacitance on the pore size. Here we show from a classical density functional theory the dependence of capacitance on the pore size from 0.5 to 3.0 nm for a model TEA-BF 4 /ACN electrolyte. We find that the capacitance−pore size curve becomes roughly flat after the first peak around the ion diameter, and the peak capacitance is not significantly higher than the large-pore average. We attribute the invariance of capacitance with the pore size to the formation of an electric double-layer structure that consists of counterions and highly organized solvent molecules. This work highlights the role of the solvent molecules in modulating the capacitance and reconciles apparently conflicting experimental reports. SECTION: Energy Conversion and Storage; Energy and Charge Transport
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