Ferroelectric Phase Behaviors in Porous Electrodes - Langmuir (ACS

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Ferroelectric phase behaviors in porous electrodes Kenji Kiyohara, Yasushi Soneda, and Kinji Asaka Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.7b01787 • Publication Date (Web): 10 Aug 2017 Downloaded from http://pubs.acs.org on August 15, 2017

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Ferroelectric phase behaviors in porous electrodes Kenji Kiyohara,∗,† Yasushi Soneda,,‡ and Kinji Asaka,† †Inorganic Functional Material Research Institute, National Institute of Advanced Industrial Science and Technology (AIST), Ikeda, Osaka 563-8577, Japan ‡Research Institute of Energy Frontier, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8564, Japan E-mail: [email protected] Abstract The phase behavior of ions in porous electrodes is qualitatively different from that in the bulk because of the confinement effect and the interaction between the electrode surface and the electrolyte ions. We found that porous electrodes of which the pore size is close to the size of the electrolyte ions can show ferroelectric phase behaviors in some conditions by Monte Carlo simulations of simple models. The phase behavior of the porous electrodes dramatically changes as a function of the pore size of the porous electrode and that is compared to the phase behavior of typical ferroelectric materials, for which the phase behavior changes as a function of temperature or the composition. The origin of the phase behavior is discussed in terms of the molecular interaction and the ionic structure inside the porous electrodes. We also found that the density of counterions and that of coions inside porous electrodes change in a non-linear fashion as a function of the applied voltage, which is in accordance with experimental results.

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Introduction The phase behavior of fluids in porous materials is qualitatively different from that in the bulk due to the confinement effect and the interaction between the fluid molecules and the surface of the pores. This issue has been intensively studied for gas adsorption of porous materials by using both experimental and theoretical techniques. 1–8 Depending on the pore size of the adsorbent material, the size of the adsorbate molecules, and the molecular interaction between them, the phase behavior inside the porous adsorbent material shows various characteristic features. Today, the accumulated knowledge of the phase behaviors of particular gases that adsorb in porous materials, for example nitrogen or carbon dioxide, plays important roles in determining the pore size distribution and surface area of porous materials. 6–8 Patterns of the phase behaviors that are characteristic of particular geometrical forms of the pores have been identified. The importance of the characteristic phase behavior of fluids in porous materials has also been noticed in the field of electrochemistry. 9–14 In the development of electrochemical double layer capacitors (EDLC), materials of large specific capacitance, which is the capacitance per volume or per weight of the electrode material, are sought for. Considering that EDLC stores electrochemical energy in the form of electrical double layers (EDL), the capacitance is a product of the surface area and the capacitance per surface area. For a given amount of the electrode material, one way to obtain a large surface area is to make the pore size small. In fact, experimental studies have shown that electrode materials with micropores, which are defined as the pores that are smaller than 2 nm by IUPAC, 6 tend to have large specific area and large specific capacitances at the same time. The capacitance per surface area has been analyzed by Chmiola et al. 13 by experiments using carbon materials of which the pore size is finely tuned at a molecular scale. They have shown that the capacitance per surface area dramatically increases as the pore size becomes smaller than 1 nm. This is an indication that, as the pore size decreases, the attractive interaction between the electrolyte ions and the surface of the electrode material is enhanced. Furthermore, this 2


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effect of interaction must surpass the opposing effect of entropy or confinement that tends to prevent the electrolyte ions from entering the pore of the electrode and decreases the capacitance as the pore size decreases. The electrode materials with such extremely small pores were then intensively studied because they were hoped to be candidates for better energy storage devices. That work has also motivated theoretical studies 15–33 to clarify the relation between specific capacitance and the pore size. On the other hand, it has also been shown that mesopores (pore size between 2 nm and 50 nm) and macropores (pore size larger than 50 nm) provide electrolyte ions with pathways for fast ion transport and contribute to the performance of EDLC. 11 Exploration of effective distribution of the pore size for EDLC is continues to be an active research field. The electrostatic interaction between an ion and a planar electrode surface of carbon is typically an order of magnitude larger than the van der Waals interaction at the minimum of the potential function. For that reason, the primitive model is often used for the model of electrolyte ions in theoretical studies. The primitive model is a hard sphere with an electrical charge embeded at the center 34–36 and has no van der Waals interaction. The structural and thermodynamic properties of ions have been studied in the bulk, near surfaces, and in porous media by using the primitive model. 18–23,33–37 The characteristic properties of ions are often considered to be captured by the primitive model. However, it needs to be clarified to what extent the effect of the van der Waals interaction can be neglected in analyzing the thermodynamic or structural properties. For the case of planar electrodes, there are several studies in literature that discussed the effect of the van der Waals interaction in ionic systems. 38–41 In this paper, we discuss the phase behavior of electrolyte ions which interact with electrostatic and the van der Waals interactions in porous electrodes of which the pore size is at the nanometer scale by using the Monte Carlo simulation. We will show that the ion model with the van der Waals interaction that we used can condense in the pore at low applied voltages when the pore size is comparable to the ion size. Furthermore, in some

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conditions, the phase behaviors are quite similar to the antiferroelectric, ferroelectric, and paraelectric phase behaviors of materials such as PbZrO3 or (Pb1−x Bax )ZrO3 . 42–46 These peculiar phase behaviors are discussed in terms of the molecular interaction and the ionic structure in porous electrodes for which the pore size is comparable to that of the electrolyte ions.

Model The molecular interaction between ions were modeled by the Lennard-Jones potential with a charge embedded at the center of the ion. The interaction between ion species i and j, uij , is expressed as

uij = 4ij

dij rij

12 −

dij rij

6 +

1 qi qj , 4π0 r rij

(1)

where ij and dij are the Lennard-Jones parameters, 0 and r are the vacuum permittivity and the relative dielectric constant of the medium, respectively. The charge qi is either +1 or −1 times the electron charge for cations or anions, respectively. The strength of the van der Waals interaction, ij , was chosen to be the same for all the combinations of i and j. The Lennard-Jones parameters for the ions were chosen to be, dc =da =0.63 nm, c =a =398K, where c and a denote the cations and the anions, respectively. These parameters can qualitatively represent those of typical ionic liquids. 47,48 The relative dielectric constant was chosen to be r = 10. We note that this value is similar to the relative dielectric constants of some room temperature ionic liquids. 49–51 For the calculation of the electrostatic interaction between ions, the Lekner-Sperb method 52,53 was used in order to take into account the long range nature of the interaction. In order to represent porous electrodes of which the pore size is at the nanometer scale, the electrode planes were placed parallel to each other and separated by the pore size W in the z-direction as in previous works. 18,23 The pore size is the separation of two electrode planes in the cathode and the two electrode planes in the anode (Fig.1). In this work, the 4


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pore size of the cathode and that of the anode were chosen to be the same. The structure of the porous electrode employed in this study is the same as that used in previous works, which is composed of six electrode planes. 23 The periodic boundary condition was applied in the x and y directions, which are parallel to the electrode planes. The interaction between ions and the electrodes was modeled by the Steele potential 3,4,54,55 and the Coulomb interaction, which mimics a planar surface of graphite on which carbon atoms are uniformly smeared. The planar electrodes are allowed to have uniform charge on the surface at the position away from the surface by the distance corresponding to the radius of a carbon atom. The parameters of the Steele potential that were used in this work are ρC =38.18 nm−2 , Δ=0.355 nm, dC =0.34 nm, and C =28 K. 3,55 With these parameters, the interaction between the electrode and an ion i is expressed as uCi = 2ρC Ci dCi

2

10 4 dCi dCi 4 2 dCi − − 5 zCi zCi 3Δ(zCi + 0.61Δ)3 +

qi σA zCi + const., 0 r

(2)

where zCi is the distance between the position of surface charge on the electrode and the position of the i-th ion, qi is the charge of the i-th ion, σ is the surface charge density on the electrode, and A is the area of the electrode surface of the simulation system. The surface charge density is uniform on the electrode planes in this model. The parameters for the van der Waals interaction between the ions and the electrode were determined by the Lorentz-Berthelot rule as Ci =

√

C ii ,

(3)

dCi = (dC + dii )/2.

(4)

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Similarly, the interaction between electrode planes was defined as uCa Cb =

2ρ2C AC dC 2

10 4 dC dC 4 2 dC − − 5 z Ca Cb z C a Cb 3Δ(zCa Cb + 0.61Δ)3 +

σ a σ b A2 zCa Cb + const., 0 r

(5)

where Ca and Cb denote two different electrode planes and zCa Cb the distance between them. The pressure-tensor component that is perpendicular to the electrode planes (zz-component) was calculated by using the formulation of Irving and Kirkwood 56,57 as a function of the position in the z-axis, which is written as the following. 1 p(z) = ρ+ (z) + ρ− (z) T − A

|zij | duij rij drij

z∈(zi ,zj )

duC i 1 a − A d|zCa i | z∈(zi ,zCa ) duCa Cb 1 . − A d|zCa Cb | z∈(zCa ,zC

b

(6)

)

where ρ+ (z) and ρ− (z) denote density of cations and that of anions, respectively, and z∈(zα ,zβ ) denotes that summation is performed over all the combinations of α and β with the condition that zα < z < zβ or zβ < z < zα . The first term in the right hand side of Eq.6 is the kinetic part of the pressure-tensor, the second term the contribution from the ion-ion interaction, the third term the contribution from the ion-electrode interaction, and the forth term the contribution from the electrode-electrode interaction. The units of the physical properties in this study are defined as the following. Length is reduced by the length d0 , which was chosen to be 0.42 nm in this study. The surface charge density, σ, is reduced by e/d20 , where e is the electron charge. Energy is reduced by the electrostatic interaction between two unit electrons separated by the cation diameter, e2 /(4π0 d0 ). Temperature, T , and voltage applied between the cathode and the anode, ΔΦ,

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are also reduced accordingly using the unit of energy and the charge. In the following, the unit of voltage is shown by φ, which is defined by φ = ΔΦ/4π. Then, surface charge density of 1μC/cm2 corresponds to σ=0.011, voltage of 1.0V to φ=0.232, temperature of 300K to T =0.075, and pressure of 1atm to 1.3×10−4 . In this study, temperature was fixed at 300K.

Monte Carlo simulation In sampling the thermodynamic states in the Monte Carlo simulations, we employed the grand-canonical constant-voltage ensemble. 18 In this ensemble, the activity of the ions and the voltage are kept constant in addition to the volume and the temperature. Use of this ensemble is advantageous for the study of electrodes, because it most closely reconstructs the experimental conditions in the sense that the surface charge density and the number of the ions in the pore are the properties to be measured, while the voltage and the activity of the ions are those to be controlled. We note that the activity of ions is closely related to the ion concentration in the electrolyte, which is controlled in experiments. The thermodynamic states were sampled with the following weighting factor, zN exp [− (U (r, σ) − ΔΦAσ+ ) /T ] , (N !)2

(7)

where z, N , U , A, and σ+ denote the activity of a pair of a cation and an anion, the number of cations or anions, the total potential energy of the system, the area of an electrode plane, and the total electrical charge pumped from the cathode to the anode per area, respectively. The square of the factorial of the number of ions in the denominator, (N !)2 , accounts for the indistinguishability of the ions. The detailed procedure taken of the Monte Carlo simulation in the grand-canonical constant-voltage ensemble was the same as in the previous works. 20–23 The area of the electrode planes, which expands in the x and y directions, was chosen to be 20 × 20 in the length unit described above. System size scaling was not performed in this study. The pore size was defined by the separation of two adjacent electrode planes in 7


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the z direction. The activity of the ions was chosen so that the resulting ion concentration in the bulk electrolyte, ρ, becomes 0.10 ion pairs per unit volume, which corresponds to 2.2 mol/dm3 . The concentration in the bulk that was used in this study is smaller than some typical ionic liquids that are used in electrochemical systems; for example, the density of 1-ethyl-3-methylimidazolium bis(fluorosulfonyl)imide 4.8mol/dm3 . It is noted, however, that the spherical shape of the model of cations effectively makes them bulkier than the real 1-ethyl-3-methylimidazolium or bis(fluorosulfonyl)imide ions, which have an elongated shape for the comparable size.

Results and discussion Surface charge density Fig.2 shows the surface charge density as a function of applied voltage for different pore sizes. The initial value of the applied voltage, φ, was zero and it was gradually raised to 0.3, and then it was gradually lowered to negative values, until the surface charge that corresponds to the negative voltage shows the same absolute value with the different sign as the surface charge density that corresponds to the positive voltage of the same absolute value. Because the potential parameters of the cations and anions are the same except for the sign of the charge and the pore size of the cathode and that of the anode were chosen to be the same, the graph of the surface charge density as a function of the applied voltage should be point symmetric with respect to the origin. Some of the points in Fig.2 that were obviously symmetric with respect to the origin were obtained by inversion of the sign for the voltage and the surface charge density at the same time. It is clearly seen from Fig.2 that the phase behavior of ions inside porous electrodes significantly changes as a function of the pore size W . For W =1.1, no surface charge is induced on the electrode for the range of the voltage, φ, from zero to about 0.1. When φ is raised over 0.1, the surface charge density, σ discontinuously increases to 0.12, showing 8


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a first order phase transition. This phase transition is similar to the one observed for the system composed of the primitive model and hard electrode planes. 20,21,23,33,60? Moreover, this phase transition is also similar to that of anitiferroelectric materials. 42–46 For W =1.2, spontaneous polarization was observed. Although the surface charge was chosen to be zero at the initial condition, as a simulation run was proceeded at zero applied voltage, there occurs spontaneous polarization between the two electrodes with the magnitude of surface charge density of approximately σ=0.13. It is simply a matter of chance which of the two electrodes is chosen to be the one with positive or negative charge under no applied voltage. Under finite applied voltages, the surface charge density gradually increases with increasing the applied voltage. In decreasing the applied voltage, the surface charge density gradually decreases and then discontinuously decreases to a negative value near the applied voltage of φ=-0.02 (see Fig.3). This is an example of ferroelectricity. It is reminded here that the present model has one pore for both the cathode and the anode. If a number of pores are prepared for the cathode and the anode, under no applied voltage, both positively and negatively polarized domains could be formed and the overall polarization could be canceled in each electrode. Even in this case, however, a global polarization would occur in each electrode by applying a large enough voltage. Such a phase transition would be similar to that of antiferroelectric materials. For W =1.4, no spontaneous polarization occurs at zero applied voltage. The surface charge density shows small values ( σ ∼ 0.01) until the applied voltage was raised to φ = 0.08, where a steep rise to σ = 0.15 was observed. The surface charge is almost saturated at this value for the higher applied voltage. In decreasing the applied voltage, a steep drop of surface charge density was observed when the voltage is changed from φ = −0.04 to −0.08. This is another example of ferroelectricity. For W =2.0, as the applied voltage increases from zero, the surface charge density starts to increase near φ = 0.04. The cause of such a delay of charging is discussed in the next section in relation to the ion density. Then the surface charge density increases with the applied voltage until it is raised to φ = 0.12, where it is almost saturated. In decreasing the voltage, the surface

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charge density gradually decreases up to φ = 0.08 and, in further decreasing the voltage, the surface charge density decreases more rapidly with the applied voltage. A clear hysteresis is observed for this case as well, but the intercept with the vertical axis in decreasing the applied voltage is rather small compared to the case of W =1.4. For W =4.0, the behavior of the φ − σ plot also shows a clear hysteresis but no surface charge is stored on the electrode surface at zero voltage. When the pore size is further raised to W =8.0, a clear ferroelectric behavior is observed again. When the applied voltage is decreased after it is increased to a value larger than φ = 0.08, the surface charge density shows a positive value until the applied voltage is decreased to φ = −0.04, where the surface charge density steeply decreases with voltage. For the largest value of the pore size that we studied, W =12, the behavior of the φ − σ plot shows almost no hysteresis, which can be classified as a paraelectric behavior. For these simulations, rather large number of Monte Carlo moves, typically 108 cycles, were attempted in decreasing the voltage, in order to make sure that the observed hysteresis is not an artifact. Calculation of the free energy as a function of the applied voltage would be useful in more precise determination of the phase diagram, but that is beyond the scope of the present study. It is interesting to compare the phase behaviors of porous electrodes as a function of the pore size to the phase behaviors of typical ferroelectric materials as functions of temperature and composition. Zhai and Chen have studied the temperature dependent behaviors of antiferroelectric to ferroelectric switching of PbZrO3 . 43 They found that, at low temperatures, the voltage dependence of polarization shows a sudden change at some high voltage, showing the antiferroelectric phase of the material. As the temperature is raised, the voltage dependence of polarization becomes ferroelectric, where spontaneous polarization is found at low applied voltage and the polarization is reversed at large enough voltage of the opposite sign. Hao and Zhai 45 have studied the composition dependent behaviors of the phase transformation behavior and electrical properties of (Pb1−x Bax )ZrO3 . They found that the material shows an antiferroelectric phase for the low concentrations of Ba2+ and a ferroelectric phase

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for the high concentrations. The voltage dependence of surface charge density of this study shows transition among antiferroelectric, ferroelectric, and paraelectric phases. The parameter that determines the phase is the pore size for the case of porous electrodes, while that parameter is the temperature for the case of PbZrO3 and and the composition for the case of (Pb1−x Bax )ZrO3 . For the case of the primitive model, some phase transitions in the surface charge as a function of the applied voltage have been reported but not the ferroelectric behavior to our knowledge. Possible reasons why ferroelectic behaviors were observed in the system that we studeid here are the following; 1) the introduction of the van der Waals attraction affects the phase behavior significantly, particularly in favor of insertion of ions into the pores, even though its magnitude is much smaller than the electrostatic interaction. 2) the soft repulsion of the Lennard-Jones interaction, as opposed to the hard sphere of the primitive model, affects the phase behavior significantly in favor of insertion of ions into the pores. 3) the chemical potential chosen for this study turned out to be effectively larger than that used in the previous studies of the primitive model, which induces the ions to condense in small pores more easily. Since all the above possibilities affect in a mixed fashion, it is difficult to point out what is the main cause of the difference of the results of this study as compared to the previous results of the primitive model at this stage of the study.

ion density In the previous section, rich phase behaviors in the relation between the applied voltage and the surface charge density on the electrode surface were shown including antiferroelectricity, ferroelectricity, and spontaneous polarization of porous electrodes. Here, the origin of those properties are discussed in terms of the ionic structure inside the porous electrodes. Fig. 4 shows the density of counterions and coions as a function of applied voltage when the voltage is gradually raised from zero. For W =1.1, the density of counterions jumps from zero to 0.22 discontinuously, which corresponds to the discontinuous change of the surface 11


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charge density as a function of the applied voltage that was discussed in the previous section. For W =1.2, only cations or only anions occupy the pore for all the voltage range studied here, including φ =0, which also corresponds to the surface charge density as a function of the applied voltage. For pore sizes of W =1.4 and larger, both counterions and coions can enter the pore. For W =1.4, the density change of counterions or coions is minor for the applied voltage in the range 0 < φ < 0.08. When the voltage is raised over φ = 0.08, the density of counterions steeply increases and that of coions steeply decreases with voltage. Similar behaviors can be found for 1.4 < W < 4.0 and the voltage at which the significant change of the densities starts becomes lower as the pore size becomes larger. For W =4.0, it is after the voltage is raised over φ=0.05 that the density of counterions increases significantly. For W =8.0, however, the density of counterions and that of coions continuously and monotonically change from φ=0. The small change in the ion density that was found for pore sizes 1.4 < W < 4.0 is similar to the experimental findings by Wang et al. 58 They conducted the electrochemical measurements and in situ NMR spectroscopy measurements for the system composed of tetraethylammonium tetrafluoroborate in acetonitrile as the electrolyte and and a porous carbon as the electrode. They measured the amount of counterions and coions and found that the density of counterions inside the electrode significantly increases only after the applied voltage is raised over 0.75V, rather than linearly changes from 0V. The confinement of the small pores would restrict the configuration of ions inside the pore and that is considered to be the cause of the peculiar voltage dependence of the ion density. There seems to be a threshold of voltage in order for significant change of ionic structure to occur. Our simulation supports the experimental findings. Fig.5 shows the density profile of ions inside the pore when the voltage was gradually increased to φ = 0.12 for different pore sizes. For W =1.1 and 1.2, only the counter ions can enter the pore and they form a single peak in the slit pore. For W =2.0, the peak split into two and, for W =4.0, there are three peaks, two for counterions and one for coions. For W =8.0, there are five peaks, three for counterions and two for coions. For W =12, the

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density in the middle of the pore shows the bulk-like structure and less defined peaks are observed near the electrodes as compared to the cases for the smaller pores. The hysteresis found in the voltage dependence of the surface charge density is related to the ion density profile: the well structured density profile for the small pores implies an interaction-driven structure, which are expected to be weakly susceptible to a change in the external field once it is formed.

pressure inside the pore The pressure-tensor component that is perpendicular to the electrode planes was calculated by using the formulation of Irving and Kirkwood (Eq.6). The value of p(z) was found to be virtually constant with in each pore. We define the average of p(z) inside each pore as the pressure inside the pore. Fig. 6 shows the pressure inside the pore when the voltage was gradually increased from φ = 0 for different pore sizes. For the larger pore sizes that we studied, W =8.0 and 12, the pressure almost monotonically increases from near zero values with the applied voltage, whereas, for small pore sizes, the pressure dramatically changes in both positive and negative directions as a function of the applied voltage. The positive pressure implies that the pore tends to expand and the negative pressure implies that the pore tends to shrink. The small pores such as those of W =1.1, 1.2, and 1.4 show large positive pressure for a wide range of the applied voltage, which is two to four orders of magnitude larger than the atmospheric pressure. Similar dramatical change of pressure as functions of the pore size and the applied voltage has been reported previously for the case of the primitive model. 24,59–61 This is caused by several different effects and they are entangled in a complicated way because of the presence of the electrode surface. The positive contributions to the pressure comes from the volume exclusion interaction, which is the repulsive part of the Lennard-Jones interaction in the present model, in addition to the contribution of the kinetic part. Although the electrostatic interaction between a pair of similarly charged objects also causes repulsive interaction, the 13


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ionic structures are naturally chosen in such a way that the effective interaction between the charged objects is overall attractive at the equilibrium. The negative contributions to the pressure comes from the attractive part of the Lennard-Jones interaction and the electrostatic interaction. The strength of the electrostatic interaction between the electrode and the ions varies with the relative size of the pores and the ions and with the applied voltage. As the equilibrium ionic structure is determined by minimizing the free energy, the resulting pressure can be positive or negative for a fixed pore size. The pressure is also plotted as a function of the pore size for different voltages φ=0. 0.08, and 0.20 (see Fig.7). This diagram shows whether the pore size tends to expand or shrink if the pore is flexible. For the case of φ=0.08, the stable values of W will be 1.5 and 2.9, where p ≈ 0 and (dp/dW ) < 0, which is a condition of stability. For the case of φ = 0.20, the pressure inside the pore is larger than the atmospheric pressure for the range of W slightly larger than 2.0. This means that if the pore size starts at, for example W =2.5, the pore tends to continuously expand at that voltage.

Conclusions We studied the phase behavior of electrolyte ions in porous electrodes of which the pore size is at the nanometer scale by using the Monte Carlo simulation. The electrolyte ions and the electrode surfaces were modeled to interact by the electrostatic interaction and the van der Waals interaction, the latter of which was modeled by the Lennard-Jones interaction. The ion model that we used in this study can show condensation in the porous electrode at low applied voltages when the pore size is comparable to the ion size. Such phase behaviors are qualitatively different from those in the previous studies of the primitive model. It is remarkable that introduction of the van der Waals interaction can dramatically change the electrochemical phase behavior of ions in porous electrodes, considering that the van der Waals interaction that was introduced in the present model is an order of magnitude weaker

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than the electrostatic interaction that was also present in the primitive model. Furthermore, the diagram of surface charge density as a function of the applied voltage showed characteristic behaviors of the antiferroelectric, ferroelectric, and paraelectric phases as well as the spontaneous polarization, depending on the pore size. Those phase behaviors are similar to the the phase behaviors of typical ferroelectric materials such as PbZrO3 or (Pb1−x Bax )ZrO3 . In the system of porous electrodes that we studied, the pore size plays the role of the parameter that determines the phase behavior, which is the temperature for the case of PbZrO3 and the composition for the case of (Pb1−x Bax )ZrO3 . The density change of counterions and coions in the porous electrodes as a function of the applied voltage is closely related to the surface charge density. For porous electrodes for which the pore size is equal to or smaller than W =1.4, significant change of ion density starts only after the applied voltage is raised to some threshold values. Similar behaviors have been experimentally found in an EDLC. This is a characteristic behavior for porous electrodes for which the pore size is comparable to the ion size. The density profile in the porous electrode shows well defined peaks for the pore sizes equal to or smaller than W =8.0, showing that the ionic structure is interaction dominated. The hysteresis characteristic of the various phase behaviors of the diagram of surface charge density as a function of the applied voltage is attributed to the strong interaction between the ions and the electrode. Pressure inside the pore was found to dramatically change as a function of the pore size and the applied voltage. The pressure can be positive or negative, showing that the pore tends to expand or shrink depending on the pore size and the applied voltage if they are flexible, and its absolute value can be orders of magnitude larger than the atmospheric pressure. For real electrodes of EDLCs, the cycling stability is a key property as well as the capacitance. The origin of degradation of EDLCs can be attributed to various mechanisms including chemical decomposition of the molecules, blocking of pores by the reaction products, and deformation of the porous structure of the electrode. The ferroelectric behavior

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in porous electrodes could be another possible mechanism for the degradation because that implies that ions that once entered in a pore could stay there even after the applied voltage has changed its sign. Generation of large pressure inside a pore could also be a possible mechanism of degradation because that would deform the electrode material. Furthermore, the pore size of real electrodes has a distribution in a wide range. In this study, a range of pore sizes were discussed but an electrode that has pores of multiple sizes was not discussed. It would be possible as a first approximation to apply the results of this study to an electrode that has a distribution of pore sizes simply by averaging the property by weighting with the corresponding pore distribution. There are some reports on disordered porous electrodes in the literature. 62,63 However, detailed analysis of the charging mechanism of disordered electrodes is still left for the future study.

Acknowledgement This work was supported by JSPS KAKENHI Grant No.24550169. Part of the computation in this work was performed using the supercomputers of Research Institute for Information Technology of Kyushu University and those of Cybermedia Center of Osaka University.

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R2

R1 ı1

ı2

+

+

ı3

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+

+ -

+

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+

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+ +

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+ -

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+

-

-

+ + -

W

W

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Figure 1: Schematic picture of the model for porous electrodes used for the Monte Carlo simulation. Six electrode plates are placed parallel to each other. Voltage is applied between three electrode planes on the left and the other three electrode planes on the right. Ions (shown in circles) are inserted into or deleted from the regions denoted R1 and R2 in this study. W and ΔΦ denote pore size and applied voltage, respectively. The electrode planes are denoted plane 1, plane 2, ..., plane 6 from left to right. The corresponding surface charge densities are denoted σ1 , σ2 , ..., σ6 , respectively. 24


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0 -0.2

0.4

(b)

0 -0.2

-0.4 0

0.2

φ

0

0.2

0 -0.2

0.2

0

φ

0.2

-0.2

0

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(g)

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-0.4 0

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0

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σ

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0

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(d)

-0.2

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-0.2

0

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-0.2

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(c) σ

0.2

σ

σ

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(a)

σ

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-0.4 -0.2

0

0.2

-0.2

φ

0

0.2

φ

Figure 2: Surface charge density of the electrode surface as a function of the applied voltage, which is the electrical potential compared to the counter electrode. (a), (b), (c), (d), (e), (f), and (g) correspond to W =1.1, 1.2, 1.4, 2.0, 4.0, 8.0, and 12.0, respectively, where W is the pore size. More than half of the points in each graph are the actually calculated values. Those points are copied to the points that are point symmetric with respect to the origin, utilizing the symmetry of the pore size of the two electrodes and the symmetry of the ion size for the cations and anions. 25


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С

С

С

Figure 3: Snapshots of a porous electrode with pore size W =1.2. Green spheres denote anions and red spheres denote cations. Starting at φ=0.02, for which the pore was occupied by the counterions (anions) the electrical potential with respect to the other electrode, was gradually decreased to the negative voltages, φ=-0.01 and then to φ=-0.02. At φ=-0.01, the 26 which is attributed to the hysteresis effect. pore was occupied only by the coions (anions), The pore is occupied only by counterions (cations) when the electrical potential was further decreased to φ=-0.02. ACS Paragon Plus Environment

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w1.1 CT w1.1 CO w1.2 CT w1.2 CO w1.4 CT W1.4 CO w2.0 CT w2.0 CO w4.0 CT w4.0 CO w8.0 CT w8.0 CO

0.2 ρs

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0.1

0

0

0.1

0.2 φ

Figure 4: Ion density of counterions (CT, filled symbols) and coions (CO, open symbols) as a function of the applied voltage (φ) for different pore sizes (W =1.1, 1.2, 1.4, 2.0, 4.0, and 8.0).

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6

(a)

2 0 z

0

0.5

(e)

-0.5

0 z

(f)

2

1 0

-2

2

2 -0.5

0 z

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0

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(g)

2

1

0 z

0

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ρ

2

2

ρ

-0.5

(d)

4

ρ 2

0

6

(c)

4

ρ

4

ρ

4

6

(b)

ρ

6

ρ

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0

1

-4 -2

0 z

2

4

0

-5

Figure 5: Ion density profile inside a pore as a function of the distance from the center of the pore when the voltage was gradually increased to φ = 0.12. (a), (b), (c), (d), (e), (f), and (g) correspond to W =1.1, 1.2, 1.4, 2.0, 4.0, 8.0, and 12.0, respectively, where W is the pore size. Density of conterions is shown in black and that of coions in red.

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5.0

w1.1 w1.2 0.2 w1.4 w2.0 0.0 w4.0 w6.0 w8.0 -0.2 0 w12.0

4.0

3.0

0.2

p

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2.0

1.0

0.0 0

0.1

0.2 φ

Figure 6: Pressure inside the pore as a function of the applied voltage for different pore sizes (W =1.1, 1.2, 1.4, 2.0, 4.0, 6.0, 8.0 and 12.0). For example, W 1.1 denotes pore size W =1.1.) Inset is the magnification of the small values of the applied voltage and the pressure.

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1.5

φ=0.0 φ=0.08 φ=0.2

1.0 p

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0.5 0.0 2

4

6 8 W

10 12

Figure 7: Pressure inside the pore as a function of the pore size for three values of the applied voltage. Black, red and green denote φ=0.00 0.08, and 0.20, respectively.

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ı WŽƌĞ^ŝǌĞу/ŽŶ^ŝǌĞ ĭ Pore Size >> Ion Size

Figure 8: TOC

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Ferroelectric Phase Behaviors in Porous Electrodes - Langmuir (ACS

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