The Influence of Co-Solutes on the Chemical Equilibrium - A Kirkwood

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C: Energy Conversion and Storage; Energy and Charge Transport

The Influence of Co-Solutes on the Chemical Equilibrium - A Kirkwood-Buff Theory for Ion Pair AssociationDissociation Processes in Ternary Electrolyte Solutions Anand Narayanan Krishnamoorthy, Christian Holm, and Jens Smiatek J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.7b12255 • Publication Date (Web): 03 Apr 2018 Downloaded from http://pubs.acs.org on April 3, 2018

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The Journal of Physical Chemistry

The influence of co-solutes on the chemical equilibrium - a Kirkwood-Buff theory for ion pair association-dissociation processes in ternary electrolyte solutions Anand Narayanan Krishnamoorthy,† Christian Holm,† and Jens Smiatek∗,†,‡ †Institute for Computational Physics, University of Stuttgart, Allmandring 3, D-70569 Stuttgart, Germany ‡Helmholtz Institute Muenster: Ionics in Energy Storage (HI MS-IEK 12), Forschungszentrum Juelich, Corrensstrasse 46, D-48149 Muenster, Germany E-mail: [email protected]

Abstract We present an application of Kirkwood-Buff theory, which is introduced to describe the influence of co-solutes on the chemical equilibrium between dissociated and associated ion pairs in ternary electrolyte solutions. Our approach makes use of KirkwoodBuff integrals and the introduction of a local/bulk partition model. The co-solute species can be either charged or uncharged, and our approach is applicable for ideal and weakly non-ideal solutions in combination with low ion concentrations. As main result, the theory reveals that differences in the local co-solute accumulation behavior around the ions induce a shift of the chemical equilibrium either to the associated or the dissociated state. The findings of our analysis are useful for a deeper understanding of electrolyte solutions in modern electrochemical storage devices. All results are

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verified by atomistic molecular dynamics simulations in terms of sodium chloride pairs in dimethylacetamide (DMAc)-water mixtures.

1. Introduction Aqueous electrolyte solutions with simple salts show a plethora of specific ion effects and ion pairing mechanisms 1,2 . With regard to several conformations, it can be distinguished between contact (CP), solvent-shared (1SP) and solvent-separated (2SP) ion pairs, whereas the theoretical description of ion pair formation processes is challenging and even nowadays under debate 1–5 . The results of computer simulations implied that most CPs in aqueous solution are marginally stable configurations with life times of several picoseconds 2 . Dynamic transitions to 1SP and 2SP ion pairs can thus be often observed, which both significantly differ from CPs due to the presence of one or two hydration shells between the ions. In consequence, the distance between the ions increases with CP < 1SP < 2SP, which also corresponds to the inverse energetic stability of the states. Furthermore, it was observed that individual ions differ in their pairing behavior, which can be rationalized by ion-specific chaotropic and kosmotropic properties 6 . As it was often pointed out 7,8 , smaller ions like fluoride reveal kosmotropic properties, meaning that they are water structure makers, whereas larger ions like iodide are chaotropes, which emphasizes their water-structure breaking effects. With regard to these assumptions, the ’law of matching water affinities’ was introduced 6 , which states that combinations of kosmotropic or chaotropic ion pairs form the most stable compounds. Although the existence of chaotropic properties is yet under debate, experimental and simulation results indeed revealed that the individual hydration properties of the ions play a decisive role in their pairing behavior 2 . Apart from aqueous systems, recent work was also spent on organic solvents and the occurrence of specific ion effects in non-aqueous solutions 9 . The rising interest in organic solvents and solvent mixtures can be rationalized by the ongoing improvement of electrolyte solutions

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for electrochemical storage devices, as dominated by lithium ion batteries (LIBs) and postlithium ion battery technologies 10–13 . In LIBs, the electrolyte, as a main component of the electrochemical cell, is usually a mixture of polar aprotic solvents like propylene carbonate (PC) or ethylene carbonate (EC), a lithium salt, usually LiPF6 , and several co-solutes, in this context also called ’additives’, like dimethylcarbonate (DMC) or further linear alkyl components 10 . As a crucial requirement for the desired functionality, the solvent must have a high dielectric constant in order to dissolve the lithium salt in high amounts, whereas the additives usually lower the viscosity of the solution, foster the formation of stable electrolyte interfaces (SEI) or induce an enhanced solvation of the ions 11 . Without proper detailed understanding of the molecular mechanisms, often solvents and additives with high donor numbers are chosen as electrolyte components, which thus favor a preferential solvation of lithium ions 14–16 . Previous experimental and simulation results highlighted the validity of the underlying donor number concept for simple one-component solvents and their influence on polyelectrolyte dissociation processes 14,17,18 , whereas the corresponding findings for solvent mixtures draw a less clear picture 19,20 . As it was observed for mixtures of aprotic solvents, the solvation behavior of ions is mainly dominated by the composition of the solution 19–21 in combination with electrostatic iondipole interactions 22 . For simple salts in water-dimethyl sulfoxide (DMSO) mixtures, it was observed that sodium ions are preferentially solvated by water molecules, whereas chloride ions favor a coordination by DMSO, which stands in sharp contrast to the corresponding donor numbers of the solvents in pure solution 23 . Furthermore, it was observed that sodium chloride reveals a homoselective preferential solvation behavior by water molecules in watermethanol 24 and DMSO-acetonitrile (AN) solutions 25,26 , as well as in water-AN mixtures 27 . For more complex organic ions, recent simulations of DMSO-water mixtures 28 also revealed a homoselective solvation mechanism, where water molecules form the first solvation shell around both ion species. Hence, all findings for ternary solutions imply a complex coordination behavior around the ions, which depends on the considered ion species, the composition

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of the solution, and the properties of the individual components. The consequences of these observations were recently discussed by us is a previous publication 28 , where we showed that the amount of ion pairs increases significantly with increasing DMSO mole fractions in water-DMSO mixtures. As a result, the ionic conductivity decreases considerably, which has important consequences for electrolyte compositions in modern battery devices 10,14 . The current lack of understanding thus reveals, that more detailed insights are needed for a proper understanding of interactions in electrolyte solutions in combination with consequences for ionic conductivities and ion pair formation processes. In this article, we present an application of the Kirkwood-Buff (KB) theory, which focuses explicitly on co-solute assisted ion pairing mechanisms in ternary electrolyte solutions. In more detail, we develop a local/bulk binding model in order to study the accumulation behavior of the co-solute molecules around the individual dissociated or associated ion reference states. All outcomes of the approach are verified by atomistic molecular dynamics (MD) simulations concerning sodium and chloride ions in water-dimethylacetamide (DMAc) mixtures. The remainder of the article is organized as follows. In section 2, we introduce the theoretical framework and discuss the main outcomes of our analysis. The simulation details are presented in the third section. All numerical results are compared to our analytical expressions in section 4. We briefly conclude and summarize in the last section.

2. Ion pair dissociation-association processes in ternary electrolyte solutions Ion pair dissociation-association equilibrium We consider a simple dissociation-association equilibrium for ion pairs in an arbitrarily chosen solvent in accordance with (X+ Y− )sol − )− −* − (X+ + Y− )sol 4


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where the associated ion pair is denoted by (X+ Y− )sol and the fully dissociated ion pair by (X+ +Y− )sol . Here, we assume an infinitesimally small ion concentration, thus having negligible influence on the overall pH value in aqueous solution, and we do not distinguish between CP, 1SP and 2SP. In fact, each ion pair representing a CP, 1SP or 2SP state is considered as an associated ion pair (X+ Y− )sol . The connection with the chemical equilibrium P constant K is established by the relations 29 ∆µ = j νj µj = 0 and ∆µ0 =

X

νj µ0j = −RT ln K,

(1)

j

with the chemical potentials µj , the standard chemical potentials µ0j , the stoichiometric coefficients νj , the molar gas constant R and absolute temperature T . With regard to Eqn. (1), the chemical equilibrium constant is defined by 29

K=

Y

ν

aj j

(2)

j

where aj denotes the corresponding chemical activity of the considered species. In more detail, the chemical activity reads aj = γj xj ,with the chemical activity coefficient γj , and the mole fraction xj = ρj /ρ, where ρj denotes the bulk number density of species j in combination with the total bulk number density ρ of the solution. In order to develop a theoretical approach for the influence of co-solutes on the chemical equilibrium, we introduce the concept of m-value analysis, which was also shown to be helpful for the understanding of protein denaturation and stabilization experiments in presence of co-solutes 30–32 . This approach assumes a linear change in several thermodynamic properties, like transfer free energies or equilibrium constants besides others, with respect to the concentration of co-solute species. Here, we study this approach for varying concentration of co-solutes from low to high molar concentration regimes. A well-known relation including

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the so-called m-value 30,31 reads

∆µcs = ∆µ0 − mRT ρ3

(3)

with the bulk number density of co-solute species ρ3 , where ∆µcs expresses the modified chemical potential difference between the dissociated and the associated ion state in presence of co-solutes. Thus, a modified version of Eqn. (1) can be formulated 32 , which reads

Kcs = exp(−∆µcs /RT ) = exp(−∆µ0 /RT ) · exp(mρ3 ) = K 0 · Kapp

(4)

Kapp = exp(mρ3 )

(5)

with

where Kapp denotes the apparent chemical equilibrium constant with regard to the influence of co-solutes.

The influence of co-solutes on the chemical equilibrium After having established the basic chemical relations, we now introduce the underlying model for our theoretical approach. In fact, a consistent description for solution effects based on thermodynamic and statistical mechanics arguments is provided by the Kirkwood-Buff (KB) theory, which was originally developed as a molecular theory of solutions and liquid mixtures 33 . In the following, we will present a simple derivation of some expressions in terms of the KB theory, whereas more rigorous formulations under consideration of statistical mechanics arguments can be found in Refs. 32–48. More specifically, we focus on the corresponding analysis of the accumulation behavior for co-solute (’j = 3’) around both ion species (as denoted by the common index ’j = 2’) in presence of water or any other solvent (’j = 1’). Here, and in agreement with standard approaches for electrolyte solutions 31,38,49–51 , we assume that

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b b a

a

K + −

+

−

Figure 1: Schematic illustration of co-solute-induced ion pair dissociation, where co-solute species are represented as black filled spheres. The cation (blue sphere) and the anion (red sphere) are located in the local region (a) and thus form an associated (either CP, 1SP or 2SP) ion pair (left side) in contrast to dissociated ions on the right side. The chemical potential in the bulk region (b) for all species remains constant for both ion states and solvent molecules are not shown for the sake of clarity.

cations and anions are indistinguishable in accordance with j = 2 ≡ ±. Furthermore, we assume a constant temperature T and a vanishing ion density ρ2 → 0 in the limit of infinite dilution. In consequence, a vanishing ion concentration and the indistinguishable ion approach are the only assumptions that are needed for our approach. Interestingly, the use of KB theory for the study of dissociable groups and the properties of electrolyte solutions was already discussed in Refs. 52–54. In seminal works, KB theory was also generalized in order to be applicable for the study of solvent effects on chemical equilibria 55,56 . Recent extensions in this direction also focus on chemotaxic effects and molecular association processes 57,58 . A schematic representation of our approach is presented in Fig. 1. The system including all species is divided into two subsystems 34 : the local system (a) with volume Va including co-solute species, water molecules and ions, and the bulk system (b) with co-solute and water molecules with volume Vb . The volume relation reads Vb Va , which also means that the ions in the local volume do not perturb the co-solute and solvent distribution in subsystem (b). With regard to thermodynamic and chemical equilibrium, one can define

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two Gibbs-Duhem relations for the systems (a) and (b) 31,34,35

−N10 dµ1 − N20 dµ2 − N30 dµ3 + (Va − N20 V2pmv )(dπ + dp) = 0

(6)

−N1 dµ1 − N3 dµ3 + Vb dp = 0

(7)

and

with the osmotic pressure π and the total pressure p, the resulting partial molar volume of the ions V2pmv and the number of molecules Nj0 (system (a)) and Nj (system (b)) with j = 1, 2, 3. Under constant pressure conditions with dp = 0, the second equation (Eqn. (7)) can be used to derive an expression for dµ1 , which can be inserted into Eqn. (6). With van’t Hoff’s law π = RT (N20 /Va ) 29 for N20 /Va → 0 and the approximation (Va − N20 V2pmv ) ≈ Va , the resulting chemical potential after division by N20 = 1, in terms of a single ion pair, reads dµ∗2 = d(µ2 − RT ln ρ02 ) with ρ02 = N20 /Va . In the limit of negligible ion concentration, the chemical potential of the ion pair further reduces to the standard chemical potential with dµ∗2 ≈ dµ02 . Differentiation with respect to the chemical potential of the co-solute finally gives ν23 = −

∂µ02 ∂µ3

= N23 − p,T,ρ2 →0

ρ3 N21 ρ1

(8)

as a definition for the preferential binding coefficient ν23 for co-solute molecules around the ions, including the new definition N2j = Nj0 and the bulk number density ρj = Nj /V ≈ Nj /Vb with V = Va + Vb . Hence, the value of Eqn. (8) can be used to determine an excess or deficit of co-solute molecules around the ions when compared to bulk solution. The connection between the preferential binding coefficient and the KB theory can be derived by the corresponding definition of the cumulative number of molecules around the ions Z N2j (r) = 4πρj

r

r0 2 g2j (r0 ) dr0

0

8


(9)

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with the pair radial distribution function between the ions and species j as defined by g2j (r) = ρ2j (r)/ρj , where ρ2j (r) denotes the local number density of species j around the ions. In more detail, Eqn. (8) can be also expressed by Kirkwood-Buff integrals in accordance with Z Gij = 4π

∞

r2 [gij (r) − 1] dr

(10)

0

which can be approximated as Z

rc

Gij = lim Gij (rc ) = 4π lim rc →∞

rc →∞

r2 [gij (r) − 1] dr

(11)

0

where rc denotes a finite cutoff distance, as defined by the relation limrc →∞ gij (rc ) = 1, such that Gij (rc ) ≈ Gij , which also yields a relation for the number of excess molecules around xs = ρj G2j 36,39,48 . The corresponding KB integrals for charged species are defined the ions N2j

by 38,49 G2j = Gj2 =

|p+ | |p− | G+j + G−j p± p±

(12)

with G2j = G±j = Gj± and j = 1, 3, where |p+ | and |p− | denote the number of charged species in the chemical reaction with p± = |p+ | + |p− |. As a side remark, several finite-size corrections for KB integrals in computer simulations were proposed in order to account for differences between closed and open ensembles 59–65 . Most of the approaches rely on the fact, that the measured finite-size KB integral Gij ≡ Gij (rc ) differs from the infinite KB integral 59,65 in open systems, as denoted by G∞ in accordance with ij

Gij = G∞ ij +

A L

(13)

where A is a constant, which does not depend on the value of the simulation box length L. As it was often observed for sufficiently large simulation box lengths including a broad bulk 59,61 region like in our simulations, the differences between Gij and G∞ , such ij are rather small

that we can ignore the correction factor for our analysis. 9



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In combination with Eqn. (9) and Eqn. (11), the resulting relation for Eqn. (8) reads

ν23 = ρ3 lim (G23 (rc ) − G21 (rc )) = ρ3 (G23 − G21 ) rc →∞

(14)

which explicitly relies on the differences between the excess volumes of co-solute and solvent molecules around the ions 38–48,66–68 . If we now assume different chemical potentials for the individual ion states in presence of co-solutes, as denoted by the superscripts A (associated) A and D (dissociated), in accordance with ∆µcs = µD cs − µcs , one can calculate the difference

in the preferential binding coefficients between the dissociated and the associated ion state in accordance with ∆ν23 = −

∂∆µcs ∂µ3

= ∆N23 − p,T,ρ2 →0

ρ3 ∆N21 = ρ3 (∆G23 − ∆G21 ) ρ1

(15)

where the corresponding differences in the number of co-solute and solvent molecules, and the differences in the KB integrals around the dissociated and the associated ion state are A A D and ∆G2j = GD − N2j defined by ∆N2j = N2j 2j − G2j , respectively. In combination with

Eqn. (4) and Eqn. (15) and the definition 29 µj = µ0j + RT ln aj , it follows ∆ν23 = RT

∂ ln Kcs ∂µ3

= p,T,ρ2 →0

∂ ln Kapp ∂ ln a3

(16) p,T,ρ2 →0

with the chemical activity aj and (∂ ln K 0 /∂ ln a3 )p,T,ρ2 →0 = 0. Thus, Eqn. (16) implies that a stronger accumulation of co-solute molecules in the local region around the ions shifts the chemical equilibrium either towards the dissociated (∆ν23 > 0) or the associated state (∆ν23 < 0). It has to be noted that comparable expressions can be also derived with regard to the solubility of species in mixed solvents 69 .

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Implications for ion pair association-dissociation equilibria In the following, we assume a simple two-step process and the existence of chemical equilibrium between the dissociated and the associated ion pair state. The associated ion state includes any CP, 1SP and 2SP pairs, and all ion pairs with a distance shorter than r ≤ lB with the Bjerrum length lB = e2 /(4π0 r kB T ), including the elementary charge e, the vacuum permittivity 0 , the dielectric constant of the solution r and Boltzmann’s constant kB . With regard to this definition, we follow previous reasonable assumptions for an appropriate definition of ion pairs as discussed in Ref. 1. Moreover, in the limit of low ion concentrations, we can neglect any twofold participation of ions in distinct ion pairs. The probability of associated ion pairs for a 1 : 1 salt is given by pA , whereas the probability of dissociated ion pairs is defined by pD = 1 − pA . If we now assume that the actual fraction of ion pairs θc coincides with the corresponding value of pA , one can define a detailed balance condition according to K0 =

1 − pA 1 − θc pD = = pA pA θc

(17)

which can be inserted into Eqn. (4) with Kapp = 1 in absence of co-solutes. The agreement between pA and θc is mainly valid for ideal conditions 29 with γA = 1 and hence K 0 = xD /xA (cf. Eqn. (2)), such that pA = xA = θc . In combination with Eqn. (5), we can rewrite Eqn. (3) in order to read ln Kcs = ln K 0 + mρ3

(18)

which allows us to introduce the m-value with the actual co-solute bulk number density ρ3 . If we now define the actual fraction of ion pairs in presence of co-solutes as θccs and in absence of co-solutes (ρ3 = 0) as θc0 , we can rewrite the above introduced relation as ln

1 − θccs θccs

= ln

11

1 − θc0 θc0

+ mρ3


(19)


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which in principle, under the assumption of a strictly linear behavior, can be used to determine the m-value from simulation and experimental data via linear regression. Furthermore, we can also find an explicit expression for the m-value after differentiation of Eqn. (18) with the co-solute bulk number density yielding m=

ln Kcs ∂ρ3

= T,p

∂ ln Kapp ∂ρ3

(20) T,p

which can be further transformed into

∂ ln Kapp ∂ ln a3

T,p

∂ ln a3 ∂ρ3

= ∆ν23 T,p

∂ ln a3 ∂ρ3

(21) T,p

and

∂ ln a3 ∂ ln ρ3

T,p

∂ ln ρ3 ∂ρ3

= T,p

a33 ρ3

(22)

in order to provide a direct relation for the m-value with the KB expressions 31 in accordance with m=

∂ ln Kapp ∂ρ3

= T,p

a33 ∆ν23 ρ3

(23)

including the derivative of the chemical activity 31,33,35,37,44 a33 =

∂ ln a3 ∂ ln ρ3

= T,p

1 1 + ρ3 (G33 − G31 )

(24)

and the corresponding KB integrals 33,37 Gij as defined in Eqn. (10). As a central quantity, it has to be mentioned that specific expressions for the derivative of the chemical activity in binary and higher order solutions can also be derived 44,69 . In consequence, after inserting Eqn. (23) into Eqn. (19), one can derive the relation ln

1 − θccs θccs

= ln

1 − θc0 θc0

12

+ a33 ∆ν23


(25)

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or written equivalently as ln Kcs = ln K 0 + a33 ∆ν23 . Solving for the chemical equilibrium constant yields

Kcs = K 0 exp(a33 ∆ν23 ),

(26)

and an expression for the fraction of associated ion pairs

θccs =

1+

K0

1 exp(a33 ∆ν23 )

(27)

which highlights the fact that the ion pair association behavior depends non-linearly on the values of a33 and ∆ν23 . Furthermore, it is convenient to express the derivative of the chemical activity (Eqn. (24)) as a33 = 1 +

∂ ln γ3 ∂ ln ρ3

(28) T,p

with the activity coefficient γ3 70 , such that Eqn. (25) can be rewritten as ln

1 − θccs θccs

= ln

1 − θc0 θc0

+

1+

∂ ln γ3 ∂ ln ρ3

! ∆ν23 .

(29)

T,p

The benefit of Eqn. (28) follows from the fact, that activity coefficients of components in simple liquid mixtures can be calculated with reasonable accuracy to experimental values in terms of well-established computational methods like the UNIFAC framework 71 . Comparable strategies were also recently used for the calculation of activity coefficients for distinct ionic species in order to parameterize KB based force fields 70,72 . With regard to Eqn. (10), the KB integrals can also be interpreted as excess volumes of the individual solvent and co-solute species around the ions. Hence, Eqn. (14) can be rewritten as ν23 = ρ3 (V23xs − V21xs ) = ρ3 ∆V31xs

13


(30)


where V21xs and V23xs denote the excess volumes of solvent and co-solute molecules around the ions. With regard to the partial molar volumes of the species V1pmv and V3pmv around the ions, it is often reasonable to assume that ν23 < 0 for V1pmv V3pmv , as it was also in more detail discussed in Ref. 41. This assumption can be mostly related to the excluded volume of species at short distances around the ions, such that large co-solute molecules reveal a more negative excluded volume when compared with smaller solvent molecules. With regard to these assumptions, one can further rationalize the linear relation ∆ν23 ∝ ∆∆V31xs with xs{D}

∆∆V31xs = ∆V31

xs{A}

− ∆V31

, such that the value for the difference of the preferential bind-

ing coefficient does not change its sign when compared to the individual preferential binding coefficients ν23 around the associated or dissociated ion state. As can be expected, preferentially excluded co-solute molecules thus enforce ion pair association due to ∆ν23 0 for V1pmv /V3pmv 1, which is in accordance with Kcs < K 0 (Eqn. (26)). In fact, such a co-solute-induced increase in the amount of associated ion species results in a decrease of the ionic conductivity 28 , which is disadvantageous for electrolytes in modern electrochemical storage devices. Despite the underlying simple assumptions, notable deviations to the aforementioned effects of excluded volume in terms of preferential attraction mechanisms were recently discussed in Ref. 73. After having established all necessary relations, we now turn to MD simulations of sodium and chloride ions in water-DMAc solutions in order to verify the basic assumptions outlined in this section.

3. Simulation Details All MD simulations were performed with the GROMACS 4.6.5 74 software package. Three simulation setups (solvent mixtures without ions, solvent mixtures with ions and solvent mixtures with fixed ion positions) of DMAc/water mixtures were studied in order to verify the relations presented in the previous section. In all simulations, we used a well-parameterized and validated KB derived force field for DMAc 75 in combination with the SPC/E water

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model 76 . Simulation runs of DMAc/water mixtures without ions were performed for different mole fractions of DMAc ranging from xDMAc = 0.1 − 0.9. These simulations were used to calculate the derivatives of the chemical activity in solutions without ions 28,44

aαα =

1 , 1 + ρα (Gαα − Gαβ )

(31)

with α, β being either DMAc or water molecules. Herewith, we intend to detect non-ideal solution effects, as it was in more detail described in Ref. 28. The second set of simulations was mainly intended to study the association-dissociation behavior of sodium and chloride ions in presence of DMAc co-solute species and to obtain reliable values for the fraction of ion pairs. We used a KB derived force field, as introduced in Ref. 72, for the ionic species. The distance-dependent fraction of associated ion pairs was calculated by the normalized cumulative number distribution function 18,28 4π ρc θc ≡ θc (lB ) = Nc

Z

lB

r2 gic (r) dr

(32)

0

with the number Nc and the number density ρc of counterions and the radial distribution functions gic (r) between co-ions (i) and counterions (c), where lB denotes the actual value of the Bjerrum length for the mixture. We randomly positioned 10 pairs of free and mobile sodium and chloride ions in DMAc-water mixtures (mole fraction ranges between xDMAc = 0 − 0.5) in a cubic simulation box. The concentration of ions was low (for all simulations c+ = c− ≤ 0.56 mol/L) in order to take the dilute concentration limit of our theory into account. More details on the final and equilibrated simulation box volumes can be found in the supplementary material. The third set of simulations was performed for a reliable evaluation of the preferential binding coefficients to associated and dissociated ion pair states. With regard to this aim, we performed constrained MD simulations for a single Na+ -Cl− pair by freezing the distance between the ions. In order to model the associated state (CP), the distance between Na+ and 15



Cl− was fixed to 0.25 nm (first maximum in the radial distribution function as obtained by the second set of simulations and shown in the supplementary material), whereas the dissociated state had a distance of 1.3 nm, which is larger than the resulting Bjerrum length of the individual mixtures (maximum values of Bjerrum length for all mixtures around 1.1 nm). All constrained simulations were performed for DMAc mole fractions xDMAc = 0.1 − 0.5. The corresponding simulation box volumes and more details on the simulation protocol are presented in the supplementary material. Electrostatic interactions for all systems were calculated by the Particle Mesh Ewald method 77 with a Verlet pair list cutoff scheme and a short-range radius of 1 nm. The same cutoff scheme was also used for the calculation of the Lennard-Jones interactions in combination with dispersion corrections. The Fourier grid spacing was 0.16 nm and all covalent bonds were constrained by the LINCS algorithm 78 . The time step in all simulations was 2 fs. All systems were equilibrated for 5 ns at 300 K and 1 bar pressure by using the Nose-Hoover thermostat 79 in combination with the Parrinello−Rahman barostat 80 . The initial box size of all systems was (4 × 4 × 4) nm3 . Pure solution mixtures and free ions in solution mixtures were simulated for more than 2 µs each in the isothermal-isobaric (NPT) ensemble. The constrained MD simulations for associated and dissociated ion states were run for 20 ns each in the constant volume-isothermal (NVT) ensemble. In total, more than 30 µs of simulation time was evaluated in order to generate a sufficient amount of data.

4. Simulation results In order to study the ideal behavior of the DMAc-water mixture without ions, we calculated the derivatives of the chemical activity for DMAc and water molecules in accordance with Eqn. (31). The corresponding results for simulations without ions are shown in Fig. 2. As it was pointed out in Ref. 28, the values for the derivatives of the chemical activity provide an estimate for the non-ideality of the solution. Thus, significant deviations from aαα = 1

16


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Page 17 of 34

20 α=DMAc α=Water

18 16 14 12 aαα

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


10 8 6 4 2 0 0.1

0.2

0.3

0.4

0.5 xα

0.6

0.7

0.8

0.9

Figure 2: Derivative of the chemical activity aαα for the individual components water (α=1) and DMAc (α=3) in DMAc-water mixtures for different mole fractions of DMAc and water. The corresponding error values are within symbol size. reveal a higher non-ideality for each component α = 1, 3 of the mixture. With regard to the results shown in Fig. 2, it becomes evident that the water chemical activity derivative reveals values between a11 = 1 − 2 for all mole fractions of water, which can be regarded as a weak non-ideal behavior. In contrast, the values for DMAc show a highly non-ideal distribution for all mole fractions xDMAc ≥ 0.3, which can be attributed to the large molecular size of DMAc when compared with water molecules. Based on these results, we thus focus in our corresponding analysis on the behavior at low DMAc mole fractions xDMAc ≤ 0.4 in order to fulfill the requirement of ideality, or at least weak non-ideality, for the derivative of the chemical activity of DMAc as co-solute in terms of constant m-values (Eqn. (23)). In order to study the spatial distribution of DMAc and water molecules around the associated and dissociated ion pairs as implemented in the constrained MD simulations, we show the radial distribution functions for a mole fraction xDMAc = 0.2 in Fig. 3. It can be clearly seen that DMAc molecules are preferentially excluded from the ion species due to larger distances of the corresponding first solvation shells when compared with water molecules. This finding

17



10

g23(dissociated) g21(dissociated) g23(associated) g21(associated)

9 8 7 g2α(r)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 34

6 5 4 3 2 1 0 0.2

0.4

0.6 r [nm]

0.8

1

1.2

Figure 3: Radial distribution functions g2α (r) for a mole fraction xDMAc = 0.2 around both ion species (as denoted by the index ’2’) for water (index ’1’, represented by black lines) and DMAc molecules (index ’3’ with red lines). The values for the dissociated state are represent as solid lines and for the associated ion state as dashed lines. is valid for the associated and the dissociated ion state and can be simply rationalized by the larger partial molar volume of DMAc when compared with water molecules. Moreover, one can see a pronounced increase in the corresponding peak heights for water molecules at short distances around the dissociated ion state due to a larger accessible volume and the corresponding small size of the water molecules. In contrast, the values for DMAc are roughly comparable for the dissociated and the associated ion state. In order to verify the validity of our approach, we also calculate the corresponding values for m in accordance with Eqn. (23) for different DMAc mole fractions. In more detail, we computed the differences in the preferential binding coefficient ∆ν23 for DMAc and water molecules around the associated and the dissociated ion pairs in the constrained MD simulations. Herewith, we avoid sampling problems of unlikely conformations and are thus able to determine the values for ∆ν23 for different co-solute concentrations in accordance with Eqn. (15) with high statistical accuracy. The numerical values for a33 in ternary solution (Eqn. (24) and Fig. 2) and the corresponding values for ρ3 and ∆ν23 , as both obtained from 18


Page 19 of 34

the constrained MD simulations, were then inserted into Eqn. (23) in order to compute the values of m for different mole fractions of DMAc. The corresponding results are depicted in Fig. 4. It can be clearly seen that the values for m are varying slightly and increase with 2 1 m-value [nm-3]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 -1 -2 -3 -4 -5 0

0.1

0.2

0.3

0.4

0.5

0.6

xDMAc

Figure 4: Values of m in accordance with Eqn. (23) for different mole fractions of DMAc. The black straight line corresponds to the mean value with m ¯ = (−1.24 ± 0.27) nm−3 . More details on the calculation can be found in the text. higher DMAc mole fractions. All results reveal negative values between m = −2.5 nm−3 and m = 0 nm−3 , which corresponds to a shift of the chemical equilibrium to the associated ion pair state in presence of DMAc (Eqn. (18)). With regard to the assumed ideal behavior of our approach in terms of constant m-values for all DMAc mole fractions, we computed the mean value of m with standard deviation for mole fractions xDMAc = 0.1 − 0.5. The corresponding value reads m ¯ = (−1.24 ± 0.27) nm−3 , and is marked as a straight black line in Fig. 4. In order to verify the constant m-value approach in the context of Eqn. (18), we plot the values for ln Kcs , as obtained by the free ion MD simulations and evaluated by Eqn. (32) against the corresponding values for the DMAc number density ρ3 . The results are presented in Fig. 5. It becomes obvious that the values of ln Kcs follow a linear decreasing trend with 19



4 3 2 ln Kcs

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 20 of 34

1 0 -1 -2 2

2.5

3

3.5

4 4.5 -3 ρ3 [nm ]

5

5.5

6

Figure 5: Logarithm of the ion pair chemical equilibrium constant Kcs for increasing values of the DMAc bulk number density ρ3 . The black straight line corresponds to the result of a linear regression analysis. All errorbars are within symbol size. increasing bulk number densities of DMAc in agreement with Eqn. (18). We are able to determine the corresponding value for pure water with ln K 0 = 6.01 ± 0.58 by a linear regression analysis of the data shown in Fig. 5, which corresponds to an association constant of θc0 ≈ 0.002 in pure water. As can be seen in the supplementary material, the regressionbased value coincides well with the corresponding value of ln K 0 as obtained by simulations without constraints for free ions in pure water. In more detail, the corresponding results from the free ion MD simulations highlight the dominance of CPs for the association behavior, as discussed in the supplementary material, whereas the contribution from 1SP and 2SP pairs is significantly smaller. In general, it can be clearly seen that an increasing DMAc bulk number density favors the association of ion species. Furthermore, the corresponding slope of the linear fit relation, as shown in Fig. 5, yields a value of m e = (−1.28±0.10) nm−3 , which is in good agreement with the obtained value from Fig. 4 with m ¯ = (−1.24 ± 0.27) nm−3 . In addition, it has to be noted that a variation of the interaction range R for θc (R) in Eqn. (32) with R < lB or R > lB does not change the corresponding values significantly. As it can

20


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be seen in the supplementary material with regard to ion radial distribution functions, and further analysis for association constants in simulations without constraints, CPs at r lB dominate, such that slight and moderate variations of r, as long as the corresponding value is larger than the effective CP distance, have a minor influence on the association behavior. The pronounced and linear negative correlation between ln Kcs and ρ3 is also verified by the Pearson correlation coefficient which reveals a value of R2 = −0.98. As a final verification, we compare the derivatives of the chemical activity, when calculated with KB integrals in terms of Eqn. (24) (denoted by a33 (KB) and already shown in Fig. 2), with the values for a33 as calculated by the linear regression analysis (Eqn. (25)) with regard to the data shown in Fig. 5, which then reads

a33 =

ln(Kcs /K 0 ) ∆ν23

(33)

for distinct values of xDMAc . For the corresponding analysis, we use a value of ln K 0 ≈ 6.1 as obtained from linear regression. The differences in the preferential binding coefficients were evaluated with regard to the dissociated and the associated CP ion pair. A comparable analysis for distinct associated ion pairs in the supplementary material highlights the fact that the chosen associated reference ion state only marginally influences the results as long as the associated ion pair significantly differs from the dissociated ion state. A perfect agreement between both values in terms of a33 = a33 (KB) would thus indicate a direct connection between the properties of the solution and the ion pair association behavior, which is one of the cornerstones of our theory. The corresponding comparison is shown in Fig. 6. A good agreement between both values, as expressed by the black solid line with unit slope, can be observed for all values a33 (KB) < 4, which corresponds to mole fractions xDMAc < 0.4. The strongly deviating value at a33 (KB) ≈ 7 corresponds to a mole fraction of xDMAc = 0.4, which clearly reveals a non-ideal behavior, as can be also seen in Fig. 2. With regard to these results, it can be concluded that our approach is verified with all of

21



20 15

0

(ln Kcs-ln K )/∆ν23

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 34

10 5 0 -5 0.5

1

2 a33(KB)

3

4

5 6 7 8

Figure 6: Values for the derivative of the chemical activity a33 (KB), as obtained from the KB approach according to Eqn. (24) in comparison to the values obtained from linear regression analysis and Eqn. (33) for DMAc mole fractions between xDMAc = 0 − 0.4. The black solid line corresponds to an exact agreement with slope equals unity as plotted with a logarithmic scale. Errorbars are within symbol size. implications. In summary, increasing DMAc mole fractions in water induce a strong tendency of Na+ -Cl− ion pair association.

5. Summary and conclusion We presented an application of KB theory, which can be applied to describe the influence of co-solutes on the chemical equilibrium of ion pair association and dissociation processes. The introduced framework relies on the evaluation of KB integrals in combination with the definition of a local and a bulk region around the ion species. The corresponding local distribution of solvent and co-solute molecules induces a shift of the chemical equilibrium towards the dissociated or the associated ion pair state. In fact, only two crucial parameters determine the new chemical equilibrium state, which are the derivative of the chemical activity of the co-solute molecules and the associated preferential binding coefficient. Our approach is

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applicable for low ion concentrations and nearly ideal or close to ideal global solution properties. Due to these reasons, the presented theoretical framework can be mostly understood as a qualitative picture for co-solute induced shifts in ion pair equilibria. As a rule of thumb, it can be concluded that the preferred ion state, either associated or dissociated, shows a stronger co-solute accumulation when compared with the disfavored state. Our findings are important for electrolyte solutions in modern electrochemical storage devices. As we already discussed before, the electrolyte in these devices can be regarded as a complex mixture including salts, organic solvents with high dielectric constants, and a significant amount of molecular additives. Often, these additives have a lower dielectric constant when compared with the main solvent, and it is thus assumed that they are preferentially excluded from the ion species 10 . In consequence, it can be concluded that the presence of those excluded additives induces a shift of the chemical equilibrium towards the associated ion state, which counteracts a beneficial increase of the ionic conductivity in combination with desired lower viscosities 11 . In order to verify our theory, we performed all-atom MD simulations of sodium and chloride ions in aqueous DMAc mixtures as model systems with varying DMAc mole fractions. The corresponding results revealed the validity of our preferential binding model and also highlighted crucial deviations for non-ideal solutions. In consequence, our approach introduces a useful framework in order to understand the influence of co-solutes on the chemical equilibrium between dissociated and associated ion pair states.

Acknowledgements We thank Torben Saatkamp and Klaus-Dieter Kreuer for useful discussions. The authors gratefully acknowledge financial funding from the Deutsche Forschungsgemeinschaft through the SimTech Cluster of Excellence (EXC 310), the Sonderforschungsbereich (SFB) 716, as well as the bw4cluster initiative for granting computing times at the JUSTUS cluster in Ulm.

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(67) Shulgin, I. L.; Ruckenstein, E. The Kirkwood-Buff theory of solutions and the local composition of liquid mixtures. J. Phys. Chem. B 2006, 110, 12707–12713. (68) Smiatek, J. Osmolyte Effects: Impact on the Aqueous Solution around Charged and Neutral Spheres. J. Phys. Chem. B 2014, 118, 771–782. (69) Smith, P. E.; Mazo, R. M. On the Theory of Solute Solubility in Mixed Solvents. J. Phys. Chem. B 2008, 112, 7875–7884. (70) Fyta, M.; Netz, R. R. Ionic force field optimization based on single-ion and ion-pair solvation properties: Going beyond standard mixing rules. J. Chem. Phys. 2012, 136, 124103. (71) Fredenslund, A.; Jones, R. L.; Prausnitz, J. M. Group-contribution estimation of activity coefficients in nonideal liquid mixtures. AIChE J. 1975, 21, 1086–1099. (72) Gee, M. B.; Cox, N. R.; Jiao, Y.; Bentenitis, N.; Weerasinghe, S.; Smith, P. E. A Kirkwood-Buff derived force field for aqueous alkali halides. J. Chem. Theory Comput. 2011, 7, 1369–1380. (73) Schroer, M. A.; Michalowsky, J.; Fischer, B.; Smiatek, J.; Gr¨ ubel, G. Stabilizing effect of TMAO on globular PNIPAM states: preferential attraction induces preferential hydration. Phys. Chem. Chem. Phys. 2016, 18, 31459–31470. (74) Pronk, S.; Páll, S.; Schulz, R.; Larsson, P.; Bjelkmar, P.; Apostolov, R.; Shirts, M. R.; Smith, J. C.; Kasson, P. M.; van der Spoel, D.; Hess, B.; Lindahl, E. GROMACS 4.5: a high-throughput and highly parallel open source molecular simulation toolkit. Bioinformatics 2013, 29, 845–854. (75) Kang, M.; Smith, P. E. A Kirkwood-Buff derived force field for amides. J. Comput. Chem. 2006, 13, 1477–1485.

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(76) Berendsen, H.; Grigera, J.; Straatsma, T. The missing term in effective pair potentials. J. Phys. Chem. 1987, 91, 6269–6271. (77) Darden, T.; York, D.; Pedersen, L. Particle mesh Ewald: An N log (N) method for Ewald sums in large systems. J. Chem. Phys. 1993, 98, 10089–10092. (78) Hess, B.; Bekker, H.; Berendsen, H. J. C.; Fraaije, J. G. E. M. {LINCS}: {A} linear constraint solver for molecular simulations. J. Comput. Chem. 1997, 18, 1463–1472. (79) Frenkel, D.; Smit, B. Understanding molecular simulation: from algorithms to applications; Academic press, 2001. (80) Parrinello, M.; Rahman, A. Polymorphic transitions in single crystals: A new molecular dynamics method. J. Appl. Phys. 1981, 52, 7182–7190.

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The Influence of Co-Solutes on the Chemical Equilibrium - A Kirkwood

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