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. 2021 Apr 30;125(19):10243–10249. doi: 10.1021/acs.jpcc.1c00624

Capillary Ionization and Jumps of Capacitive Energy Stored in Mesopores

Carolina Cruz †, Svyatoslav Kondrat †,‡,§,*, Enrique Lomba ∥, Alina Ciach †
PMCID: PMC8282200  PMID: 34276858

Abstract

graphic file with name jp1c00624_0005.jpg

We study ionic liquid–solvent mixtures in slit-shaped nanopores wider than a few ion diameters. Using a continuum theory and generic thermodynamic reasoning, we reveal that such systems can undergo a capillary ionization transition. At this transition, the pores spontaneously ionize or deionize upon infinitesimal changes of temperature, slit width, or voltage. Our calculations show that a voltage applied to a pore may induce a capillary ionization, which—counterintuitively—is followed by a re-entrant deionization as the voltage increases. We find that such ionization transitions produce sharp jumps in the accumulated charge and stored energy, which may find useful applications in energy storage and heat-to-energy conversion.

Introduction

Ionic liquids (ILs) under confinement play a key role in science and technology, exhibiting remarkable properties and finding applications in energy storage,1−4 capacitive deionisation,5−7 heat-to-energy conversion,8−10 etc. For instance, subnanometer pores filled with an electrolyte provide the highest achievable capacitance11−13 and stored energy,14 though sluggish dynamics.15−18 Electrodes with mesoscale pores avoid a typically poor interpore connectivity of microporous electrodes,19 enabling faster charging.20,21 Using neat ILs, which have large electrochemical windows,22−24 enhances the electrical energy stored in micro- and mesopores.25 Unfortunately, neat ILs exhibit slow dynamics; mixing ILs with solvents, such as acetonitrile or water, enhances the IL conductivity26−29 and speeds up the charging kinetics.30

Previous work has focused on micro- and mesopores filled with IL–solvent mixtures far from phase transitions. However, confined fluids show an exciting physics close to state transformations. A classic example is a capillary condensation,31,32 which has numerous practical applications, particularly in determining the pore-size distribution of micro- and mesoporous materials.33−35 In this article, we study IL–solvent mixtures liable to phase separation, confined in pores substantially wider than the ion diameter. Using a simple continuum theory and general thermodynamic arguments, we demonstrate that the pores can become spontaneously ionized or deionized in response to small changes in temperature or the applied potential difference. We show that such capillary ionization goes along with abrupt changes in charge and energy storage, which may find practical applications in electrochemical energy storage and generation.

Model

We consider a mixture of a room-temperature ionic liquid and solvent confined into a slit mesopore of a supercapacitor’s electrode, with the potential difference U applied to the pore walls with respect to the bulk electrolyte (Figure 1). We model solvent implicitly and assume that the system is translational invariant in the latteral x, y directions. This system can be described by the grand potential36,37

graphic file with name jp1c00624_m001.jpg 1

where β = 1/(kBT) (kB is the Boltzmann constant and T is temperature), A is the surface area, w is the slit width, ρ± are the cation and anion densities, and μ is the chemical potential. The charge density is c = ρ+ – ρ– (in units of the elementary charge e) and the total ion density ρ = ρ+ + ρ–.

Figure 1.

Figure 1

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Ionic liquid–solvent mixture in a slit mesopore. The pore walls are separated by a distance w. The ion diameter a is the same for cations (light blue spheres) and anions (dark blue spheres). Orange spheres represent solvent molecules, modeled implicitly in eq 1. hs is the surface field that describes the preference of pore walls for ions or solvent. The potential difference U is applied to the pore walls with respect to the bulk electrolyte (not shown). The system is translational invariant in the lateral x, y directions.

The first term in eq 1 is the entropic contribution, which consists of the ideal gas entropy and the free energy density due to excluded volume interactions, fex, approximated here by the Carnahan–Starling expression,

graphic file with name jp1c00624_m002.jpg 2

where η = πa3ρ/6 is the ion packing fraction and a is the ion diameter. The second term in eq 1 is the electrostatic energy, where u is the local electrostatic potential and ϵr is the relative dielectric constant. At room temperature, ϵr ranges from 80 for water down to about 10 for less polar solvents like alkohols and increases monotonically with increasing temperature.38−40 Correspondingly, the Bjerrum length λB = e2/(ϵrkBT) acquires a relatively weak dependence on temperature and can increase or decrease with temperature. For simlicity, however, we assume here a temperature-independent Bjerrum length λB = a (corresponding to ϵr ≈ 80 at room temeprature) and note that the temperature dependence and the choice of ϵr do not affect our results qualitatively (cf. Figure S5).

The third term in eq 1 arises due to van der Waals interactions, with K measuring the strength of the interactions and ξ0 their spatial extension.36 The parameter K sets a temperature (energy) scale, which we express via the critical temperature of a bulk system, as described below. The parameter ξ0 is comparable to the molecular dimension and we set it equal to the ion diameter in all calculations.

Ionophilicity hs in eq 1 describes the preference of pore walls for ions or solvent. A negative hs means that the pore walls favor solvent, while hs > 0 means that the walls prefer ions, and we assume that this preference is the same for cations and anions. We focus on ionophilic and weakly ionophobic (hs ≈ 0) pore walls and note that even for an “ionophilic” wall, the ion density at the wall can be lower than in the bulk, provided the bulk ion density ρb > hs/ξ0.36

The equilibrium properties of the system are determined by a minimum of Ω. Minimization of Ω with respect to ρ± and u leads to differential equations which we solved numerically (Section S1).

Results and Discussion

Bulk System

In a bulk system, that is, outside of the pores, eq 1 simplifies to βΩ(ρ̅b)/A = −βKρ̅b2/2 + ρ̅b ln(ρ̅b/2) + βρ̅bfex(ρ®b) – βμρ̅b, where ρ̅b= a3ρ. The equilibrium condition, ∂Ωb/∂ρ̅b = 0, leads to a nonlinear equation, which we solved numerically. The solution reveals the existence of two phases, enriched in ions and solvent, which we call IL-rich and IL-poor phases. Figure 2a (solid black lines) shows that there is a region of temperatures and IL densities, where an IL–solvent mixture separates into the IL-poor and IL-rich phases. This region shrinks for increasing temperature and ends at a critical point T̅c = kBTca3/K ≈ 0.09 and ρca3 ≈ 0.25.36

Figure 2.

Figure 2

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Capillary ionization of uncharged slit mesopores. (a) Phase diagram in the temperature–ion density plane. The solid black line shows the bulk diagram and the circle denotes the critical point. The open squares show the results for a slit of width w = 20a obtained by numerically minimizing eq 1, and the blue lines show the results obtained by the Kelvin equation, eq 4. The triangles and the dashed line show the temperature T/Tc = 0.74 and the densities at coexistence obtained at the chemical potential μ/kBTc = −4.57; this value of the chemical potential is used in the other panels. Phase diagrams in the chemical potential–temperature and ionophilicity–temperature planes are shown in Figures S1 and S2. (b) In-pore ion density profiles at the coexistence indicated in panel (a). (c) Diagram showing capillary phase transitions in the plane of temperature and inverse slit width. The vertical dash line indicates the slit width used in the other panels. (d) Amount of IL adsorbed in the pore, Γ, as a function of temperature for the slit width w = 20a. Γ is given by eq 3. In all plots the ionophilicity a3hs/ξ0 = 0.25, where ξ0 is the bare correlation length and a the ion diameter.

The bulk phase diagram shown in Figure 2a is typical for IL–solvent mixtures.41,42 Various imidazolium tetrafluoroborate ILs in arenes, alkohols, and water have been reported to have critical temperatures in the range from 300 to 400 K and critical IL mole fractions from as low as 0.02 to about 0.125.42 A popular 1-hexyl-3-methylimidazolium tetrafluoroborate (C6mim-BF4) has the critical temperature Tc ≈ 326 K and critical mole fraction xc ≈ 0.125 in alkohol (C6OH) and Tc ≈ 331 K and xc ≈ 0.04 in water.42 However, aqueous bistriflimide (TFSI)-based ILs have higher critical concentrations (between 0.2 and 0.3), closer to our model, with the critical temperature ranging from 400 to 420 K.41

Capillary Ionization of Uncharged Pores

The bulk phase behavior of IL–solvent mixtures translates into a similar behavior in uncharged mesopores, where the phase separation region is shifted (symbols in Figure 2a). Examples of the in-pore ion density profiles at coexistence are shown in Figure 2b. The phase diagram in the plane of temperature and inverse slit width consists of a line of first-order phase transitions between the IL-poor and IL-rich phases (Figure 2c). Thus, a phase transition can be induced by changing temperature or slit width. A transition as a function of temperature is illustrated in Figure 2d. This figure shows the amount of an IL adsorbed in the mesopore,

graphic file with name jp1c00624_m003.jpg 3

and demonstrates a capillary ionization obtained upon decreasing temperature; that is, that the amount of the IL adsorbed in the pore increases abruptly at the transition as the temperature is decreased.

The location of a capillary ionization transition can be estimated using the Kelvin equation,43,44 which in our case reads (Section S2)

graphic file with name jp1c00624_m004.jpg 4

where μbulk is the chemical potential of the bulk transition, Δρ is the jump of the IL density at the transition, and Δγ is the difference in the wall–fluid surface tensions of the IL-poor and IL-rich phases. To calculate the surface tensions, we used a semi-infinite system consisting of a single flat electrode. The prediction from eq 4 is shown by blue solid line in Figure 2a and demonstrates good quantitative agreement with the full numerical calculations (see also Figure S1).

Voltage-Induced Capillary Ionization

The black line in Figure 3a shows that the region of the IL-rich phase widens as a voltage is applied to a pore with respect to the bulk electrolyte. The applied potential creates favorable conditions for the counterions to reside inside the pore, which bring along the co-ions (Figure 3e). At high voltages, this region shrinks. The difference between the ion structures near the pore walls in the IL-rich and IL-poor phases decreases for increasing voltage (Figures S6 and S7). Thus, the thermodynamic state becomes determined more and more by the in-pore bulk region, favoring the IL-poor phase at given thermodynamic conditions (Figure S1).

Figure 3.

Figure 3

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Voltage-induced capillary ionization and chargingof slit mesopores. (a) Phase diagram in the temperature–voltage plane. The thick black line shows a line of first-order transitions betweenthe IL-rich and IL-poor phases. The thin vertical lines indicate temperatures used inthe other panels. The horizontal lines show the values of voltage used inFigure 4. (b) Amount of IL adsorbed in the pore (eq 3), (c) charge accumulated in the pore (eq 5), (d) differential capacitance (eq 6), (e) charging parameter XD (eq 1) and (f) energy stored inthe pore (eq 4) as functions of the applied voltage for three values of temperature indicated in (a). The capacitance forT/Tc = 0.78 and T/Tc = 0.838 diverges at the transitions, as schematically denoted by upward pointing arrows. In all plots the pore width w = 20a, chemical potential μ/kBTc = −4.57 and ionophilicity a3hs/ξ0 = 0.25, where ξ0 is the bare correlation length and a isthe ion diameter.For typical values of the ion diametera = 0.7 nm and room temperature for Tc, the various units are thermal voltage e/kBTc ≈ 26 mV for voltage, e/a2 ≈ 2 e nm–2 ≈ 32 μC cm–2 for accumulated charge, thermal electric capacitance e2/(kBTca2) ≈ 620 μF cm–2 for capacitance, and kBTc/a2 ≈ 0.84 mJ cm–2 ≈ 0.23 nW cm–2 for energy. The color and line codes are the same in panels (b)–(f). Phase diagrams in the chemical potential–temperature plane for a few voltages are shown in Figure S3.. The diagrams for a few other values of the chemical potential are presented in Figure S4 and for a few values of the Bjerrum length in Figure S5. Examples of the total and charge density profiles at the coexistence are shown in Figures S6 and S7.

Outside of any transition, ion adsorption (Γ, eq 3), accumulated charge,

graphic file with name jp1c00624_m005.jpg 5

and the differential capacitance,

graphic file with name jp1c00624_m006.jpg 6

are all continuous functions of voltage (the blue dashed lines in Figure 3b–d). At low temperatures, the system is in the IL-rich state, characterized by a high ion density, and hence the capacitance has a bell shape (T/Tc = 0.72 in Figure 3d), in accord with numerous studies.45−53 At higher temperatures, the IL-poor phase becomes stable and the shape of the capacitance becomes bird-like (T/Tc= 0.78 and T/Tc = 0.838 in Figure 3d).36,54 The charging mechanisms in these two cases differ significantly, as demonstrated by the charging parameter55,56

graphic file with name jp1c00624_m007.jpg 7

shown in Figure 3e. In the IL-poor phase at low voltages (T/Tc = 0.72 and T/Tc = 0.78 in Figure 3e), XD quickly becomes larger than unity, which means that charging proceeds by adsorption of both co-ions and counterions. In the IL-rich phase (T/Tc = 0.838 in Figure 3e), we have 0 < XD < 1 in the whole range of voltages and hence charging is a combination of counterion adsorption and co-ions swapping for counterions.

Applying a voltage to a mesopore can induce a capillary ionization transition (Figure 3a,b). This transition is accompanied by a sharp increase of the charge accumulated in the pore (Figure 3c), which has important consequences for capacitance and energy storage. To analyze charging in this case, we write for the accumulated charge

graphic file with name jp1c00624_m008.jpg 8

where Qrich(U) and Qpoor(U) are the charge accumulated in the pore in the IL-rich and IL-poor phases, respectively, Uci is the transition voltage, and θ(x) is the Heaviside step function, equal to unity for x > 0 and zero otherwise. Correspondingly, the differential capacitance diverges at the transition, viz., C(Uci) = Cpoor(Uci) + ΔQciδ(U – Uci), where δ(x) is the Dirac delta function, Cpoor(U) = dQpoor/dU is the capacitance in the IL-poor phase, and

graphic file with name jp1c00624_m009.jpg 9

is the jump of the accumulated charge at the transition voltage Uci. Thus, the energy stored in the pore,

graphic file with name jp1c00624_m010.jpg 10

acquires an additional contribution at the transition, ΔEci = UciΔQci, which appears as a jump in the stored energy (Figure 3f).

Of course, the jumps of the accumulated charge obtained upon voltage increase are always positive (Figure 3c). Because of the re-entrant behavior, therefore, a jump ΔQci from the IL-poor to the IL-rich phase (eq 9) is positive for U < Ub and negative for U > Ub, where Ub ≈ 32kBTc/e is the voltage at which the transition line bends (Figure 3a). In Figure 4a,b, we show examples of the accumulated charge and stored energy density as functions of temperature for U < Ub and U > Ub. It is worth noting that there is no monotonous relation between the accumulated charge Q (or integral capacitance CI = Q/U) and the stored energy density E, as one might expect from the E = CIU2/2 expression. In particular, an increase in the stored energy may accompany a decrease (Figure 4a) or an increase (Figure 4b) in the accumulated charge. Clearly, the CIU2/2 equation is only valid in the linear regime (Q ∼ U), while the actual charge–voltage relation is more complex.

Figure 4.

Figure 4

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Charge and energy storage in slit mesopores. (a) Accumulated charge (top) and stored energy (bottom) as functions of temperature for appliedvoltage eU/kBTc = 20. (b) Same as in (a) but for eU/kBTc = 46. The shaded areas show the regions with one and two transitions occurring at voltages below those indicated onthe plots. The number of transitions in each region is also indicated on the plots. (c) Jumps in the accumulated charge (top) and stored energy (bottom) along the transition line as functions of the transition voltageUci. The symbols correspond to the jumps shown in (a) and (b). Chemical potential μ/kBTc = −4.57, slit width w = 20a, and ionophilicity a3hs/ξ0 = 0.25, where ξ0 is the bare correlation length and a the ion diameter. For typical values of the ion diameter a = 0.7 nm and room temperature for Tc, the various units are thermal voltage e/kBTc ≈ 26 mV for voltage, e/a2 ≈ 2 e nm–2 ≈ 32 μC cm–2 for accumulated charge, andkBTc/a2 ≈ 0.84 mJ cm–2 ≈ 0.23 nW cm–2 for energy. For the phase diagram, seeFigure 3a.

In Figure 4c, we summarize this behavior by plotting ΔQci and ΔEci = UciΔQci along the transition line of Figure 3a. Remarkably, the magnitudes of ΔQci and ΔEci increase steeply with voltage at high voltages.

Conclusion

We have studied ionic liquid–solvent mixtures in slit mesopores. We revealed that a pore could become spontaneously ionized as a function of temperature, slit width, or applied voltage, as manifested by jumps in the amount of an ionic liquid adsorbed in the pores. Voltage-induced capillary ionization transitions are exciting as they can be re-entrant and create jumps in the accumulated charge and stored energy density. The possibility to obtain a sharp increase in the stored energy by minute changes of voltage or temperature is spectacular and may find useful technological applications.

Our predictions are based on a simple mean-field model and ideal slit mesopores. In real systems, the locations of the transitions might be shifted by fluctuations, high-density steric repulsions,57 and chemical complexity of ion–solvent interactions, all neglected in this work. Furthermore, the distribution of pore sizes and shapes in real electrodes may smear out the jumps at the transitions.14,58,59 However, the rapid progress in low-dimensional carbon materials provides hope for engineering highly monodisperse porous structures, e.g., based on MXene60,61 or graphene,62−64 which would facilitate experimental observation of capillary ionization transitions and enable their application in heat-to-electricity conversion and energy storage.

Acknowledgments

We thank Mathijs Janssen (University of Oslo) for critical reading of the manuscript and insightful comments and suggestions. This work received funding from the European Unions Horizon 2020 research and innovation programmes under the Marie Skłodowska-Curie grant agreement No. 711859 to C.C. and No. 734276 to A.C. and S.K. Additional funding was received from the Ministry of Science and Higher Education of Poland for the implementation of the international cofinanced projects No. 734276 in the years 2017-2021 and No. 711859 in the years 2017–2022.

Supporting Information Available

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acs.jpcc.1c00624.

  • Details of the numerical approach, the Kelvin equation, supplementary plots of phase diagrams and of ion and charge densities (PDF)

The authors declare no competing financial interest.

Supplementary Material

jp1c00624_si_001.pdf (566.1KB, pdf)

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