Free Energy Plots Research Articles

Assessing the stability of binary copper oxide photocathodes and augmenting CuO water reduction performance: Comprehensive experimental and theoretical study

Using photocathodes such as Cu2O and CuO is an effective strategy for direct hydrogen production in photoelectrochemical (PEC) water splitting. The theoretical photocurrent densities of Cu2O and CuO are ∼14.7 and ∼35 mA cm−2, respectively, suggesting the superiority of CuO over Cu2O. However, the foremost drawback is the poor stability in aqueous solution under illumination. From this perspective, the PEC performance of CuO and Cu2O is compared via experimental and theoretical approaches. In this case, CuO outperformed Cu2O in terms of both PEC performance and stability, and the density functional theory (DFT) studies also evidently showed the same. Further, the AlOx and RuO2 were also loaded over CuO, and the CuO/AlOx/RuO2 delivered a photocurrent density of ∼2 mA cm−2 (@ +0.3 VRHE). The free energy plot shows the lower energy barrier of CuO/AlOx/RuO2 than CuO/AlOx and CuO, and it is present near and left side of the standard volcano plot, which further gives evidence for higher hydrogen evolution reaction (HER) performance.

Folding Free Energy Surfaces from Differential Scanning Calorimetry.

Protein folding/unfolding processes involve a large number of weak, non-covalent interactions and are more appropriately described in terms of the movement of a point representing protein conformation in a plot of internal free energy versus conformational degrees of freedom. While these energy landscapes have an astronomically large number of dimensions, it has been shown that many relevant aspects of protein folding can be understood in terms of their projections onto a few relevant coordinates. Remarkably, such low-dimensional free energy surfaces can be obtained from experimental DSC data using suitable analytical models. Here, we describe the experimental procedures to be used to obtain the high-quality DSC data that are required for free-energy surface analysis.

Exploring the Ion Solvation Environments in Solid-State Polymer Electrolytes through Free-Energy Sampling

The success of poly(ethylene oxide) (PEO) in solid-state polymer electrolytes for lithium-ion batteries is well established. Recently, in order to understand this success and to explore possible alternatives, we studied polyacetal electrolytes to deepen the understanding of the effect of the local chemical structure on ion transport. Advanced molecular dynamics techniques using newly developed, tailored interaction potentials have helped elucidate the various coordination environments of ions in these systems. In particular, the competition between cation–anion pairing and coordination by the polymer has been explored using free-energy sampling (metadynamics). At equivalent reduced temperatures, with respect to the polymer-specific glass-transition temperature, two-dimensional free-energy plots reveal the existence of multiple coordination environments for the lithium (Li) ions in these systems and their relative stabilities. Furthermore, we observe that the Li-ion movement in PEO follows a serial, stepwise pathway when moving from one coordination state to another, whereas this happens in a more continuous and concerted fashion in a polyacetal such as poly(1,3-dioxalane) [P(EO-MO)]. The implication is that interconversion between coordination states of the Li ions may be easier in P(EO-MO). However, the overarching observation from our free-energy analysis is that Li-ion coordination is dominated by the polymer (in either case) and contact-ion pairs are rare. We rationalize the observed higher increase in glass-transition temperature (Tg) with salt loading in polyacetals as due to intermolecular Li-ion coordination involving multiple polymer chains, rather than just one chain for PEO-based electrolytes. This interchain coupling in the polyacetals, resulting in the higher Tg, works against any gains due to variations in Li-ion coordination that might enhance transport processes over PEO. Further research is required to overcome the interdependence between local coordination and macroscopic properties to compete with PEO electrolytes at the same absolute working temperature.

Gasification of Waste Tyres: Thermodynamic Attainable Region Approach

The innovative G-H graphical technique, a plot of Enthalpy vs Gibbs free energy was utilized to obtain a thermodynamically attainable region (AR) for the gasification of waste tyres. The AR is used to examine the interaction between the competing reactions in a gasifier and used to identify optimal targets for the conversion of waste tyres. The objective is to investigate the effect of temperature on the product selectivity. a temperature range of 25-1500°C at 1 bar was used for the analysis. The results show that at temperatures from 200°C to 600°C methane and carbon dioxide are dominant products at minimum Gibbs free energy. However, as the temperature increases, methane production decreases and hydrogen production become more favourable. Between 600°C and 700°C, carbon dioxide and hydrogen are dominant products. The AR results show that the products of gasification (CO and H2) are preferred products at minimum Gibbs free energy only at temperatures from 800°C to 1500°C, when both water and oxygen are used as oxidants. Therefore, syngas production from tyres is only feasible at high temperatures. Temperatures above 1000°C are recommended to prevent the formation of intermediate radicals.

Microfluidic investigation of the effect of graphene oxide on mechanical properties of cell and actin cytoskeleton networks: experimental and theoretical approaches

Biomechanical and morphological analysis of the cells is a novel approach for monitoring the environmental features, drugs, and toxic compounds’ effects on cells. Graphene oxide (GO) has a broad range of medical applications such as tissue engineering and drug delivery. However, the effects of GO nanosheets on biological systems have not been completely understood. In this study, we focused on the biophysical characteristics of cells and their changes resulting from the effect of GO nanosheets. The biophysical properties of the cell population were characterized as follows: cell stiffness was calculated by atomic force microscopy, cell motility and invasive properties were characterized in the microfluidic chip in which the cells are able to visualize cell migration at a single-cell level. Intracellular actin was stained to establish a quantitative picture of the intracellular cytoskeleton. In addition, to understand the molecular interaction of GO nanosheets and actin filaments, coarse-grained (CG) molecular dynamics (MD) simulations were carried out. Our results showed that GO nanosheets can reduce cell stiffness in MCF7 cells and MDA-MB-231 cell lines and highly inhibited cell migration (39.2%) in MCF-7 and (38.6%) in MDA-MB-231 cell lines through the GO nanosheets-mediated disruption of the intracellular cytoskeleton. In the presence of GO nanosheets, the cell migration of both cell lines, as well as the cell stiffness, significantly decreased. Moreover, after GO nanosheets treatment, the cell actin network dramatically changed. The experimental and theoretical approaches established a quantitative picture of changes in these networks. Our results showed the reduction of the order parameter in actin filaments was 23% in the MCF7 cell line and 20.4% in the MDA-MB-231 cell line. The theoretical studies also showed that the GO nanosheet–actin filaments have stable interaction during MD simulation. Moreover, the 2D free energy plot indicated the GO nanosheet can induce conformational changes in actin filaments. Our findings showed that the GO nanosheets can increase the distance of actin-actin subunits from 3.22 to 3.5 nm and in addition disrupt native contacts between two subunits which lead to separate actin subunits from each other in actin filaments. In this study, the biomechanical characteristics were used to explain the effect of GO nanosheets on cells which presents a novel view of how GO nanosheets can affect the biological properties of cells without cell death. These findings have the potential to be applied in different biomedical applications.

Study and characterization of potential adsorbent materials for the design of the hydrogen isotopes extraction and analysis system

One of the most challenging tasks in the design of the inner fuel cycle system lies in the effective design of Tritium Extraction System (TES), which involves proper extraction and purification of tritium in a fusion reactor. A prototype Hydrogen Isotopes Recovery System (HIRS) is being developed to validate the concepts for tritium extraction by adsorption mass transfer mechanism. The two main systems of HIRS are Atmospheric Molecular Sieve Bed (AMSB) adsorber and Cryogenic Molecular Sieve Bed (CMSB) adsorber. AMSB removes ppm levels of water vapour while CMSB removes ppm levels of hydrogen isotopes, oxygen and nitrogen gas from Helium purge gas. Selection of appropriate adsorbents for the HIRS is important for its efficient functioning. Adsorbents, namely, Zeolites 3A, 4A, 5A, 13X and Activated Carbon have been studied in detail at cryogenic temperatures to understand their surface characteristics and adsorption potential. In this work, we have generated the adsorption isotherms for hydrogen isotopes on potential adsorbents using a volumetric adsorption apparatus for a wide range of pressure from 1 Pa to 105 Pa. The BET and DFT models are applied for the determination of the textural properties of the adsorbents and the Langmuir-Freundlich composite model is used to fit the isotherm data. The parameters of the model were determined using nonlinear regression analysis. The enthalpy of adsorption is determined and is in the range of 8–11.5 kJ/mol for the adsorbate loadings corresponding to the partial pressure of hydrogen isotopes of ∼ 50–500 Pa in the TES. At low pressures of hydrogen isotopes, the free energy plot followed Henry’s law. The isotherms and enthalpy of adsorption clearly indicate reversible physisorption and monolayer formation of hydrogen isotopes on the potential adsorbents. Amongst hydrogen and deuterium, hydrogen is more strongly adsorbed on Zeolites 4A, whereas deuterium is more strongly adsorbed on Zeolites 13X. In case of activated carbon, no isotopic selectivity was observed. This study facilitates the selection of potential adsorbents for the CMSB and quantification of the parameters associated with the adsorbents.

Thermodynamics of cavity formation in different solvents: Enthalpy, entropy, and the solvophobic effects

For a long time thermodynamics of cavity formation in non-aqueous solvents remained virtually unexplored. We evaluated the entropic and enthalpic contributions into the Gibbs free energy of cavity formation in 22 structurally different organic solvents and water at 298 K. The calculations were carried out using 50–100 ns long molecular dynamics trajectories of solvents and Widom test particle insertion method with up to 1011 insertions per trajectory. It is shown that protic solvents with three-dimensional hydrogen bond networks, namely water, formamide, glycerol, ethylene glycol, propylene glycol, and diethylene glycol, behave differently from other considered solvents. They are characterized with lower values of the entropy of cavity formation than other considered solvents, which also leads to the higher Gibbs free energies. Water has a particularly low entropy as well as the lowest enthalpy of cavity formation at room temperature. We point out the similar tendencies observed for the thermodynamic functions of the cavities and solvation of real molecules in the considered solvents using entropy vs enthalpy and Gibbs free energy vs enthalpy plots. The cavity term governs the difference in solvation properties of protic hydrogen-bonded and aprotic solvents giving rise to the hydrophobic and more general solvophobic effects.

Application of Thermodynamic Phase Diagrams and Gibbs Free Energy of Mixing for Screening of Polymers for Their Use in Amorphous Solid Dispersion Formulation of a Non-Glass-Forming Drug.

The objective of the present investigations was to assess the use of thermodynamic phase diagrams and the Gibbs free energy of mixing, ΔGmix, for the screening of the polymeric carriers by determining the ideal drug-loading for an amorphous solid dispersion formulation and optimum processing temperature for the hot-melt extrusion of a non-glass-forming drug. Mefenamic acid (MFA) was used as a model non-glass-forming drug and four chemically distinct polymers with close values of the solubility parameters, viz. Kollidon® VA64, Soluplus®, Pluronic® F68, and Eudragit® EPO, were used as carriers. The thermodynamic phase diagrams were constructed using the melting point depression data, Flory-Huggins theory, and Gordan-Taylor equation. The Gibbs free energy of mixing was estimated using the values of the drug-polymer interaction parameter, χ, and Flory-Huggins theory. The rank order miscibility of MFA in the four polymeric carriers estimated based on the difference in the values of their solubility parameters, Δδ, did not correlate well with the thermodynamic phase diagrams and Gibbs free energy plots. The study highlights the limitation of using the solubility parameter method in screening the polymeric carriers for poorly glass-forming drugs and reiterates the applicability of thermodynamic phase diagrams and Gibbs free energy plots in determining the ideal drug-loading and optimum processing temperature for hot-melt extrusion.

Single Si‐doped fullerene as a catalyst in the oxygen reduction reaction: A quantum chemical insight

AbstractThe ability of the Si‐doped fullerene C59Si to catalyze the oxygen reduction reaction (ORR) in acidic and alkaline media has been studied by density functional theory calculations. The active sites for oxygen adsorption were determined from the distribution of the charge density difference analysis. The Si‐doped fullerene C59Si has the global minimum at the structure HO*O*C59Si generated by only one hydrogen transfer [H+ + e−] at the equilibrium potential, but shifts to the structure HO*C59Si at overpotentials higher 0.8 V or below 0 V in acidic or alkaline media, respectively. The comparison of the free energy plots for the 2e and 4e pathways of ORR showed the preference of the latter over the former due to the formation of stable intermediates suffering the OO bond breaking.

Quantum corrections to AdS black hole in massive gravity

In this paper, we investigate the effect of quantum corrections on the thermodynamics of AdS black hole in massive gravity. We compute the leading order corrections to the entropy of AdS black hole and then plot the entropy as a function of event horizon radius so as to have a comparative analysis between the corrected and the uncorrected entropy densities. Furthermore, we also evaluate the first-order leading corrections to other thermodynamic quantities like free energy, internal energy, pressure, enthalpy, and Gibbs free energy and later plot these quantities against event horizon radius for different values of correction parameter in order to have a qualitative picture of the effect that quantum corrections lead to the thermodynamics of massive AdS black hole.

Bubble Formation in a Finite Cone: More Pieces to the Puzzle.

We investigate the stability of bubble formation, starting with a convex or a concave meniscus, from a liquid solution (of water and a dissolved gas) inside a finite cone at constant temperature and constant liquid pressure (above the saturation pressure of the pure solvent). It is assumed that the dissolved gas (nitrogen) forms a dilute solution at equilibrium, which can be described by Henry's law. The number and nature of equilibrium states are determined with Gibbsian composite-system thermodynamics, both from the intersection of the equilibrium Kelvin radius with the geometry radius and from the extrema in the plot of free energy of the system versus size of the new phase. Bubble stability is studied along the whole growth path, as the bubble grows inside, gets pinned, and grows further outside the finite cone. The changes in the concentration of the liquid bulk phase and the vapor phase during the growth of the bubble are carefully incorporated in the equations. The effects of various parameters, including cone apex angle, cone half mouth radius, contact angle, total number of moles, and initial degree of saturation, on the stability of the bubble are also investigated. Stability of bubble formation from a liquid solution inside a confined geometry such as a finite cone is of interest in areas such as restoring underwater superhydrophobicity and adhesion of particles to the roughness of synthetic biomaterials.

Electronic properties and oxygen reduction reaction catalytic activity of h-BeN2 and MgN2 by first-principles calculations

Currently, identifying suitable oxygen reduction reaction (ORR) catalysts in novel two-dimensional (2D) materials has attracted more and more research attention. Here, we have studied the catalytic activities of 2D h-BeN2 and MgN2 monolayers for ORR by using first-principles calculations. The calculated results reveal that the direct quasiparticle bandgap of BeN2 monolayer is 3.32 eV, and the indirect bandgap of MgN2 is 3.42 eV. 2D h-BeN2 and MgN2 exhibit high exciton binding energies of 1.07 and 0.83 eV respectively, and their optical properties are determined by bound exciton transitions due to the strong quantum confinement effects. Importantly, h-BeN2 and MgN2 monolayers with positive-charged (+1.6 e) metal atom (Be/Mg) on the surface exhibit excellent adsorption ability for O2 and ORR intermediates, and show better CO tolerance than Pt(111). The calculated free energy plots are always downhill for ORR catalyzed by BeN2 in both acid and alkaline environments, and by MgN2 in alkaline environments. The detailed reaction mechanism analyses show that high-efficient four-electron pathway is the optimal pathway for ORR catalyzed by BeN2 in acid environments. Surprisingly, there is a low overpotential of 0.45 eV for ORR catalyzed by BeN2 in the acid solution and no overpotential in the alkaline solution. Our studies found for the first time that 2D h-BeN2 shows huge potential as a non-precious metal ORR catalyst in acid and alkaline environments.

How to Boost the Activity of the Monolayer Pt Supported on TiC Catalysts for Oxygen Reduction Reaction: A Density Functional Theory Study.

Developing the optimized electrocatalysts with high Pt utilization as well as the outstanding performance for the oxygen reduction reaction (ORR) has raised great attention. Herein, the effects of the interlayer ZrC, HfC, or TiN and the multilayer Pt shell on the adsorption ability and the catalytic activity of the TiC@Pt core-shell structures are systemically investigated by density functional theory (DFT) calculations. For the sandwich structures, the presence of TiN significantly enhances the adsorption ability of the Pt shell, leading to the deterioration of the activity whilst the negligible influence of the ZrC and HfC insertion results the comparable performance with respect to TiC@Pt1ML. In addition, increasing the thickness of the Pt shell reduces the oxyphilic capacity and then mitigates the OH poisoning. From the free energy plots, the superior activity of TiC@Pt2ML is identified in comparison with 1ML and 3ML Pt shell. Herein, the improved activity with its high Pt atomic utilization makes the potential TiC@Pt2ML electrocatalyst for the future fuel cells.

Developing Hispolon-based novel anticancer therapeutics against human (NF-κβ) using in silico approach of modelling, docking and protein dynamics

Hispolon is a polyphenolic compound derived from black hoof mushroom (Phellinus linteus) or shaggy bracket mushroom (Inonotus hispidus) which induces the inhibition of cancer-promoting nuclear factor-kappa beta (NF-κβ) complex. To develop more potent lead molecules with enhanced anticancer efficiency, the mechanism of hispolon-mediated nuclear factor-κβ inhibition has been investigated by molecular modelling and docking. Ten derivatives of hispolon (DRG1-10) have been developed by pharmacophore-based design with a view to enhance the anticancer efficacy. Hispolon and its derivatives were further screened for different pharmacological parameters like binding free energy, drug likeliness, absorption–digestion–metabolism–excretion (ADME), permeability, mutagenicity, toxicity and inhibitory concentration 50 (IC50) to find a potent lead molecule. Based on pharmacological validation, comparative molecular dynamics (MD) simulations have been performed for three lead molecules: Hispolon, DRG2 and DRG7 complexed with human NF-κβ up to 50 ns. By analysing different factors like root mean square deviation (RMSD), root mean square fluctuation (RMSF), radius of gyration (Rg), solvent-accessible surface area (SASA) and principal component analysis (PCA), Gibb’s free energy plots DRG2 have more binding efficiency compared to hispolon and DRG7. In RMSD plot, hispolon-bound NF-κβ has the most deviation within a range between 0.125 and 0.45 nm, and DRG2-bound complex showed the range between 0.125 and 0.25 nm. The residues of NF-κβ responsible for hydrophobic interactions with ligand, e.g. Met469, Leu522 and Cys533, have the lowest fluctuation values in DRG2-bound complex. The average Rg fluctuation for DRG2-bound NF-κβ has been recorded under 2.025 nm for most of the simulation time which is much less compared to hispolon and DRG7. Gibb’s free energy plots also define the highest stability of DRG2-bound NF-κβ.Communicated by Ramaswamy H. Sarma

Tuning Enzyme/α-Zr(IV) Phosphate Nanoplate Interactions via Chemical Modification of Glucose Oxidase.

Using glucose oxidase (GOx) and α-Zr(IV) phosphate nanoplates (α-ZrP) as a model system, a generally applicable approach to control enzyme-solid interactions via chemical modification of amino acid side chains of the enzyme is demonstrated. Net charge on GOx was systematically tuned by appending different amounts of polyamine to the protein surface to produce chemically modified GOx(n), where n is the net charge on the enzyme after the modification and ranged from -62 to +95 electrostatic units in the system. The binding of GOx(n) with α-ZrP nanosheets was studied by isothermal titration calorimetry (ITC) as well as by surface plasmon resonance (SPR) spectroscopy. Pristine GOx showed no affinity for the α-ZrP nanosheets, but GOx(n) where n ≥ -20 showed binding affinities exceeding (2.1 ± 0.6) × 106 M-1, resulting from the charge modification of the enzyme. A plot of GOx(n) charge vs Gibbs free energy of binding (ΔG) for n = +20 to n = +65 indicated an overall increase in favorable interaction between GOx(n) and α-ZrP nanosheets. However, ΔG is less dependent on the net charge for n > +45, as evidenced by the decrease in the slope as charge increased further. All modified enzyme samples and enzyme/α-ZrP complexes retained a significant amount of folding structure (examined by circular dichroism) as well as enzymatic activities. Thus, strong control over enzyme-nanosheet interactions via modulating the net charge of enzymes may find potential applications in biosensing and biocatalysis.

Coalescence of Nanoclusters Analyzed by Well-Tempered Metadynamics. Comparison with Straightforward Molecular Dynamics.

The coalescence process of two nanoparticles to yield a core-shell structure is analyzed by a well-tempered metadynamics procedure. This methodology has been shown to be useful in understanding the present phenomenon in terms of two collective variables: the distance between the center of mass of the coalescing particles and the gyration radius of the resulting core element. The free-energy contour plots clearly show that the coalescence process involves the deformation of the core material, which is manifested in the residence of the system in regions with a larger gyration radius. Results from molecular dynamics for the same system were found helpful to reach the definition of this second collective variable. The advantages and limitations of the latter approach are discussed.

Understanding Chemical Equilibrium: The Role of Gas Phases and Mixing Contributions in the Minimum of Free Energy Plots

The use of free energy plots to understand the concept of thermodynamic equilibrium has been shown to be of great pedagogical value in materials science. Although chemical equilibrium is also amenable to this kind of analysis, it is not part of the agenda of materials science textbooks. Something similar is found in chemistry branches, where free energy plots in the context of chemical equilibrium are occasionally addressed, in qualitative fashion, and with a main focus on gas phase reactions. With the aim of providing a more complete perspective on the topic, free energy plots in several reactive systems that include condensed and gas phase components are analyzed. Free energy functions of the reactive systems are assembled using expressions of chemical potentials as building blocks, a useful approach to articulate several layers of concepts (fugacity coefficients, activity coefficients, solution thermodynamics) developed in earlier stages of thermodynamic courses. The examples presented highlight the in...

Applying thermodynamics to digestion/gasification processes: the Attainable Region approach

The research shows theoretical calculations on the thermodynamics of digestion/gasification processes where glucose is used as a surrogate for biomass. The change in Enthalpy (∆H) and Gibbs Free Energy (∆G) is used to obtain the Attainable Region (AR) that shows the overall thermodynamic limits for digestion/gasification from 1 mol of glucose. Gibbs Free Energy and Enthalpy (G–H) plots were calculated for the temperature range 25–1500 °C. The results show the effect of temperature on the AR for the processes when water is in both liquid and gas states using 25 °C, 1 bar as the reference state. The AR results show that the production of CO, H2, CH4 and CO2 are feasible at all temperatures studied. The minimum Gibbs Free Energy becomes more negative from −418.68 kJ mol−1 at 25 °C to −3024.34 kJ mol−1 at 1500 °C while the process shifts from exothermic (−141.90 kJ mol−1) to endothermic (1161.80 kJ mol−1) for the respective temperatures. Methane and carbon dioxide are favoured products (minimum Gibbs Free Energy) for temperatures up to about 600 °C, and this therefore includes Anaerobic Digestion. The process is exothermic below 500 °C, and thus Anaerobic Digestion requires heat removal. As the temperature continues to increase, hydrogen production becomes more favourable than methane production. The production of gas is endothermic above 500 °C, and it needs a supply of heat that could be done, either by combustion or by electricity (plasma gasification). The calculations show that glucose conversion at temperatures around 700 °C favours the production of carbon dioxide and hydrogen at minimum G. Generally, the results show that the gas from high-temperature gasification (>~800 °C) typically carries the energy mainly in syngas components CO and H2, whereas at low-temperature gasification (<500 °C) the energy is carried in CH4. The overall analysis for the temperature range (25–1500 °C) also suggests a close relationship between biogas production/digestion and gasification as biogas production can be referred to as a form of low-temperature gasification.

Intricate Phase Behavior and Crystal Structure Features of Chiral para-Methoxyphenyl Glycerol Ether Forming Continuous and Partial Solid Solutions

Heterogeneous equilibria, crystallization, and polymorphism of chiral para-methoxyphenyl glycerol ether 1 have been inspected, and, as a result, the binary phase diagram and the Gibbs free energy vs temperature plot were constructed and analyzed. This enantiomeric system forms a stable racemic compound, which turns into an almost ideal continuous solution of the enantiomers in the crystalline phase at elevated temperatures. At room temperature the system represents a stoichiometric racemic compound and two symmetrical eutectoid invariants with partial solid solutions based on the enantiomers. Crystal structures of the true racemate, the pseudoracemate, and the pure enantiomer were investigated by single crystal X-ray diffraction. The true racemate crystallizes in the Pc space group with Z′ = 2. The pseudoracemate was solved in the Pbcn group with the only independent molecule equally disordered into two mirror-related positions. The enantiomeric crystals belong to the P21212 group and are characterized by six symmetry independent molecules (Z′ = 6), two of which undergo disordering. We also discussed possible connection between the phase behavior features and the details of the crystal structure, in particular, bilayer supramolecular organization, pseudosymmetry, high Z′, and disordered packing. General considerations about the crystalline nature of solid solutions of enantiomers were also made.

Thermodynamics of Surface Nanobubbles.

In this paper, we examine the thermodynamic stability of surface nanobubbles. The appropriate free energy is defined for the system of nanobubbles on a solid surface submerged in a supersaturated liquid solution at constant pressure and temperature, under conditions where an individual nanobubble is not in diffusive contact with a gas phase outside of the system or with other nanobubbles on the time scale of the experiment. The conditions under which plots of free energy versus the radius of curvature of the nanobubbles show a global minimum, which denotes the stable equilibrium state, are explored. Our investigation shows that supersaturation and an anomalously high contact angle (measured through the liquid) are required to have stable surface nanobubbles. In addition, the anomalously high contact angle of surface nanobubbles is discussed from the standpoint of a framework recently proposed by Koch, Amirfazli, and Elliott that relates advancing and receding contact angles to thermodynamic equilibrium contact angles, combined with the existence of a gas enrichment layer.