Summary_eng (1136181), страница 6
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Occurrence of the satellite maximum atthe positive electrode potentials is determined by the fact, that in the case of both hydrophillicand hydrophobic IL, increase of the electrode potential leads to the abrupt increase in the localconcentration of the water molecules. However, signicantly lower value of the water concentrationon the electrode in comparison to the concentration in the bulk phase at small negative potentialsand slow growth of the water concentration at large enough potentials in its absolute value leads tothe standard bell-shaped prole of the DEC.It is established, that increase of the water concentration in the bulk phase leads to the continuous growth of the water concentration on the electrode and more pronounced satellite maximumon the DC curves. It is noted that such behavior of DC is in qualitative agreement with the experimental behavior for the IL with small additive of water, presented in recent works [Wippermannet al., Phys.
Chem. Chem. Phys., 2017; Friedl et al, ChemElectroChem, 2017].Then, it has been shown that increase in the electrode hydrophobicity W0 leads to the quiteexpected electrosorption curves behavior: continuous decrease of the water concentration on the15electrode at all values of the electrode potential. The same trends are seen for hydrophobic IL.Finally, it has been discussed how specic adsorption of the ions to the electrode inuences on theDEC. For simplicity a case of the IL without impurities is investigated, with account of the specicadsorption of the cations to the electrode.
It has been established that increase (in absolute value)of the energy of the specic adsorption of ions to the electrode (Wc < 0) leads to the shift of theDC maximum into the region of the positive potential values. It can be explained by the fact thatspecic adsorption of the cations to the electrode leads to the fact, that, even at zero potential, theelectrode charge will not be equal to zero. Such a behavior of the DC is in the full agreement withthe results of self-consistent eld theory, proposed recently in the paper [Uematsu, et al. J. Phys.:Condens. Matter, 2018].In the conclusion of the Chapter 2 it is noted that formulated phenomenological theory willserve as a foundation for the following experimental and theoretical investigations of the ILs, mixedwith polar solvents, located on the charged electrodes, and will be useful for the interpretations ofthe future experimental and numerical results.Chapter 3The existing statistical theories of polar uids describe molecules as the point dipoles [Abrashkinet al., Phys.
Rev. Lett., Coalson, Duncan, J. Phys. Chem., 1996] or hard spheres with the pointdipole in its center [Levin, Phys. Rev. Lett., 1999]. As is well known, disregard of the details of theinternal electrical structure of the polar molecules in the calculations of the electrostatic free energyof the polar uid from the rst principles of statistical mechanics leads to its ”ultraviolet divergence”,which makes researches resort to the articial ”ultraviolet cut-o”, while integrating over the vectorsof the reciprocal space [Levy, et al., Phys.
Rev. Lett., 2012; Budkov et al., Eur. Phys. J. E, 2017].On the other hand, a ”local” theory without consideration of the charge distribution inside the polarmolecule, does not allow one to describe the behavior of the point charge potential, surrounded bythe polar molecules, at the distances of its linear size order. One can expect that ”non-local”statistical theory, taking such details into account, will be spared from the ultraviolet divergenceof the free energy, taking place in the local theories, and will allow one to investigate the behaviorof the electrostatic potential at the distances from the test point charge, that are comparable withthe characteristic distances between charged centers of the polar particles. On the other side, thelatest achievements in the experimental studies of the dierent organic compounds (such as proteins,betains, zwitterionic liquids, etc.), containing long enough polar groups (∼ 5 − 20 nm), challengestatistical physics to develop analytical approaches for the modelling of the polar particles not aspoint ones, but as a combination of the charge centers with xed or uctuating distances betweenthem.With reference to the above mentioned, a non-local statistical theory of the polar molecules,located in the electrolyte solution medium is formulated in the Chapter 3.
The electrolyte solution(further, for simplicity, salt) is investigated with N+ point cations with charges q+ > 0, N− pointanions with charges q− < 0 and Nd dipolar particles with point ionic groups with charges ±e (e isthe elementary charge), conned in the volume V at temperature T . Let the number of ions N±satisfy condition of the electrical neutrality q+ N+ + q− N− = 0. It is considered, that particles arelocated in solvent, which is modelled as a continuous dielectric medium with the constant dielectricpermittivity ε. It is postulated, that there is a probability distribution function g(r), associatedwith each dipolar particle, of the distance between charge centres.
Taking into account mentionedmodel assumptions, a congurational integral of the mixture is written in the following form:ZZQ = dΓd dΓs exp [−βH] ,(25)whereZdΓd (·) =Z Y(+) (−)Nddrj drjj=1V16(+)g(rj(−)− rj )(·)(26)(±)is the measure of integration over the coordinates rj of the ionic groups with the mentioned aboveprobability distribution function g(r), which, from its denition, must satisfy the normalizationcondition:Zg(r)dr = 1,(27)and1ZdΓs (·) =VZ YN+N+ +N−(+)drlZ YN−l=1(−)(28)drk (·)k=1is the integration measure over the coordinates of the salt ions; β = 1/kB T , kB is the Boltzmannconstant. Energy of the electrostatic interactions is written in the form:ZZ11dr dr0 ρ̂(r)G0 (r − r0 )ρ̂(r0 ) = (ρ̂G0 ρ̂) ,(29)H=22where G0 (r − r0 ) = 1/(ε|r − r0 |) is the Green function of the Poisson equation andρ̂(r) = eNdX(δ(r −(+)rj )− δ(r −(−)rj ))+ q+j=1N+Xδ(r −(+)rk )k=1+ q−N−X(−)δ(r − rl) + ρext (r)(30)l=1is the full charge density of the system, consisting of the microscopic charge density of the polarparticles and ions (rst, second and third terms) and charge density of the external charges (forthterm).
Also note, that in the work the contribution of the repulsion forces between the particles isneglected. It can be justied by the rather small concentrations of salt and polar particles. Usingthe standard Hubbard-Stratonovich transformation, in the thermodynamical limitV → ∞, Nd,± → ∞, Nd,± /V → nd,± ,the following representation of the congurational integral it the form of the functional integral hasbeen obtained:ZDϕQ=exp {−S[ϕ]} ,(31)Cwhere the following functional is introduced:ZZβS[ϕ] = (ϕG−1ϕ)−iβ(ρϕ)−ndrdr0 g(r − r0 )(exp iβe(ϕ(r) − ϕ(r0 )) − 1)extd02ZZ−n+ dr(exp [iβq+ ϕ(r)] − 1) − n− dr(exp [iβq− ϕ(r)] − 1).(32)Further, it is shown, that obtained integrand functional S[ϕ] in the case of the polar particles withhard geometrical structure passes into the well-known Poisson-Boltzmann-Langevin functional [referto Abrashkin et al., PRL, 2007; Coalson, et al., J.
Phys. Chem., 1996].Then, in the mean eld approximation the self-consistent eld equationZεe(ψ(r0 ) − ψ(r))− ∆ψ(r) = 2nd e dr0 g(r − r0 ) sinh4πkB T+q+ n+ exp [−βq+ ψ(r)] + q− n− exp [−βq− ψ(r)] + ρext (r)(33)for the electrostatic potential ψ(r) and the expression for the electrostatic free energy(M F )Fel= kB T S[iψ](34)have been obtained.Then, within the approximation of the weak electrostatic interactions eψ(r)/kB T 1, solvingthe self-consistent eld equation (33), the expression for the electrostatic potential of the test pointion with the charge q was obtainedZ4πqdkeikrψ(r) =,(35)ε(2π)3 k 2 + κ 2 (k)17whereκ 2 (k) = κ2s +is a screening function, and8πnd e2(1 − g(k))εkB TZg(k) =dre−ikr g(r)(36)(37)2 n + q2 n2is the characteristic function of the distribution; I = q++− − /2e is the ionic strength ofthe solution.
For the model characteristic functiong(k) =11+k 2 l26,(38)determining the following distribution function√ !36r,g(r) =exp −22πl rl(39)where r = |r|, in the absence of the ions (n± = 0), following expression for the electrostatic potentialof the point charge was obtained:qψ(r) =,(40)εloc rwhereεloc (r) =ε(1 + yd )1 + yd exp − lrs(41)2 is the dimensionless parameter,is the local dielectric permittivity; yd = 4πnd e2 l2 /(3εkB T ) = l2 /6rDpdetermining a ”strength” of the dipole-dipole electrostatic interaction and ls = l/ 6ε(1 + yd );−1/2rD = 8πnd e2 /εkB Tis the eective Debye screening length, associated with the chargedcenters of the polar particles.
Thus, the length ls determines a radius of the sphere around thepoint charge, within which the dielectric permittivity is smaller then its bulk value, determined bythe relation4πnd e2 l2εb = ε(1 + yd ) = ε +.(42)3kB TThus, it is found, that length ls can be interpreted as the eective solvation radius of the pointcharge, surrounded by the dipoles, within the linear response theory.With the presence of the salt ions in the bulk phase of the solution following expression for theelectrostatic potential has been obtainedq ψ(r) =u(yd , ys )e−κ1 (yd ,ys )r + (1 − u(yd , ys )) e−κ2 (yd ,ys )r ,(43)εrwhere√ 1/2p3κ2 l 2κ1,2 (yd , ys ) =1 + ys + yd ± (1 + ys + yd )2 − 4ys, ys = s ,l6pys + yd + (1 + ys + yd )2 − 4ys − 1pu(yd , ys ) =,2 (1 + ys + yd )2 − 4ys(44)(45)which transforms into the expression (40) at ys = 0.
In the absence of the polar particles (yd = 0)we obtain standard Debye-Hukkel (DH) potential:ψ(r) =q −κs re.εr(46)Then, after simple algebraic transformations it has been shown that mean-eld electrostatic freeenergy within the linear approximation takes the form:Z1(M F )=drρext (r)ψ(r).(47)Fel218To obtain the expression for the solvation free energy of the test ion, one has to subtract its selfenergy from the electrostatic free energy, i.e.Z1q2∆Fsolv =drρext (r) (ψ(r) − ψext (r)) = − (u(yd , ys )(κ1 (yd , ys ) − κ2 (yd , ys )) + κ2 (yd , ys )) .22ε(48)It has been shown, that in the absence of the salt ions (ys = 0) in the solution, the expression forthe solvation free energy (48) of the test charge takes the form∆Fsolv = −yq2√ d ,2εR 1 + yd(49)√where R = l/ 6 is the mean-square distance between the ionic groups of the dipolar particles.Basing on the obtained analytical results, the numerical analysis of the point charge potential,surrounded by the dipolar particles only, and with the presence of the salt ions in the bulk solutionhas been carried.
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