Summary_eng (Статистическое описание многочастичных эффектов в классических ион-молекулярных системах), страница 5

It is assumed, that IL, conned in the area between two electrodes,does not participate in the electrochemical reactions, if it is used as an inert electrolyte for theinitiation of the electrocatalytic reactions of the dissolved in it molecules. However, if there arewater molecules in the form of impurity, they can participate in extraneous electrolysis reactions,which can interfere one in the investigation of the main electrochemical processes, proceeding onthe electrodes. Such processes would take place in supercapacitors. In such a manner, researches,oriented at the practical use of IL, in the rst place would like to know how the water molecules,absorbed by IL, would be distributed in the vicinity of the charged electrode.

Thus, two fundamentalquestions arise:ˆ Will the water molecules, located in the IL volume, be adsorbed to the surface of the chargedelectrode?ˆ How will adsorption (electrosorption) be inuenced by the electrode potential?With the reference to the above mentioned, a classical density functional theory (DFT) of theIL with a small amount of the molecules of polar solvent is formulated in the Chapter 2. In contrastto the existing theories, in addition to the electrostatic interactions between the species, the shortrange specic interactions have been taken into account. Formulated theory has been applied to thedescription of the at EDL at the metal/IL interface. It has been shown, that unlike the classicalself-consistent eld theories for EDL without accounting for the specic short-range interactions,current theory allows one to describe DC behavior with the increase of the water concentration inthe IL, recently discovered experimentally.It is assumed that ions have charges ±e (e is the elementary charge), whereas the molecules ofthe polar solvent carry the permanent dipole moment p.

Then, the grand thermodynamic potentialof the system is written in the formZZZε (∇ψ(r))2sinh βp|∇ψ(r)|Ω=−dr + ρc (r)ψ(r)dr − kB T drc0 (r) ln8πβp|∇ψ(r)|12−12ZdrXA−1ij φi (r)φj (r) +ZdrXi,jZci (r)φi (r) +driXci (r)Wi (r)iZ+(f (c+ (r), c− (r), c0 (r)) − µ+ c+ (r) − µ− c− (r) − µ0 c0 (r)) dr,(16)where ε is the dielectric permittivity of the pure ionic liquid, c± (r) are the local ionic concentrations,ρc (r) = e(c+ (r) − c− (r)) + ρext (r) is the local charge density, c0 (r) is the local concentration of thepolar solvent. First two terms determine the electrostatic energy of the ionic liquid. Third one is afree energy of the ideal gas of the dipolar molecules in the electric eld with the strength −∇ψ(r)according to the Langevin theory, β = 1/kB T , kB is the Boltzmann constant, T is the absolutetemperature; the forth and fth terms describe the free energy of the short-range specic interactionsbetween the components of the mixture, described by the ”interaction matrix” Aij (i, j = ±, 0) andctitious ”potentials” φj (r) of the specic interactions (j = ±, 0).

The sixth term describes theenergy of the mixture particles interactions with the external elds with potential energies Wi (r).In the seventh term f determines the free energy density of the reference system, which in our caseis the ternary mixture of the particles without specic and electrostatic interactions. Then, usinglocal Legendre transformation as in the chapter 1 and varying the grand thermodynamic potentialover the potentials ψ and φi , the following system of the self-consistent equations has been obtained:∇((r)∇ψ(r)) = −4πe (c̄+ (r) − c̄− (r)) ,(17)XAij c̄j (r),(18)4πp2L(βp|∇ψ(r)|)c̄0 (r)kB Tβp|∇ψ(r)|(19)andφi (r) =jwhere(r) = ε +is the local dielectric permittivity; L(x) = coth x − 1/x is the standard Langevin function;c̄± =∂P∂P, c̄0 =∂µ±∂µ0(20)are the local concentrations of the ions and solvent molecules.

Potentials of the specic interactionsφi,b in the bulk phase are calculated with the aid of the expression (18) for the concentrations ofthe mixture components in the bulk phase cj,b .Then, the case of the at electrode is considered. For the basis system two independent latticegases with equal eective cell volume v are chosen, so the mentioned above dependence of thepressure on the chemical potentials of species can be written in the form:P (µ+ , µ− , µ0 ) =kB T ln 1 + eβµ+ + eβµ− + ln 1 + eβµ0 .v(21)It has been noted, that such selection of the reference system is due to the fact that, generally,size of the molecules of the polar solvents (such as water or acetonitrile), added to the IL, haveeective size much less than the size of the ions.

This allows one to simulate solvent molecules asthe particles of the lattice gas, lattice sites of which are located in the space between the IL ions. Ithas been also noted, that presence of the large space between the ions in IL is indirectly conrmedby the data of the small molecules solubility in them, such as carbon dioxide.Parameters of the specic interactions are introduced using next expressions: A++ = vAc ,A−− = vAa , A+− = A−+ = vAca , and A00 = vAs , A0+ = A+0 = vBcs è A0− = A−0 = vBas . It isnoted, that the solution of the equation system (18) allows one to obtain potentials of the specicinteractions as the functions of the local electrostatic potential ψ(z) and electric eld strengthE = −ψ 0 (z), i.e.

φi = φi (ψ, E). Knowing these dependencies, one can calculate the componentconcentrations, using expressions (20). Then, the self-consistent eld equation could be written inthe following form:13Figure 4: Schematic illustration of the ionic liquid model with a small addition of the polar solvent.2πeγeβ (φ+,b −φ+ −eψ−W+ ) − eβ (φ−,b −φ− +eψ−W− )d(z)ψ 0 (z) = −,dzv 1 − γ + γ [eβ (φ+,b −φ+ −eψ−W+ ) + eβ (φ−,b −φ− +eψ−W− ) ](22)2where the following local dielectric permittivity(z) = ε +G(βpE)eβ (φ0,b −φ0 −W0 )4πp2 γskB T v 1 − γ 1 − sinh(βpE) eβ (φ0,b −φ0 −W0 )sβpE(23)is introduced, containing the auxiliary function G(x) = (sinh(x)L(x))/x2 ; γ = 2cv is the ion packingparameter [Kornyshev, J.

Phys. Chem. B, 2007], γs = c0,b v is the packing parameter of the polarsolvent molecules [Budkov, et al., Electrochimica Acta, 2018]. Equations (22) are complementedwith the following boundary conditions:ψ(0) = ψ0 , ψ(∞) = 0,(24)where ψ0 is the electrode potential.Surface charge density of the electrode is determined by the standard relation: σ = el Es /4π ,where el = (0) è Es = E(0) are values of the dielectric permittivity and the electric eld atthe electrode, respectively. DC is calculated by the conventional approach: C = dσ/dψ0 .

For thenumerical calculations it is assumed that W± (z) = Wc/a H(l − z) and W0 (z) = W0 H(l − z), whereH(z) is the Heaviside step-function and Wc/a ≤ 0 are energies of the specic adsorption of the ionsto the electrode; W0 is solvophobicity of the electrode, if W0 > 0 or the solvophilicity, if, on theopposite, W0 < 0; l is the characteristic thickness, in the range of which the specic interactionsof the ions and molecules with the electrode are relevant.

For the simplicity we neglect specicinteractions between the molecules of the polar solvent, i.e. consider As = 0. The latter condition ismotivated by the fact that at small enough concentrations of the polar solvent, which are realizablein practice, specic interactions (such as hydrogen bonding in the case of water or alcohols) have alittle eect on the DC.Turning to the discussion of the numerical results, it is considered that the thickness, withinwhich the specic interactions of the particles with the electrode are signicant, is approximatelyequal to l ' v 1/3 , i.e.

is of the molecular size order. For numerical calculations, following physicalparameters are chosen ε = 3.5, p = 1.8 D, T = 338 K , v 1/3 = 0.7 nm, γ = 0.8, γs = 0.025,Ac = Aa = 50 kB T and Aca = 0, which are roughly equal to the parameters of the real IL withwater addition.

It is mentioned, that the values of the parameters of the cation-cation and anionanion specic interactions are chosen in virtue of the adjusting theoretical values of the DC withexperimental ones for the pure ionic liquid [Goodwin et al., Electrochimica Acta, 2017].14Three physically interesting cases are investigated (g. 5): (1) IL without specic interactions(Bcs = Bas = 0), (2) hydrophobic IL (Bcs > 0, Bas = 0) and (3) hydrophillic IL (Bcs = 0,Bas < 0). It is shown, that in the absence of the specic ion-water interactions, the electrosorptioncurve is symmetric.

In that case, an electric eld gradient, occurring due to the screening of theelectrode charge by the oppositely charged ions, leads to the attraction of the polar molecules tothe electrode (so called dielectrophoretic force), so the increase in the electrode potential alwaysleads to the continuous growth of the water concentration. However, the hydrophobic propertiesof the cation and hydrophillic properties of the anion result in strongly asymmetrical behavior ofthe DC. Indeed, at the positive electrode potentials, water concentration on the electrode increasesin both cases, whereas at the negative potentials water concentration decreases, attains minimumand, then, begins to slowly grow.Figure 5: Dependence of the normalized water concentration (on the left) c0,s /c0,b at electrode(electrosorption curves) and dierential electric capacitance (on the right) on the electrode potentialfor three dierent cases: (1) ionic liquid without specic interactions (red line), (2) hydrophobicionic liquid (black line) and (3) hydrophillic ionic liquid (blue line).

The data is shown for γ = 0.8,γs = 0.025, Ac = Aa = 50 kB T , Acs = 0, As = 0, W0 = 2 kB T .Then, it is described, how behavior of the local water concentration inuences on the DC. Inthe case, when anion-water and cation-water specic interactions are absent, the graph for the DCdependence on the electrode potential presents a standard bell-shaped form (g. 5).

However,inclusion of the strong short-range specic interactions of the ions with the water molecules changesthe behavior of the DC qualitatively. In both cases of the hydrophobic and the hydrophillic ILthere is a pronounced satellite maximum on DC in the region of the positive electrode potentials.However, at the negative electrode potentials, presence of the specic interactions between theions and water molecules does not lead to new eects.