Confinement Induced Dilution: Electrostatic Screening Length

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Cite This: J. Phys. Chem. C XXXX, XXX, XXX−XXX

Confinement Induced Dilution: Electrostatic Screening Length Anomaly in Concentrated Electrolytes in Confined Space Jun Huang* College of Chemistry and Chemical Engineering, Central South University, Changsha 410083, P.R. China ABSTRACT: For electrolytes confined in nanoscale space, a growing body of experimental evidence suggests that the electrostatic screening length (or Debye length) increases with the electrolyte concentration in the concentrated regime (e.g., ionic liquids). This electrostatic screening length anomaly can be understood under the assumption of an effectively dilute electrolyte due to massive ion-pairing, which, however, contradicts with the conventional wisdom that ionic liquids are highly dissociated, given the negative free energy of dissociation. To resolve this puzzle, a phenomenological theory is developed to understand ion-pair dissociation in electrolytes in conf ined space. Nanoconfinement suppresses lattice expansion upon ion-pair dissociation, shifting the effective free energy of dissociation toward positive values, favoring massive ion-pairing, and rendering a long electrostatic screening length. The dependence of the electrostatic screening length on the temperature and the ionic size is discussed. It is of high priority to investigate the nanoconfinement effect of the differential double-layer capacitance Cdl to further corroborate the confinement induced dilution phenomenon.

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where ϵr is the relative permittivity, ϵ0 the vacuum permittivity, cbi and qi the bulk concentration and charge number of an ion i, respectively. It is very important to note that cbi should be the concentration of participating ions, which can effectively screen the electrical field produced by the central charge. Due to ionpairing, cbi is usually smaller than the concentration when the electrolyte is completely dissociated, denoted as ce. This difference is, unfortunately, not well appreciated in the literature. λD is around 10 nm for a 1 mM aqueous NaCl solution (assuming complete dissociation). Moreover, it is readily seen from eq 4 that λD decreases with the increase of cbi . The fundamental limitation of the DH theory is that the electrostatic correlation expressed in eq 2 is local, namely wij(r) is solely determined by ϕi(r), while the nonlocality of electrostatic correlations is not accounted for. Consequently, the DH theory is rigorously valid only for infinitely dilute electrolytes. There is a long history of recognizing its limitations for concentrated electrolytes (say >0.1 M),3,4 which is further triggered by the recent observation that the electrostatic screening length, namely the Debye length λD, anomalously increases with the electrolyte concentration ce in the concentrated regime.5−8 This phenomenon is termed the electrostatic screening length anomaly. The electrostatic screening length anomaly shall not be surprising as nonlocal electrostatic correlations become remarkable for concentrated electrolytes. By accounting for the strong nonlocal electrostatic correlations, Kjellander4,9 and Gavish et al.10 have successfully rationalized the nonmonotonic dependence of λD on cbi . Considering an electrolyte near a charged surface, they revealed that the potential profile

INTRODUCTION The physics of ions in electrolyte underpin a plethora of engineering applications such as energy storage technologies including supercapacitors, batteries and fuel cells.1 The classical theory was proposed by Debye and Huckel2 nearly a century ago and remains in wide use ever since. The fundamental insight of the Debye−Huckel (DH) theory2 is to realize that there exist strong positional correlations between the cations and anions in electrolyte; the correlation function is written as gij(r ) = e−βwij(r)

(1)

with β = 1/RT and wij(r) being the work required to bring ions i and j from infinity to separation r in electrolyte. Debye and Huckel assumed that wij(r ) = qjFϕi(r )

(2)

where qj is the charge number of ions j and ϕi(r) is the electrostatic potential at distance r from an ion i. Regarding a central charge at r = 0, the electrostatic potential produced by the central charge can be solved by invoking the Poisson equation3 ϕi(r ) ∝

⎛ r ⎞ 1 exp⎜ − ⎟ r ⎝ λD ⎠

(3)

with λD being the Debye length, namely the characteristic length of the electrostatic screening effect exerted by the surrounding ionic cloud. λD is expressed as ⎛ ϵ ϵ RT ⎞1/2 λD = ⎜⎜ r 0 b 2 ⎟⎟ ⎝ ∑i ci qi ⎠

Received: November 9, 2017

(4) © XXXX American Chemical Society

A

DOI: 10.1021/acs.jpcc.7b11093 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C stretching from the interface toward the bulk exhibits a transition from exponential to spatially oscillatory decay with increasing cbi . The decaying oscillations effectively elongate λD. While it may be relevant for certain cases, such a long-ranged oscillatory decay is at odds with several experiments.7,8 Surface force balance results exhibit oscillations only in the near-surface region ( 0 and θT > 0. Moreover, δλ should be systemdependent. It is expected that δλ decreases, corresponding to suppressed lattice expansion, in confined space due to the volume constraint. On the basis of eq 11, δμ̃ ex will be more positive, hence favoring massive ion-pairing in confined space. In the limit of massive ion-pairing limit, χ ≪ 1, eq 9 is simplified to an explicit analytical expression, χ∼

χγ 2

0

with δc = lc/λD. eq 17 is refined expression of eq 6 in the BSK theory by considering the size difference and the partial dissociation of ion-pairs.17 The boundary conditions to close eq 17 are as follows: (i) at the electrode surface, x = 0, ϕ̃ = ϕ̃ 0; (ii) following the BSK theory, assuming ∂3ϕ̃ /∂x̃3 = 0 at x = 0, meaning that the meanfield charge density is flat near the surface;17 and (iii) at the symmetric plane, x̃ = lD̃ , symmetrical conditions, give ∂ϕ̃ /∂x̃ = 0, and ∂2ϕ̃ /∂x̃2 = 0. As ∂3ϕ̃ /∂x̃3 = 0, thereby, σs = −ϵ∇ϕ. After determining the distribution of electrostatic potential by solving eq 17, the distribution of cation and anion concentration can be calculated using eq 16. The differential double layer capacitance, Cdl, defined as, Cdl = ∂σs/∂ϕ, can also be calculated and its dependence on the electrode potential is to be discussed.

with δλ = 1 + λ− − λsp being the dimensionless lattice expansion upon ion-pair dissociation. Molecular dynamic simulation by Akbarzadeh et al. indicates that the internal pressure of ionic liquids increases and the separation of molecules decreases at high ionic concentrations.19 As a result, it is expected that δλ decreases at higher γ0. In addition, it is assumed that δλ increases at higher temperatures due to the elevated thermal energy. As a first approximation, we have, 0 δλ = δλ 0 − θγ + θT(T − T °) c

sinh(ϕ)̃

=−

0 exp((ln N + 2)(−θγ + θT(T − T °))) c

γ0

(13)

We have (ln N + 2) ≈ 50 for ionic liquids. Equation 13 indicates that χ decreases with γ0 when θc > 0 and increases with T when θT > 0. The relation between λD and the ionic concentration, or equivalently γ0, is calculated using eq 4 by substituting cbi with Nχγ0/2NA. χ is determined by eq 9. NA is Avogadro constant. Using Euler−Lagrange (EL) equation, eq 5 leads to a modified fourth-order Poisson equation,17 ϵ(lc2∇4 ϕ − ∇2 ϕ) = ρ

Figure 1. Participating-ion fraction (viz. the dissociation degree) χ as a function of the concentration of the electrolyte solution ce (left). The screening length λD as a function of ce (right). Experimental data (circles for NaCl in water and squares for [C4C1Pyrr][NTf2] in propylene carbonate) are reproduced from Perkin et al.21

(14)

with the boundary condition at the electrode surface, x = 0, n ⃗ ·ϵ(lc2∇2

− 1)∇ϕ = σ

s

is calculated using eq 4 by substituting cbi with ceχ, as shown in Figure 1 (right). Encouraging agreement between model and experiment is obtained in terms of the λD ∼ ce relation (Figure 1). Model parameters to produce Figure 1 are listed in Table 1. δμ̃in = −10 for [C4C1Pyrr][NTf2] and δμ̃in = 1.6 for NaCl are taken from the literature.13,20 δλ = 0.8 − 1.3γ0 for [C4C1Pyrr][NTf2] and δλ = 1.0 − 2.0γ0 for NaCl are fitted according to experimental data of Perkin et al.,21 Figure 1 (right). It is found that χ decreases with increasing ce. In the high ce regime (ce > 1.0 M), χ drops exponentially, see the inset of Figure 1 (left). Correspondingly, a nonmonotonic relation between λD and ce is obtained. In the low ce regime (ce < 0.4 M), the electrolyte is almost completely dissociated, χ ≈ 1, hence, λD decreases with ce according to λD ∼ ce−1/2. In the high ce regime (ce > 0.4 M), there is massive ion-pairing in the electrolyte, χ ∼ e−ce, hence, λD increases with ce. The scaling relation between λD/λD′ and ac/λD′ with λD′ being the Debye length when the electrolyte is completely dissociated is shown in Figure 2. The scaling

(15)

with σs being the surface charge density on the electrode surface. In addition, taking the variation of f v with respect to NI yields the electrochemical potential of the ith species, which should be equal to its value at the symmetric plane under equilibrium. Consequently, the number density of cations and anions normalized to N can be obtained, b N±̃ = N±̃

exp( ∓ϕ)̃ 1+

χγ 2

0

(exp( −ϕ)̃ + λ− exp(ϕ)̃ − 1 − λ−) (16)

with ϕ̃ = zFϕ/RT. Equation 16 is reduced to the well-known expression in ref 18 when λ− = 1. Substituting eq 16 into 14 gives, C

DOI: 10.1021/acs.jpcc.7b11093 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C Table 1. Model Parameters category common

[C4C1Pyrr][NTf2] in propylene carbonate

NaCl in water

variable T, temperature λsp, relative size of ion aggregates λsl, relative size of solvent molecules Ñ bsl, compacity of solvent ac, mean ion diameter ϵr, relative permittivity δμ̃, dimensionless free energy of ion-pair dissociation δλ, reduced volume expansion upon ionpair dissociation ac, mean ion diameter ϵr, relative permittivity δμ̃, dimensionless free energy of ion-pair dissociation δλ, reduced volume expansion upon ionpair dissociation

value

source

294 K 1.5

ref 21 assumed

1

assumed

0.7

estimated

0.4 nm

ref 21

30 −10

ref 21 ref 13

0.8 − 1.3γ0

fitted

0.294 nm 70 1.58

ref 21 ref 21 ref 20

1 − 2γ0

fitted

Figure 3. Left panel: participating ion fraction as a function of the dimensionless free energy of dissociation (intrinsic term only) for the case of λ− = λsp = 1. Right panel: comparison between the present theory with δλ = 0.8 − 1.3γ0, the conventional theories with δλ = 1, and the special case with δλ = 0, when δμ̃ in = −10. Other parameters have their values given in Table 1.

between λD and ce is examined in Figure 4 (left). The model indicates that in the high ce regime (ce > 0.4 M) λD decreases

Figure 4. Temperature dependence of the nonmonotonic relation between the electrostatic screening length λD and the ionic concentration ce. The right panel shows the comparison between the model results (solid line) and experiment data (squares) taken from Gebbie et al.6

Figure 2. Scaling relation between λD/λD′ and ac/λD′ using parameters of [C4C1Pyrr][NTf2] as listed in Table 1. Note that λD′ is the Debye length when the electrolyte is completely dissociated.

with increasing T. At very high temperatures, say 1000 °C, ions are expect to be nearly completely dissociated, like in molten salts. Consequently, a monotonic relation between λD and ce as described by the original Debye length can be seen in Figure 4 (left). Figure 4 (right) shows the model fitting of the temperature dependence of the electrostatic screening length λD at ce = 2M. Experiment data are taken from Gebbie et al. using [C2min][NTf2].6 The linear relation between λD and 1/T is well reproduced by the model. In calculating the model results, the following parameters are used/extracted: ac = 5 Å, ϵr = 12.3, δμ̃in = −10, γ0 = 0.18 (1.2 M), and δλ = 0.8 − 1.3γ0 + 10−3(T − 295). Other parameters have their values in Table 1. The ion size ac and the dielectric constant ϵr are taken from the original paper. δμ̃in = −10 is consistent with the data collected in the review.13 Gebbie et al. did not give the concentration of [C2min][NTf2] in the original paper.6 Herein, a value of 1.2 M, corresponding to γ0 = 0.18, is used. θT = 10−3 in the expression for δλ is fitted from the experimental data. It is noted that the slope of the curve is largely dictated by θT, while the vertical position of the curve is very sensitive to γ0. We further look into the size dependence of the nonmonotonic relation. Figure 5 shows that λD increases more sharply with ce in the high ce regime when the ionic size is larger. Moreover, the transition between two regimes is advanced when the ionic size is larger.

relation was first uncovered by Perkin et al. by organizing experimental data.8 We can see a transition from a plateau to an oblique curve at ac ≈ λD′, which is in line with the analysis by Perkin et al.8 In previous theories (e.g., in the literature14,18), it is a tacit assumption that electrolyte molecules and dissociated ions all have the same size, that is, λ− = λsp = 1. In this case, δλ = 1 + λ− − λsp = 1. Consequently, δμ̃ ex = 0, and χ is exclusively determined by δμ̃in according to eq 9, as shown in Figure 3 (left). It is found that χ ≈ 1 for δμ̃in < 0 and massive ion-pairing is possible only when δμ̃ in > 0, for the case of δλ = 1. Given δμ̃ in = −10, Figure 3 (right) compares three cases, including the conventional case with δλ = 1, the special case with inhibited lattice expansion on ion-pair dissociation (δλ = 0), and the present theory with δλ = 0.8 − 1.3γ0, corresponding to suppressed lattice expansion upon ion-pair dissociation. It is found that the suppression of the lattice expansion upon ionpair dissociation favors ion-pairing by shifting the effective free energy of dissociation toward positive values, as seen from eq 11. It is observed in experiments that λD decreases at elevated temperatures and there is an approximately linear relation between λD and 1/T.6,8 By taking θT to be 0.001, the temperature dependence of the nonmonotonic relation D

DOI: 10.1021/acs.jpcc.7b11093 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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The Journal of Physical Chemistry C

magnitude of Cdl decreases. It is not surprising because the electrolyte is actually dilute due to massive ion-pairing at γ0 = 0.2. Such a camel shape has been reported in experimental1,22 and computational studies.1,23 However, the model-calculated range between 2 and 5 μF cm−2 for the magnitude of Cdl is at odds with the experimentally observed range between 5 and 20 μF cm−2 for ionic liquids (see the detailed discussion by Fedorov and Kornyshev,1 and Goodwin and Kornyshev14). Moreover, it has not escaped our notice that several experiments reported a bell shape,1,24 which is at variance with the presented theory. The controversies concerning Cdl can be interpreted from two aspects. On one hand, it may result from the insufficient description of the strong overscreening in concentrated electrolytes in the otherwise elegant and influential BSK theory. In this regard, future efforts are required to refine the double layer theory. On the other hand, it should be noticed that the experimentally observed range between 5 and 20 μF cm−2 was usually measured in a large volume of ionic liquid, in which case the suppression of lattice expansion upon ion-pair dissociation is significantly weakened compared with that in a confined space. As a result, the ionic liquids are likely to be highly dissociated in these experiments, resulting in a bell shape Cdl profile. As an important implication from the presented theory, it is of high priority for experimentalists to look into the nano confinement effect on the differential double layer capacitance of concentrated electrolytes.

Figure 5. Size dependence of the nonmonotonic relation between the electrostatic screening length λD and the ionic concentration ce.

Figure 6 exhibits Cdl (left) and ionic concentration distribution (right) for the case of [C4C1Pyrr][NTf2] in

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Figure 6. Differential double layer capacitance (left) and ionic concentration distribution (right) for the case of [C4C1Pyrr][NTf2] in propylene carbonate.

CONCLUSION A phenomenological theory has been developed to understand ion-pair dissociation in confined electrolyte solutions, and to decipher the electrostatic screening length anomaly. The key and novel element in the present theory is the refinement of the lattice entropy term in the free energy expression by accounting for the size difference between electrolyte molecules, dissociated ions and holes, and its dependence on the electrolyte concentration, the temperature, and the confinement. The key finding is that the suppression of the lattice expansion upon ion-pair dissociation, induced by higher the confinement, effectively shifts the free energy of dissociation toward positive values, hence, favoring massive ion-pairing and rendering a long-ranged electrostatic screening length. In short, nanoconfinement makes ion-pair dissociation unfavorable. As an important implication from the present study, confinement shall dramatically change the differential double layer capacitance curve as a function of the electrode potential. This effect can be harnessed to test the present theory.

propylene carbonate. The coexistence of participating ions and spectating ion-pairs complicates the nonlocal electrostatic correlations. Hence, δc shall also depend on γ0. As a first approximation, δc is taken to be 0.5. For γ0 = 0.02, ϕ̃ 0 = −20, the model predicts oscillations in the ionic concentration of cations and anions near the surface, Figure 6 (right), which is consistent with the near-surface oscillation observed in surface force measurement. Overcrowding of cations takes place at the electrode surface. As near-surface cations deliver more countercharge than the surface charge, anions are attracted in the next layers. This phenomenon is the so-called overscreening.1,17 However, only one oscillation is seen in Figure 6 (left), while multiple oscillations are observed in surface force measurement. This disparity reflects the intrinsic limitation of the BSK theory. Readers are directed to the work of Bazant, Storey, and Kornyshev for detailed elucidation on the competition between overcrowding and overscreening.17 In the classical Gouy−Chapman-Stern double layer model, the Cdl vs ϕ curve exhibits a U-shape with the minimal obtained at the potential of zero charge (ϕ = 0). By accounting for the ion size effect, Kornyshev revealed that the Cdl vs ϕ curve manifests a camel shape, and Cdl ∼ |ϕ|−1/2 in the two wings due to overcrowding.18 Moreover, Kornyshev profoundly found a transition from the camel shape to a bell shape when the ionic concentration increases.18 Noteworthy, the complete dissociation, χ = 1, was implicitly assumed in the Kornyshev theory.18 As shown in Figure 6, the Cdl vs ϕ curve exhibits a camel shape for γ0 = 0.02 corresponding to 0.26 M. Asymmetry is observed for the two wings, which is ascribed to the size difference between cations and anions. When γ0 is increased to 0.2 (2.6 M), we find that the camel shape is retained, which is in contrast with the Kornyshev theory. Furthermore, the

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AUTHOR INFORMATION

Corresponding Author

*(J.H.) E-mail: [email protected]. ORCID

Jun Huang: 0000-0002-1668-5361 Notes

The author declares no competing financial interest.

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ACKNOWLEDGMENTS The author is indebted to Professor Zhangquan Peng at Changchun Institute of Applied Chemistry (CIAC), Chinese Academy of Sciences, for his hospitality during my sabbatical in his research group. Financial support from the starting fund for E

DOI: 10.1021/acs.jpcc.7b11093 J. Phys. Chem. C XXXX, XXX, XXX−XXX

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(23) Vatamanu, J.; Borodin, O.; Smith, G. D. Molecular Insights into the Potential and Temperature Dependences of the Differential Capacitance of a Room-Temperature Ionic Liquid at Graphite Electrodes. J. Am. Chem. Soc. 2010, 132 (42), 14825−14833. (24) Islam, M. M.; Alam, M. T.; Ohsaka, T. Electrical Double-Layer Structure in Ionic Liquids: A Corroboration of the Theoretical Model by Experimental Results. J. Phys. Chem. C 2008, 112 (42), 16568− 16574.

new faculty members of Central South University (502045001) is appreciated.

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REFERENCES

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DOI: 10.1021/acs.jpcc.7b11093 J. Phys. Chem. C XXXX, XXX, XXX−XXX