Summary_eng (Статистическое описание многочастичных эффектов в классических ион-молекулярных системах), страница 3

These two main restrictions motivated theorists over the last couple ofdecades to discover new statistical-mechanical approaches to modify the PB equation. It is worthnoting, that achieved modications have suciently pushed forward the theory of electrolytes,which, in turn, allowed one to calculate physical-chemical characteristics of the real solutions, suchas solution dielectric permittivity and activity coecient.Most of the theoretical research are devoted to the investigation of the dependence of the different microscopic parameters of ions on the macroscopic quantities of the electrical double layer6(EDL), such as proles of the ionic concentrations near the charged electrodes, disjoining pressureand dierential capacitance (DC). The latest is the most important characteristic for the modernelectrochemical applications, in particular, for the designing of the electrochemical capacitors.

Forthe present moment it is well enough studied how characteristics of the ions (charge, size, polarizability, dipole moment) inuence on the properties of the EDL, appearing at the metal/liquid electrolyteinterface. However, until recently, it has been unexplored, how properties of the molecules of cosolvent impurity in the electrolyte solution have impact on the behavior of the DC.

This questionis one of the most important for the modern electrochemical technologies, where one of the mainproblems is to achieve the maximum DC at the minimum applied voltage. In this regard, from thefundamental point of view, two natural questions arise:ˆ How will DC change with the addition of the impurity molecules (co-solvent), having electronicpolarizability or permanent dipole moment, to the solution?ˆ How much is the quantitative dierence between the eects of the dipole moment and polarizability of the co-solvent molecules on the DC value?Answering the raised questions, in the rst part of the Chapter 1 eld-theoretical approach,allowing one to obtain the self-consistent eld equation, taking into account explicitly co-solventmolecules with nonzero dipole moments and/or polarizability tensor, has been formulated.

Usingobtained equation, the classical Gouy-Chapman EDL theory at the metal/electrolyte solution interface was generalized for the case, when the polar/polarizable co-solvent is added to the bulksolution.Electrolyte solution is examined, containing N+ point ions with a charge q+ (q+ > 0), N−ions with a charge q− (q− < 0), and solvent, described as a continuous dielectric medium witha constant dielectric permittivity ε. Besides, it is supposed that solution contains N0 co-solventmolecules, having dipole moment, equivalent to p and polarizability tensor γ̂c . It is also supposed,that solution is in the equilibrium in the volume V with the temperature T .

For simplicity, all ofthe interactions, except electrostatic ones, are neglected.Then, bearing in mind all stated above model assumptions and using standard HubbardStratonovich transformation, the representation of the conguration integral of the system at thethermodynamic limit in the form of the functional integral has been obtained. Further, within themean-eld approximation, i.e.

equaling variational derivative of the integrand functional to zero,the following self-consistent eld equation in ψ(r), generalizing classical PB equation in the case ofthe presence of the polarizable/polar solvent molecules in the electrolyte solution, has been obtainedq ψ(r)q ψ(r)− −− +kB TkB T∇ (ε̂(r)∇ψ(r)) = −4πρext (r) − 4π q− c− e+ q+ c+ e,(1)where the tensor of the local dielectric permittivity is introduced∇ψ(r)γ̂c ∇ψ(r) sinh βp|∇ψ(r)|p2 L(βp|∇ψ(r)|)2kB Tε̂(r) = εI + 4πc0 eγ̂c + Iβp|∇ψ(r)|kB T βp|∇ψ(r)|(2)with the Langevin function L(x) = coth x − 1/x; I is the unity tensor. Besides this, the generalexpression for the electrostatic free energy of the mixture within the mean-eld approximation hasbeen obtained.As an application of the obtained self-consistent eld equation (1), a generalization of the classical Gouy-Chapman theory of the EDL, appearing at the metal/electrolyte solution interface, wasformulated.

Considering the system, containing charged electrode, described by a at surface witha surface charge density σ , point ions of 1 : 1 electrolyte (q+ = −q− = e, e is the elementary charge)and point isotropically polarizable co-solvent molecules with dipole moment p and scalar polarizability tensor γ̂c = γc I, choosing axis z being normal to the electrode and placing there origin, onecan write a self-consistent eld equationd(z)ψ 0 (z) = 8πce sinh (βeψ(z)),dz7(3)6060Γ=0Γ = 0.1Γ=140γ̃c = 0γ̃c = 0.001γ̃c = 0.00540C̃C̃2020000510152005u0101520u0Figure 1: Dierential capacitance C̃ as a function of the electrode potential u0 at dierent Γ andxed parameters γ̃c = 10−3 , θ = 10−3 è p̃ = 0 (on the left).

Dierential capacitance C̃ as a functionof the electrode potential u0 at dierent γ̃c and xed parameters Γ = 0.1, θ = 10−3 and p̃ = 0 (onthe right).where the local dielectric permittivity is introducedγc (ψ 0 (z))2 sinh(βpψ 0 (z))p2 L(βpψ 0 (z))(z) = ε + 4πc0 expγc +,2kB Tβpψ 0 (z)kB T βpψ 0 (z)(4)and it has been assumed that far from the electrode (in the bulk solution) the average concentrationsof the ions are the same, i.e. c+ = c− = c.Then, using the boundary condition ψ 0 (∞) = 0, the rst integral of the equation (3) can beobtained, which denes the condition of the mechanical equilibrium of the solution and allows us,in turn, to obtain an electrostatic potential prole ψ(z).

Further, using the boundary condition−(0)ψ 0 (0) = 4πσ,(5)rst integral of the equation (3) at z = 0 and denition of the DCC=dσ,dψ0(6)where ψ0 = ψ(0) is the electrode potential, following analytical expressionC=ec cosh(βeψ0 )el2πσ(7)is obtained. The latter relation determines that within the self-consistent eld approximation theDC of the at EDL is proportional to the local dielectric permittivity el = (0) on the electrode.Before turning to the numerical results, the following dimensionless parameters were introduced3 ε, θ = l3 c, Γ = l3 c , ãäå l2p̃ = p/elB , γ̃c = γc /lBB = e /εkB T is the Bjerrum length.

ApartBB 0from that, the dimensionless potential u = βeψ , electric eld strength Ẽ = EβelB , surface chargedensity σ̃ = σβelB /ε, DC C̃ = dσ̃/du0 (u0 = u(0)), and distance from the electrode z̃ = z/lB wereintroduced. Then, it is established, that in the case of non-polar co-solvent (p̃ = 0) with nonzeropolarizability (γ̃c 6= 0) increase in both the co-solvent concentration Γ and the polarizability γ̃cleads to the sucient increase in DC at rather big electrode potentials (g.

1a,b). In the caseof the polar co-solvent at the zero electronic polarizability, in contrast, in the region of physicallyrealizable parameters, increase of the dipole moment leads to the slight increase in DC.Finally,inuence of the polarizable co-solvent on the electrostatic potential prole u(z̃) is discussed.

It isdetermined, that increase of the polarizable particles concentration leads to the slowdown of theelectrostatic potential decrease when moving away from the electrode. It is noted, that the eect ofthe polarizable co-solvent additive to the solution on the electrostatic potential prole is quite weakand becomes noticeable at suciently big values of the parameter Γ, that physically corresponds tothe case of big co-solvent concentrations or low enough temperatures.8Part 1.2As it was clearly shown by Kornyshev [Kornyshev, J.

Phys. Chem. B., 2007] within the latticegas model and later by him and coauthors using the MD simulations, the eects of the excludedvolume of ions must have a strong impact on the DC behavior at the large electrode potentials.Therefore, consideration of the excluded volume eects are extremely important for the correctdescription of the DC behavior at suciently high electrode potentials. As one can understand fromthe general considerations, at large enough polarizability and electrode potentials (eψ0 /kB T 1), asignicant amount of co-solvent molecules could accumulate near its surface.

Thereby, accumulationof the excess amount of the co-solvent molecules near the electrode surface would take place due tothe dielectrophoretic force. So, even if the co-solvent concentration is low far from electrode, nearits surface co-solvent concentration might become signicant, so that the excluded volume eectwill play an important role. Apart from that, also from the general considerations, it is clear thatat rather big polarizabilities, co-solvent molecules would compete with the ions for the ”space” nearthe charged electrode. So, the natural question arises: ”How will simultaneous account for both thepolarizability and excluded volume of the co-solvent molecules aect on DC behavior at the highelectrode potentials?”As such, in the second part of Chapter 1 classical density functional theory (DFT) of theelectrolyte solution, mixed with the polarizable molecules of the co-solvent, allowing one to considerboth the polarizability of the co-solvent molecules and excluded volume of the mixture particles,is formulated.

Using the local Legendre transformation, the grand potential of the mixture as afunctional of the electrostatic potential ψ(r) has been obtained:!Zε (∇ψ)2γc2Ω[ψ] =−+ ρext ψ − P µ+ − q+ ψ, µ− − q− ψ, µ0 +(∇ψ)dr.(8)8π2It is noted, that if the dependence of the particles pressure on the chemical potential is knownP = P (µ+ , µ− , µ0 ), then one can obtain the self-consistent eld equation, varying the functional (8)over the potential ψ(r).