Theory of Phase Separation and Polarization for Pure Ionic Liquids

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Theory of Phase Separation and Polarization for Pure Ionic Liquids Nir Gavish, and Arik Yochelis J. Phys. Chem. Lett., Just Accepted Manuscript • DOI: 10.1021/acs.jpclett.6b00370 • Publication Date (Web): 08 Mar 2016 Downloaded from http://pubs.acs.org on March 9, 2016

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The Journal of Physical Chemistry Letters

Theory of Phase Separation and Polarization for Pure Ionic Liquids Nir Gavish∗,† and Arik Yochelis∗,‡ †Department of Mathematics, Technion - IIT, Israel ‡Department of Solar Energy and Environmental Physics, Swiss Institute for Dryland Environmental and Energy Research, Blaustein Institutes for Desert Research (BIDR), Ben-Gurion University of the Negev, Sede Boqer Campus, Midreshet Ben-Gurion 84990, Israel E-mail: [email protected]; [email protected] Phone: +972 (4)8294181; +972 (8)6596794. Fax: +972 (4)8294181; +972 (8)6596736

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Abstract Room temperature ionic liquids are attractive to numerous applications and particularly, to renewable energy devices. As solvent free electrolytes, they demonstrate a paramount connection between the material morphology and Coulombic interactions: the electrode/RTIL interface is believed to be a product of both polarization and spatiotemporal bulk properties. Yet, theoretical studies have dealt almost exclusively with independent models of morphology and electrokinetics. Introduction of a distinct Cahn-Hilliard-Poisson type mean-field framework for pure molten salts (i.e., in the absence of any neutral component), allows a systematic coupling between morphological evolution and the electrokinetic phenomena, such as transient currents. Specifically, linear analysis shows that spatially periodic patterns form via a finite wavenumber instability and numerical simulations demonstrate that while labyrinthine type patterns develop in the bulk, lamellar structures are favored near charged surfaces. The results demonstrate a qualitative phenomenology that is observed empirically and thus, provide a physically consistent methodology to incorporate phase separation properties into an electrochemical framework.

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Keywords room temperature ionic liquid, phase separation, gradient flow, charge transport, charge layering, pattern formation



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High energy consumption stresses the need for the development of renewable and clean devices that will ease the transition from reliance on fossil fuels towards independent alternatives. Room temperature ionic liquids (RTILs) are attractive to several technological applications and in particular to renewable energy devices, 1–5 due to their high charge density and tunable anion/cation design, low vapor pressure, and wide electrochemical windows. RTILs are molten salts and thus propose superior properties to electrolyte solutions when incorporated into devices such as batteries, supercapacitors, dye-sensitized solar cells. Even though RTILs resemble highly concentrated electrolytes, they also exhibit fundamental physicochemical differences: (i) Electrical double layer (EDL) structure exhibits in many cases charge layering, crowding and over-screening effects which do not arise in traditional (dilute) electrolytes; 6–10 (ii) Important classes of RTILs exhibit also a nano-structure which displays lamellar, bicontinuous, and sponge-like morphologies with polar and apolar domains of order of nm size in the bulk. 11–16 These ordered nano-domains apparently arise from electrostatic interaction between the oppositely charged ions and/or from interactions between the large functional groups, e.g., hydrogen bonding or solvophobic/solvophillic interactions. 13,14,17,18 The bulk and interfacial nano-structuring of the RTIL is fundamental for device efficiency as it controls the time scales of ion transport (e.g., conductivity) and charge transfer 19 or capacitance. 20 The current theoretical studies of RTIL mostly devote their efforts to either interfacial nano-structuring by electrode polarization 21–23 or to bulk nano-structuring. 24 However, experimental data shows that interfacial RTIL nano-structure can be also a consequence of both surface-specific and bulk liquid interactions. 25 Thus, together with additional empirical and theoretical evidences, a combined theory of RTIL bulk and interfacial nanostructuring is required, cf. 26 and the references therein. Atomistic (force field) methods such as molecular dynamics, allow access to relatively realistic properties of RTILs. These methods, however, are limited to relatively small systems due to a finite number of molecules that can be traced simultaneously. 27,28 On the other hand, although mean–field formulations do not offer atomistic insights they are amenable to extended analytical and numerical compu-


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tations and thus may provide fundamental understanding of the system behavior at larger scales. Respectively, mean-field modeling of bulk nano-structure was advanced by formulating the problem via a gradient flow approach, such as Flory-Huggins. 29,30 These models predict amphiphilic-type bulk structures, in a qualitative agreement with empirical data. 11,12,15,16 Nevertheless, these models do not account for an external electrical field and thus do not provide insights to the interfacial nano-structure. On the other hand, charge layering of the EDL was captured via incorporation of the over–screening effect 31,32 through the so called Bazant-Story-Kornyshev (BSK) framework, but while assuming a structureless bulk. These equations allowed insights into the ion-pairing evolutions puzzle 33,34 and moreover, stressed the need in connecting the EDL structure with the bulk nature. 35,36 In this letter, we develop a mean-field theory that combines phase separation and ionic transport via an Onsager framework, 37,38 while incorporating finite size (a.k.a steric) effects and Coulombic interactions. The resulting model bears similarity to the Ohta-Kawasaki formulation for morphology development driven by competing short-range, and long-range nonlocal interactions, 39–41 and in fact is complimentary to the BSK approach 31,32 in the limit of absence of solvent or other neutral subsets, such as ion pairs. 34 Using linear stability analysis, we show that the bulk morphology emerges via a finite wavenumber instability type. 42 Below the onset and upon applied potential, we observe also the crowding effect, that is consistent with the BSK model. Moreover, numerical simulations show that the same phenomenology persists in higher space dimensions. Finally, we address implications to empirical observations and outlook other electrochemical systems. Coupling phase separation and polarization – Pure RTILs (i.e., fully dissociated molten salts) are fundamentally distinct from dilute electrolytes, which obey the Poisson–Nernst– + – −− Planck description, due to the absence of a solvent, such as water for the KCl ) −* − K + Cl

salt. Consequently, we start by considering a system of a symmetric RTIL of monovalent anions (n) and cations (p), confined in between two flat parallel electrodes. The salt ions are



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assumed be fully dissociated and thus, their local volume fractions, 0 ≤ (p, n) ≤ 1, preserve the uniform density constraint: n(x) + p(x) = 1,

(1)

where x ∈ R3 . In the absence of a solvent (namely, large concentration gradients) and since the molecules are charged, the driving forces for ionic transport and structural evolution of the RTIL are attributed to short-range interactions and to long-range Coulombic interactions. Thus, the mean-field free energy of the system is given by

E = ECH + EC ,

(2a)

where E0 κ2 |∇p|2 + |∇n|2 dx, ECH = c¯ fm (p, n) + 4 Z 1 EC = q¯ c(p − n)φ − |∇φ|2 dx, 2 Z

(2b) (2c)

and h p ni fm (p, n) = kB T p ln + n ln + βnp. 2 2

(2d)

Here, c¯ is the concentration of the anions and of the cations in the mixture, kB is the Boltzmann constant, T is the temperature, β is interaction parameter for the anion/cation mixture and has units of energy, E0 κ2 /4 is the gradient energy coefficient where E0 has units of energy and κ has units of length. We note that entropic terms describing the solvent, voids, or neutral subsets for concentrated electrolytes, such as the Bikerman term 31,43–45 ∝ (1−p−n) ln(1−p−n), are commonly used to account for steric effects in BSK and other meanfield models, but are not appropriate for pure molten salts due to complete absence of any subsets besides charged ions, see (1). Therefore, higher order terms are required to describe ion-ion steric effects. Indeed, in (2a), the first term ECH stands for the Cahn-Hilliard energy 6 ACS Paragon Plus Environment

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and accounts for the energetic cost of short range interactions including anion/cation mixing and for composition inhomogeneities. 46–48 When β > βc = 2kB T , the function fm (p, n ≡ 1 − p) takes the form of a double well potential, thus driving phase separation. The second component, EC of (2a), accounts for long-range Coulombic interactions where φ is the electric potential, q is the elementary charge, and is the permittivity. Requiring that φ is a critical point of the action yields Poisson’s equation, δE = q¯ c(p − n) + ∇2 φ = 0. δφ

(3)

Notably, Eqs. (2b) and (3), comprise the Ohta-Kawasaki free energy, a nonlocal Cahn-Hilliard model for the structure of diblock copolymers mixtures. 39–41 The uniform density constraint (1) for RTILs, implies that ion transport is governed by an inter-diffusion process and not by a standard diffusion process, i.e., not by a random walk of isolated ions in a solvent that leads to Einstein-Stokes relations. Following Onsagers’ framework, 37 the Cahn-Hilliard-Poisson (CHP) equations of motion for cations and anions read:





 p  ∂t   = ∇ · n

δE δE L(x) ∇ , ∇ δp δn

T ! ,

(4a)

where L(x) is the 2 × 2 matrix of coefficients and the superscript T stands for transpose. Onsagers’ reciprocal relations combined with (1), lead to the choice of the Onsager matrix 38  L(x) =



M p(x)n(x)  1 −1   , c¯ −1 1

(4b)

where M is the mobility coefficient. 49,50 Introduction of the standard dimensionless variables

φ˜ =

q kB T

φ,

x ˜= , x λ

s λ=

kB T , 2q 2 c¯

t t˜ = , τ


τ=

λ2 , M kB T


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leads, after omitting the tildes, to the dimensionless form of (3) and (4):

∇2 φ = 1 − 2p,

(5a)

and

∂t p = −∇ · J,

∂t n = −∂t p, J = (p − 1) p∇ −σ∇2 p + log

Here, σ =

E0 κ2 kB T λ2

(5b) p 1−p

+ χ(1 − 2p) + 2φ .

controls the competition between short- and long-range interactions,

and χ = β/(kB T ) is the Flory parameter. For consistency with traditional dimensionless analysis, we have chosen to scale x by the Debye-like scale λ while noting that this choice does not reflect a typical electric screening length, as for dilute electrolytes. We further note that the resulting system (5) is distinct from the Otha-Kawasaki model by its dissipation mechanism. Eqs. (5) are supplemented with boundary conditions of fixed potential at the electrodes and Neumann (i.e., no-flux) for p and n:

φ(x = 0) = −V /2, φ(x = d) = V /2, and J|∂Ω = 0,

(6)

where ∂Ω is the volume boundary and d is the distance between electrodes; at the boundaries of the rest of the volume (that is in the y−z planes), we take a Neumann boundary condition, i.e., ∇φ|∂Ωy−z = 0. Bulk stability, polarization and electrokinetics– For the sake of analysis, we consider first an infinite one-space dimensional domain for which the electrode polarization effects vanish. Under such conditions, Eq. 5b reduces to

∂t p = ∂x (1 − p) p −σ∂x3 p − 2χ∂x p + 2∂x φ + ∂x2 p.


(7)

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Figure 1: Dispersion relations (10), showing the finite wavenumber instability onset at σ = σc = 1/4 (solid line), below the onset σ = 0.255 > σc (bottom dashed line), and above the onset σ = 0.24 < σc (top dashed line); in all three simulations χ = 3. Next, we substitute the expansions

p = p0 +

∞ X

i

ε pi ,

i=1

φ=

∞ X

εi φi ,

(8)

i=1

into (7) and collect terms up to the first order in ε 1. Noting that terms of the type (∂x p)(∂x φ) ∼ o(ε2 ), we identify using p1 = est+ikx + c.c.,

(9)

the stability properties of the equilibrium uniform state (p0 , φ0 ) = (1/2, 0); s is the temporal growth rate of periodic perturbations associated with wavenumbers k. 42 The resulting dispersion relation reads: s = −1 +

σ − 1 k2 − k4. 2 4

χ

(10)

The uniform state is stable if for all k, s < 0, while the instability corresponds to a band of wavenumbers for which growth rate becomes s > 0; the dispersion relation here is always real. The instability onset is obtained by seeking for a critical wavenumber for which



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s(k = kc ) = 0, s(k 6= kc ) < 0 and ds/dk = 0. Accounting these conditions, we find that for χ > 2 and σ = σc the instability is of finite wavenumber type with

σc =

(χ − 2)2 , 4

2 and kc = √ , χ−2

(11)

as demonstrated in Fig. 1. In fact, finite wavenumber instability is known to arise in the Otha-Kawasaki type models 51 but it is distinct from generalized Poisson-Boltzmann type systems which are characterized by a stable uniform bulk. 36 In the latter case, structure formation may be driven by boundary conditions or external forces, e.g., the formation of the double layer structure near a charged wall in the Poisson-Boltzmann model. Next, we consider a finite domain and apply boundary conditions (6). Indeed, numerical integrations of (7) show that, in addition to the EDL crowding (plateau region near the boundaries) and over–screening effects (spatially decaying oscillations), as shown in Fig. 2(a), there is a bifurcation to a periodic structure, as shown in Fig. 2(b). The bifurcation is of a √ supercritical type, i.e., the amplitude of the emerging solutions scales as p ∼ σc − σ (details will be given elsewhere), and emerges also without any potential difference at the walls. 42 Notably, finite wavenumber bifurcation cannot mathematically arise in BSK model 31,32,36 (which belongs to the Poisson-Boltzmann class), namely, the bulk will preserve spatial symmetry. The typical length-scale of the spatial oscillations is attributed to the molecular characteristic sizes and interactions that are being introduced to the free energy, see (2b). Consequently, the lower bound that results here is about angstroms where upon increasing, for example χ, the spatial oscillations become of the order of nano-meter, namely, the spatial oscillations (decaying or persisting) near the interface have the same quantitative characteristics as for the BSK model equations. 31 More generally, at large applied voltages the BSK model is only capable to describe interfaces (heteroclinic-type connections) between the plateau region (crowding) and the uniform bulk, 36 while the CHP equations describe in addition a transition between heteroclinic connections to fixed points below the onset and


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(a)

(b) Figure 2: Asymptotic numerical solutions of model equations (5) in 1D, below (a) and above (b) the instability onset, showing the asymptotic states for χ = 3 and (a) σ = 0.255 > σc = 1/4, (b) σ = 0.24 < σc ; the initial state was taking as spatially uniform (p, φ) = (0.5, 0). The boundary conditions are no–flux for p and a fixed potential difference of 20 (V = 20) for φ, see (6). Both cases demonstrate in addition, crowding and over–screening effects near the boundaries. The integration was performed using the commercial software COMSOL 5.2 with equally discretized grid ∆x = 0.05.



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(a)

(b) Figure 3: Transient currents evaluated numerically by integrating (7) at (a) σ = 0.3, below the onset and (b) σ = 0.2 above the onset. The initial conditions are similar to the asymptotic solutions in Fig. 2, respectively, and for both cases the applied voltage was reversed, V = −20. Other parameters and numerical conditions are as in Fig. 2. once the onset is exceeded, forming heteroclinic connections between a fixed point (crowding) and a (spatial) limit cycle (bulk region). The latter property implies that the interface width dictates the extent of the electrode polarization impact. Morphology existence within the bulk has also a distinct impact on the dynamical properties, such as charging/discharging of the EDL. To demonstrate the differences we compute the transient currents via voltage reversal below and above the instability onsets, as shown in Fig. 3(a) and (b), respectively. The currents are computed via numerical integration of


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the flux over the entire domain 35 1 I (t) = L

ZL/2

(1 − p) p −σ∂x3 p − 2χ∂x p + 2∂x φ + ∂x p dx,

(12)

−L/2

using profiles that are obtained also numerically at each time step from Eqs. 7. Below the instability onset (σ > σc ) the current shows a standard exponential decay, as shown by the semi-log plot in Fig. 3(a). Above the onset (σ < σc ), the primary exponential decay is followed by a slow relaxation, see times 160 < t < 1000. This additional relaxation is not exponential and is due to weak modulations of the bulk structure before reaching an equilibrium. In particular, in the absence of bulk instability such additional relaxations cannot appear and thus, this prediction suggests a new empirical method for detecting the presence of bulk nano-structure. Finally, we note that as expected the finite wavenumber instability persists also in higher space dimensions (with d λ). Here, however, the 1D stripe pattern becomes a lamellar one and thus is subjected to a transverse secondary phase instability of zig-zag, similarly as in the Ohta-Kawasaki model. 51 Consequently, since the equations are derived for a symmetric case, random initial conditions may result in labyrinhtine type patterns depending on the distance from the primary onset, i.e., σ = σc . The stripes become sensitive to zig-zag instability with an increasing distance from onset σ = σc , where the dependence can be obtained via the construction of the Busse balloon, 42 and will be detailed elsewhere. Figure 4, shows three asymptotic lameller (Fig. 4a), mixed (Fig. 4b), and labyrinthine (Fig. 4c) patterns which dominate even in the EDL region, at three distances from the onset σ < σc (χ = 4) = 1. We further note that, integration of p(x, y) over y will result in a vanishing charge density (e.g., Fig. 4c), so that the resulting p¯(x) profile will by resemblance be assumed as 1D asymptotic solution below the instability onset with no morphology, e.g., Fig. 2a. However, even though the asymptotic solutions may resemble the 1D case and also the profiles obtained via BSK theory, it is a distinct solution with distinct properties, for example the respective time



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(a)

(b)

(c)

Figure 4: Numerical solution of (5) in 2D at increasing distance from the onset σc = 1. Graphs presents p(t, x, y) at t = 500000 for χ = 4 with potential φ(±50, y) = ∓V /2 = ∓10: (a) σ = 0.9, (b) σ = 0.6, and (c) σ = 0.4; all three solutions arise from the same random initial data, and have reached steady state. Light and dark colors mark the upper (p = 1) and the lower (p = 0) limits, respectively. The integration was performed using the commercial software COMSOL 5.2 with minimum finite triangular element size of 0.1 on a physical domain size of 100 × 100. scales upon approaching them will differ. Recovering these distinct time scales in (5) is a prerequisite to a systematic comparison with empirical observations and for determining strategies to tailor by-demand RTIL compositions. Discussion– Electrical diffuse layer in RTILs displays a puzzling charge layering near the liquid/solid interface. This unique property stimulated recently intensive studies and distinct approaches ranging from electrode polarization effects to bulk morphology based descriptions. 21,24 However, since understanding of EDL properties is required to control and optimize charge transport and transfer in numerous energy conversion devices, a study of the device-wise mechanisms governing RTIL structure was moved to a spotlight 26 and the references therein. To capture both the polarization effects and the bulk properties, we consider a mean-field framework which couples short-range interactions and long-range Coulombic interactions between the ions. The approach keeps a qualitative fidelity to the physicochemical empirical observations, within the thermodynamic Onsager system framework. Specifically, we use phase separation of Cahn-Hilliard type coupled to Poisson equation to unfold the origin of the emerged spatially periodic and isotropic bulk morphology, which becomes then ordered (anisotropic) near the electrode surface (see Fig. 4), cf. 24,26 As expected, the results


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show tight coupling between bulk-nanostructure and EDL structure. For example, zig-zag instability of the bulk nano-structure impacts and may even dominate the EDL structure itself, see Figure 4c. Our approach considers the limit of pure molten salts which is complementary to the BSK approach 31,32 that is appropriate in the presence of additional neutral subsets, 31,43,44 e.g. ion pairing or dilution by organic solvents. Secondly, the model predicts the emergence of bulk nano-structure. Notably, as shown in, 35,36 bulk nano-structure cannot arise in the BSK model. It is possible to further generalize the framework by merging the approach presented here with the BSK model to address for example ion intercalation. 52 However, the short range electrostatic correlation length scale that was introduced as a fourth order derivative in the Poisson’s equation, contributes only at higher orders and thus will have no qualitative impact on the results of the CHP model. Nevertheless, it may have a significant role once dilution is considered. Thirdly, while in the existing literature mean-field models of RTILs the ions are assumed to satisfy the Einstein-Stokes relations, 31,32 in purely molten salts, the absence of “solvent” inherently relates ion transport to inter-diffusion while Coulombic interactions enter naturally through Poissons’ equation. However, the detailed effects of inter-diffusion and bulk nanostructure on bulk transport and molecular properties in RTILs are beyond the scope of this study and will be addressed elsewhere. Fourthly, we have suggested a mean field formulation however, as also in other contexts, our model equations miss an atomistic interpretations which are essential to quantitative estimation of parameters in (2) and a detailed comparison to available experiments. 24 We believe that designing molecular dynamics force fields will allow bridging and better understanding of the nature of the ion-ion interactions in RTIL, and the mechanisms that dominate in pure RTILs, such as chemical reactions that were advanced by Bazant. 52 In general, our formulation bears similarity to the well-studied Ohta-Kawasaki energy for diblock copolymer mixtures. Specifically, a similar instability have been also studied for the Ohta-Kawasaki model and Newell-Whitehead-Segel equation together with the respective



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Busse balloon have been indeed derived. 51 The presence of the zig-zag instability is responsible for labyrinthine type bulk morphology in 1:1 symmetric RTIL. Once this symmetry will be broken, e.g., due to different ion sizes, we expect that also other morphologies that where observed in Ohta-Kawasaki model will form, 53 such as spheres, circular tubes, and bicontinuous gyroids patterns. In particular, since RTIL ions are not chemically bonded like the diblock copolymers, the interplay between morphology and long-range interactions (electrokinetics in the case of RTILs) is richer and will be studied in detail elsewhere. As such, we expect that the platform developed here, can be extended to a much wider range of material science and energy conversion applications that involve coupling between material nano-structure, electrostatics, and electro-diffusion, such as packed colloidal media. 54

Acknowledgement We thank Martin Bazant for helpful comments and discussions. This research was done in the framework of the Grand Technion Energy Program (GTEP) and of the BGU Energy Initiative Program, and supported by the Adelis Foundation for renewable energy research. N.G. acknowledges the support from the Technion VPR fund and from EU Marie–Curie CIG grant 2018620.

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Theory of Phase Separation and Polarization for Pure Ionic Liquids

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