Capillary Ionization and Jumps of Capacitive Energy
Stored in Mesopores
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<!>Model<!><!>Model<!>Bulk System<!><!>Bulk System<!>Capillary Ionization of Uncharged Pores<!><!>Voltage-Induced Capillary Ionization<!><!>Voltage-Induced Capillary Ionization<!>Conclusion<!>
<p>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</p><p>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.</p><!><p>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 potential36,371where 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–.</p><!><p>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.</p><!><p>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,2where 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 Î»B = 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).</p><p>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.</p><p>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</p><p>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).</p><!><p>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</p><!><p>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.</p><!><p>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</p><!><p>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,3and 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.</p><p>The location of a capillary ionization
transition can be estimated using the Kelvin equation,43,44 which in our case reads (Section S2)4where Î¼bulk 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).</p><!><p>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.</p><!><p>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).</p><p>Outside
of any transition, ion adsorption (Γ, eq 3), accumulated charge,5and the differential
capacitance,6are 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 parameter55,567shown 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.</p><p>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 charge8where 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, and9is the jump of
the accumulated charge at the
transition voltage Uci. Thus, the energy
stored in the pore,10acquires an additional contribution
at the
transition, ΔEci = UciΔQci, which appears
as a jump in the stored energy (Figure 3f).</p><p>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.</p><!><p>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.</p><!><p>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.</p><!><p>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.</p><p>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.</p><!><p>Details of the numerical
approach, the Kelvin equation,
supplementary plots of phase diagrams and of ion and charge densities
(PDF)</p><p>jp1c00624_si_001.pdf</p><p>The authors declare no
competing financial interest.</p>