The dual nature of metal halide perovskites.

Metal halide perovskites have brought about a disruptive shift in the field of third-generation photovoltaics. Their potential as remarkably efficient solar cell absorbers was first demonstrated in the beginning of the 2010s. However, right from their inception, persistent challenges have impeded the smooth adoption of this technology in the industry. These challenges encompass issues such as the lack of reproducibility in fabrication, limited mid- and long-term stability, and concerns over toxicity. Despite achieving record efficiencies that have outperformed even well-established technologies, such as polycrystalline silicon, these hurdles have hindered the seamless transition of this technology into industrial applications. In this Perspective, we discuss which of these challenges are rooted in the unique dual nature of metal halide perovskites, which simultaneously function as electronic and ionic semiconductors. This duality results in the intermingling of processes occurring at vastly different timescales, still complicating both their comprehensive investigation and the development of robust and dependable devices. Our discussion here undertakes a critical analysis of the field, addressing the current status of knowledge for devices based on halide perovskites in view of electronic and ionic conduction, the underlying models, and the challenges encountered when these devices are optoelectronically characterized. We place a distinct emphasis on the positive contributions that this area of research has not only made to the advancement of photovoltaics but also to the broader progress of solid-state physics and photoelectrochemistry.

Understanding equivalent circuits in perovskite solar cells. Insights from drift-diffusion simulation.

Perovskite solar cells (PSCs) have reached impressively high efficiencies in a short period of time; however, the optoelectronic properties of halide perovskites are surprisingly complex owing to the coupled ionic-electronic charge carrier dynamics. Electrical impedance spectroscopy (EIS) is a widely used characterization tool to elucidate the mechanisms and kinetics governing the performance of PSCs, as well as of many other semiconductor devices. In general, equivalent circuits are used to evaluate EIS results. Oftentimes these are justified via empirical constructions and the real physical meaning of the elements remains disputed. In this perspective, we use drift-diffusion numerical simulations of typical thin-film, planar PSCs to generate impedance spectra avoiding intrinsic experimental difficulties such as instability and low reproducibility. The ionic and electronic properties of the device, such as ion vacancy density, diffusion coefficients, recombination mechanism, etc., can be changed individually in the simulations, so their effects can be directly observed. We evaluate the resulting EIS spectra by comparing two commonly used equivalent circuits with series and parallel connections respectively, which result in two signals with significantly different time constants. Both circuits can fit the EIS spectra and by extracting the values of the elements of one of the circuits, the values of the elements of the other circuit can be unequivocally obtained. Consequently, both can be used to analyse the EIS of a PSC. However, the physical meaning of each element in each circuit could differ. EIS can produce a broad range of physical information. We analyse the physical interpretation of the elements of each circuit and how to correlate the elements of one circuit with the elements of the other in order to have a direct picture of the physical processes occurring in the device.

Specific cation interactions as the cause of slow dynamics and hysteresis in dye and perovskite solar cells: a small-perturbation study.

Hysteresis is one of the most remarkable features of perovskite solar cells; however, it is also present in other kinds of devices such as dye-sensitized solar cells. Hysteresis is due to underlying slow dynamic processes that interfere with the process of charge separation which depends critically on the selective contacts used. In this work we focus on the low-frequency (0.1-10 Hz) dynamics using impedance and intensity-modulated photocurrent spectroscopy and found that both perovskite solar cells (PSCs) and "viscous electrolyte containing" dye-sensitized solar cells (DSSCs) can be described on the same fundamental grounds. By comparing different electrolyte compositions in DSSCs and both methylammonium and formamidinium-based PSCs, we find a connection between the polar nature of the cations and the low-frequency component of these solar cells. There is evidence that in both cases ion transport and specific chemical interactions with the TiO2 surface give rise to the slow dynamics and the hysteresis. This is mainly inferred from the slope of the capacitance vs. applied voltage which shows accumulation behavior for the formulations with higher dipole moments only.

Molecular dynamics simulations of organohalide perovskite precursors: solvent effects in the formation of perovskite solar cells.

The stability and desirable crystal formation of organohalide perovskite semiconductors is of utmost relevance to ensure the success of perovskites in photovoltaic technology. Herein we have simulated the dynamics of ionic precursors toward the formation of embryonic organohalide perovskite CH3NH3PbI3 units in the presence of solvent molecules using Molecular Dynamics. The calculations involved, a variable amount of Pb(2+), I(-), and CH3NH3(+) ionic precursors in water, pentane and a mixture of these two solvents. Suitable force fields for solvents and precursors have been tested and used to carry out the simulations. Radial distribution functions and mean square displacements confirm the formation of basic perovskite crystalline units in pure pentane - taken as a simple and archetypal organic solvent. In contrast, simulations in water confirm the stability of the solvated ionic precursors, which prevents their aggregation to form the perovskite compound. We have found that in the case of a water/pentane binary solvent, a relatively small amount of water did not hinder the perovskite formation. Thus, our findings suggest that the cause of the poor stability of perovskite films in the presence of moisture is a chemical reaction, rather than the polar nature of the solvents. Based on the results, a set of force-field parameters to study from first principles perovskite formation and stability, also in the solid phase, is proposed.

Mechanisms of electron transport and recombination in ZnO nanostructures for dye-sensitized solar cells.

ZnO is an attractive material for applications in dye-sensitized solar cells and related devices. This material has excellent electron-transport properties in the bulk but its electron diffusion coefficient is much smaller in mesoporous films. In this work the electron-transport properties of two different kinds of dye-sensitized ZnO nanostructures are investigated by small-perturbation electrochemical techniques. For nanoparticulate ZnO photoanodes prepared via a wet-chemistry technique, the diffusion coefficient is found to reproduce the typical behavior predicted by the multiple-trapping and the hopping models, with an exponential increase with respect to the applied bias. In contrast, in ZnO nanostructured thin films of controlled texture and crystallinity prepared via a plasma chemical vapor deposition method, the diffusion coefficient is found to be independent of the electrochemical bias. This observation suggests a different transport mechanism not controlled by trapping and electron accumulation. In spite of the quite different transport features, the recombination kinetics, the electron-collection efficiency and the photoconversion efficiency are very similar for both kinds of photoanodes, an observation that indicates that surface properties rather than electron transport is the main efficiency-determining factor in solar cells based on ZnO nanostructured photoanodes.

Comparison of TiO2 and ZnO solar cells sensitized with an indoline dye: time-resolved laser spectroscopy studies of partial charge separation processes.

Time-resolved laser spectroscopy techniques in the time range from femtoseconds to seconds were applied to investigate the charge separation processes in complete dye-sensitized solar cells (DSC) made with iodide/iodine liquid electrolyte and indoline dye D149 interacting with TiO2 or ZnO nanoparticles. The aim of the studies was to explain the differences in the photocurrents of the cells (3-4 times higher for TiO2 than for ZnO ones). Electrochemical impedance spectroscopy and nanosecond flash photolysis studies revealed that the better performance of TiO2 samples is not due to the charge collection and dye regeneration processes. Femtosecond transient absorption results indicated that after first 100 ps the number of photoinduced electrons in the semiconductor is 3 times higher for TiO2 than for ZnO solar cells. Picosecond emission studies showed that the lifetime of the D149 excited state is about 3 times longer for ZnO than for TiO2 samples. Therefore, the results indicate that lower performance of ZnO solar cells is likely due to slower electron injection. The studies show how to correlate the laser spectroscopy methodology with global parameters of the solar cells and should help in better understanding of the behavior of alternative materials for porous electrodes for DSC and related devices.

Control of the recombination rate by changing the polarity of the electrolyte in dye-sensitized solar cells.

Recombination in Dye-sensitized Solar Cells (DSCs) is an electron transfer process critical for high efficiency. The chemical nature of the electron acceptor is known to have an important impact on recombination and, hence, limits the choice of hole conductors in DSCs and related solar cells. In this respect, Room Temperature Ionic liquids (RTILs) have been recognized as an alternative to volatile organic solvents due to their negligible vapor pressure, which offers the chance for long-term stability. However, RTIL-based electrolytes lead to lower performance, a feature that has been attributed to the high viscosity of ionic liquids and the mass-transport limitation associated with it. In this work we show that the origin of the lower performance is also related to an increase in the recombination loss due to the polar nature of the RTIL and the influence of the reorganization energy of the electron acceptor in a polar environment. To investigate this chemical effect, different mixing ratios of RTILs and an organic solvent (acetonitrile) have been considered. The fabricated devices have been characterized by small-perturbation techniques (Impedance Electrochemical Spectroscopy and Intensity-Modulated Photovoltage and Photocurrent Spectroscopies) and Open-Circuit Voltage Decay measurements, which have been used to extract electron lifetimes at different applied voltages. Two different ruthenium dyes (hydrophilic N719 and hydrophobic Z907) and two different cations in the RTIL (imidazolium- and pyrrolidinium-based) have been considered. The results obtained show that for pure ionic liquids the lifetime-voltage curve is exponential, which is a signature of large reorganization energies for electron transfer. In contrast, pure acetonitrile exhibits a non-exponential behavior, which is consistent with relatively low reorganization energy. Interestingly, and as a general rule, we find that recombination is faster in systems with higher reorganization energies. This is interpreted as a consequence of the availability of more acceptor states for electron transfer. In addition, it is found that mixing RTILs and acetonitrile is an interesting strategy to increase the stability of DSCs without significant recombination losses, provided that the right dye and RTIL, in particular, a pyrrolidinium component, are used.

Charge separation at disordered semiconductor heterojunctions from random walk numerical simulations.

Many recent advances in novel solar cell technologies are based on charge separation in disordered semiconductor heterojunctions. In this work we use the Random Walk Numerical Simulation (RWNS) method to model the dynamics of electrons and holes in two disordered semiconductors in contact. Miller-Abrahams hopping rates and a tunnelling distance-dependent electron-hole annihilation mechanism are used to model transport and recombination, respectively. To test the validity of the model, three numerical "experiments" have been devised: (1) in the absence of constant illumination, charge separation has been quantified by computing surface photovoltage (SPV) transients. (2) By applying a continuous generation of electron-hole pairs, the model can be used to simulate a solar cell under steady-state conditions. This has been exploited to calculate open-circuit voltages and recombination currents for an archetypical bulk heterojunction solar cell (BHJ). (3) The calculations have been extended to nanostructured solar cells with inorganic sensitizers to study, specifically, non-ideality in the recombination rate. The RWNS model in combination with exponential disorder and an activated tunnelling mechanism for transport and recombination is shown to reproduce correctly charge separation parameters in these three "experiments". This provides a theoretical basis to study relevant features of novel solar cell technologies.

Conditions for diffusion-limited and reaction-limited recombination in nanostructured solar cells.

The performance of Dye-sensitized solar cells (DSC) and related devices made of nanostructured semiconductors relies on a good charge separation, which in turn is achieved by favoring charge transport against recombination. Although both processes occur at very different time scales, hence ensuring good charge separation, in certain cases the kinetics of transport and recombination can be connected, either in a direct or an indirect way. In this work, the connection between electron transport and recombination in nanostructured solar cells is studied both theoretically and by Monte Carlo simulation. Calculations using the Multiple-Trapping model and a realistic trap distribution for nanostructured TiO2 show that for attempt-to-jump frequencies higher than 10(11)-10(13) Hz, the system adopts a reaction limited (RL) regime, with a lifetime which is effectively independent from the speed of the electrons in the transport level. For frequencies lower than those, and depending on the concentration of recombination centers in the material, the system enters a diffusion-limited regime (DL), where the lifetime increases if the speed of free electrons decreases. In general, the conditions for RL or DL recombination depend critically on the time scale difference between recombination kinetics and free-electron transport. Hence, if the former is too rapid with respect to the latter, the system is in the DL regime and total thermalization of carriers is not possible. In the opposite situation, a RL regime arises. Numerical data available in the literature, and the behavior of the lifetime with respect to (1) density of recombination centers and (2) probability of recombination at a given center, suggest that a typical DSC in operation stays in the RL regime with complete thermalization, although a transition to the DL regime may occur for electrolytes or hole conductors where recombination is especially rapid or where there is a larger dispersion of energies of electron acceptors.

NiOx Passivation in Perovskite Solar Cells: From Surface Reactivity to Device Performance.

Nonstoichiometric nickel oxide (NiOx) is one of the very few metal oxides successfully used as hole extraction layer in p-i-n type perovskite solar cells (PSCs). Its favorable optoelectronic properties and facile large-scale preparation methods are potentially relevant for future commercialization of PSCs, though currently low operational stability of PSCs is reported when a NiOx hole extraction layer is used in direct contact with the perovskite absorber. Poorly understood degradation reactions at this interface are seen as cause for the inferior stability, and a variety of interface passivation approaches have been shown to be effective in improving the overall solar cell performance. To gain a better understanding of the processes happening at this interface, we systematically passivated specific defects on NiOx with three different categories of organic/inorganic compounds. The effects on NiOx and the perovskite (MAPbI3) deposited on top were investigated using X-ray photoelectron spectroscopy (XPS), X-ray diffraction (XRD), and scanning electron microscopy (SEM). Here, we find that the perovskite's structural stability and film formation can be significantly affected by the passivation treatment of the NiOx surface. In combination with density functional theory (DFT) calculations, a likely origin of NiOx-perovskite degradation interactions is proposed. The surface passivated NiOx layers were incorporated into MAPbI3-based PSCs, and the influence on device performance and operational stability was investigated by current-voltage (J-V) characterization, impedance spectroscopy (IS), and open circuit voltage decay (OCVD) measurements. Interestingly, we find that a superior structural stability due to interface passivation must not relate to high operational stability. The discrepancy comes from the formation of excess ions at the interface, which negatively impacts all solar cell parameters.

Surface properties of anatase TiO2 nanowire films grown from a fluoride-containing solution.

Controlling the surface chemistry of nucleating seeds during wet-chemical synthesis allows for the preparation of morphologically well-defined nanostructures. Synthesis conditions play a key role in the surface properties, which directly affect the functional properties of the material. Therefore, it is important to establish post-synthesis treatments to facilitate the optimization of surface properties with respect to a specific application, without losing the morphological peculiarity of the nanostructure. We studied the surface properties of highly crystalline and porous anatase TiO2 nanowire (NW) electrodes, grown by chemical-bath deposition in fluoride-containing solutions, using a combined electrochemical and spectroscopic approach. As-deposited films showed low capacity for catechol adsorption and a poor photoelectrocatalytic activity for water oxidation. Mild thermal annealing at 200 °C resulted in a significant improvement of the electrode photoelectrocatalytic activity, whereas the bulk properties of the NWs (crystal structure, band-gap energy) remained unchanged. Enhancement of the functional properties of the material is discussed on the basis of adsorption capacity and electronic properties. The temperature-induced decrease of recombination centers, along with the concomitant increase of adsorption and reaction sites upon thermal annealing are called to be responsible for such improved performance.

Spectroscopic properties of electrochemically populated electronic states in nanostructured TiO2 films: anatase versus rutile.

A thorough characterization of nanostructured materials under application-relevant conditions is a prerequisite for elucidating the interplay between their physicochemical nature and their functional properties in practical applications. Here, we use a spectroelectrochemical approach to study the population of electronic states in different types of nanostructured anatase and rutile TiO2 films in contact with an aqueous electrolyte. The spectroscopic properties of the two polymorphs were addressed under Fermi level control in the energy range between the fundamental absorption threshold and the onset of lattice absorption (3.3-0.1 eV). The results evidence the establishment of an equilibrium between localized Ti(3+) centers absorbing in the vis/NIR and shallow (e(-))(H(+)) traps absorbing in the MIR upon electron accumulation in anatase electrodes. The absence of the MIR-active (e(-))(H(+)) traps on all rutile electrodes points to a crystal structure-dependent electron population in the films.

IR-spectrophotoelectrochemical characterization of mesoporous semiconductor films.

A combined IR-spectroscopic and electrochemical approach for the study of photo- and bias-induced reactions at the semiconductor/electrolyte interface is presented. Information on the electronic structure of a mesoporous semiconductor nanoparticle network, concretely the density and distribution of band gap states, as well as the nature of solution species are analyzed in situ. It has been shown that under appropriate conditions the electrode potential determines the quasi-Fermi level throughout the mesoporous film and thus the occupation of IR-active band gap states, independently of the type of external perturbation, i.e., application of a bias voltage or electrode exposure to photons exceeding the semiconductor band gap at open circuit. Importantly, electronic properties of the semiconductor and vibrational properties of solution species can be addressed simultaneously by IR-spectroscopy. In addition, electrochemical methods provide a means for the active manipulation (in potentiostatic measurements) or the passive tracking (during open circuit potential decay) of the quasi-Fermi level in the mesoporous film together with the possibility of electron quantification (by charge extraction experiments).

A continuity equation for the simulation of the current-voltage curve and the time-dependent properties of dye-sensitized solar cells.

A numerical model that simulates the steady-state current-voltage curve and the time-dependent response of a dye-sensitized solar cell with a single continuity equation is derived. It is shown that the inclusion of the multiple-trapping model, the quasi-static approximation and non-linear recombination kinetics leads to a continuity equation for the total electron density in the photoanode with an electron density-dependent diffusion coefficient and a density-dependent pseudo-first order recombination constant. All parameters in the model can be related to quantities accessible experimentally. The required power exponents are taken from impedance spectroscopy measurements at different voltages. The model provides new insights into the physical interpretation of the power exponents. Modeling examples involving a high-efficiency TiO(2)-based dye solar cell and a ZnO-based dye solar cell are presented. It is demonstrated that the model reproduces the transient behavior of the cell under small perturbations. The spatial dependence of the recombination rate and the influence of film thickness and of voltage dependent injection efficiency on cell performance are studied. The implications of the model are discussed in terms of efficiencies potentially attainable in dye-sensitized solar cells and other kinds of solar cells with a diffusional mechanism of charge transport.

Photochromic spiro-indoline naphthoxazines and naphthopyrans in dye-sensitized solar cells.

Photochromic dyes possess unique properties that can be exploited in different domains, including optics, biomedicine and optoelectronics. Herein, we explore the potential of photochromic spiro-indoline naphthoxazine (SINO) and naphthopyran (NIPS) for application in photovoltaics. We designed and synthesized four new photosensitizers with a donor-pi-acceptor structure embedding SINO and NIPS units as photochromic cores. Their optical, photochromic and acidochromic properties were thoroughly studied to establish structure-properties relationships. Then, after unravelling the possible forms adopted depending on the stimuli, their photovoltaic properties were evaluated in DSSCs. Although the photochromic behavior is not always preserved, we elucidate the interplay between photochromic, acidochromic and photovoltaic properties and we demonstrate that these dyes can act as photosensitizers in DSSCs. We report a maximum power conversion efficiency of 2.7% with a NIPS-based dye, a tenfold improvement in comparison to previous works on similar class of compounds. This work opens new perspectives of developments for SINO and NIPS in optical and photovoltaic devices, and it provides novel research directions to design photochromic materials with improved characteristics.

Photochromic naphthopyran dyes incorporating a benzene, thiophene or furan spacer: effect on photochromic, optoelectronic and photovoltaic properties in dye-sensitized solar cells.

We recently demonstrated that diaryl-naphthopyran photochromic dyes are efficient for sensitization of TiO2 mesoporous electrodes, thus allowing the fabrication of photo-chromo-voltaic cells that can self-adapt their absorption of light and their generation of electricity with the light intensity. Herein we report the synthesis, the characterisation of two novel photochromic dyes based on diaryl-naphthopyran core i.e NPI-ThPh and NPI-FuPh for use in Dye Sensitized Solar Cells (DSSCs). Compared to our reference dye NPI, the molecules only vary by the nature of the spacer, a thiophene or a furan, connecting the photochromic unit and the phenyl-cyano-acrylic acid moiety used as the anchoring function. We found that swapping a phenyl for a thiophene or a furan leads to an improvement of the absorption properties of the molecules both in solution and after grafting on TiO2 electrodes, however their photochromic process becomes not fully reversible. Despite better absorption in the visible range, the new dyes show poorer photochromic and photovoltaic properties in devices compared to NPI. Thanks to UV-Vis spectroscopy, DFT calculation, electrical characterization of the cells, and impedance spectroscopy, we unravel the factors limiting their performances. Our study contributes to better understand the connection between photochromic and photovoltaic properties, which is key to develop better performing molecules of this class.

Influence of Redox Couple on the Performance of ZnO Dye Solar Cells and Minimodules with Benzothiadiazole-Based Photosensitizers.

ZnO-based dye-sensitized solar cells exhibit lower efficiencies than TiO2-based systems despite advantageous charge transport dynamics and versatility in terms of synthesis methods, which can be primarily ascribed to compatibility issues of ZnO with the dyes and the redox couples originally optimized for TiO2. We evaluate the performance of solar cells based on ZnO nanomaterial prepared by microwave-assisted solvothermal synthesis, using three fully organic benzothiadiazole-based dyes YKP-88, YKP-137, and MG-207, and alternative electrolyte solutions with the I-/I3 -, Co(bpy)3 2+/3+, and Cu(dmp)2 1+/2+ redox couples. The best cell performance is achieved for the dye-redox couple combination YKP-88 and Co(bpy)3 2+/3+, reaching an average efficiency of 4.7% and 5.0% for the best cell, compared to 3.7% and 3.9% for the I-/I3 - couple with the same dye. Electrical impedance spectroscopy highlights the influence of dye and redox couple chemistry on the balance of recombination and regeneration kinetics. Combined with the effects of the interaction of the redox couple with the ZnO surface, these aspects are shown to determine the solar cell performance. Minimodules based on the best systems in both parallel and series configurations reach 1.5% efficiency for an area of 23.8 cm2.

Illumination Intensity Dependence of the Recombination Mechanism in Mixed Perovskite Solar Cells.

Recombination mechanisms in solar cells are frequently assessed through the determination of ideality factors. In this work we report an abrupt change of the value of the "apparent" ideality factor (nAP ) in high-efficiency FA0.71 MA0.29 PbI2.9 Br0.1 based mesoscopic perovskite solar cells as a function of light intensity. This change is manifested as a transition from a regime characterized by nAP ∼1.8-2.5 at low light intensities (<10 mWcm-2 ) to one characterized by nAP ∼1. This transition is equally observed in the recombination resistance extracted from open-circuit impedance measurements. We use drift-diffusion simulations with explicit consideration of ion migration to determine the origin of this transition. We find that a change ofrecombination mechanism concurrent with a modification of the concentration of ionic vacancies is the most likely explanation of the observed behaviour. In the drift-diffusion simulations we show that the apparent ideality factor is in fact affected by the ion vacancy concentration so it is not the optimal parameter to assess the dominant recombination mechanism. We argue that a procedure based on a recently derived "electronic" ideality factor obtained from the high frequency feature of the impedance spectrum is better suited to determine the recombination route that dictates the photovoltage.

Characterization of Photochromic Dye Solar Cells Using Small-Signal Perturbation Techniques.

Photochromic dye-sensitized solar cells (DSSCs) are novel semi-transparent photovoltaic devices that self-adjust their optical properties to the irradiation conditions, a feature that makes them especially suitable for building integrated photovoltaics. These novel solar cells have already achieved efficiencies above 4%, and there are multiple pathways to improve the performance. In this work, we conduct a full characterization of DSSCs with the photochromic dye NPI, combining electrical impedance spectroscopy (EIS) and intensity-modulated photocurrent spectroscopy (IMPS). We argue that the inherent properties of the photochromic dye, which result in a modification of the functioning of the solar cell by the optical excitation that also acts as a probe, pose unique challenges to the interpretation of the results using conventional models. Absorption of light in the visible range significantly increases when the NPI dye is in the activated state; however, the recombination rate also increases, thus limiting the efficiency. We identify and quantify the mechanism of enhanced recombination when the photochromic dye is activated using a combination of EIS and IMPS. From the comparison to a state-of-the-art reference dye (RK1), we were able to detect a new feature in the IMPS spectrum that is associated with the optical activation of the photochromic dye, providing a useful tool for assessing the electronic behavior of the device under different conditions of light excitation. This study provides guidelines to adequate characterization protocols of photochromic solar cells and essential insights on the interfacial electronic processes.

Internal and free energy in a pair of like-charged colloids: Monte Carlo simulations.

The effective interaction between two colloidal particles in a bath of monovalent co- and counterions is studied by means of lattice Monte Carlo simulations with the primitive model. The internal electrostatic energy as a function of the colloid distance is studied fixing the position of the colloids. The free energy of the whole system is obtained introducing a bias parabolic potential, that allows us to sample efficiently small separations between the colloidal particles. For small charges, both the internal and free energy increase when the colloids approach each other, resulting in an effective repulsion driven by the electrostatic repulsion. When the colloidal charge is large enough, on the other hand, the colloid-ion coupling is strong enough to form double layers. The internal energy in this case decreases upon approaching the colloids because more ions enter the double layer. This attractive contribution to the interaction between the colloids is stronger for larger charges and larger ionic concentrations. However, the total free energy increases due to the loss of ionic entropy, and resulting finally in a repulsive interaction potential driven by the entropic contributions. The loss of ionic entropy can be almost quantitatively reproduced with the ideal contribution, the same level of approximation as the Derjaguin-Landau-Verwey-Overbeek (DLVO) theory. The overall behavior is captured by the DLVO theory qualitatively, and a comparison is made with the functional form predicted by the theory, showing moderate agreement.