Summary_eng (1136181), страница 7
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It has been shown, that presence of the long enough dipolar particles in thesolution leads to the signicant deviation of the electrostatic potential from the Coulomb (in theabsence of salt) and DH (in the presence of salt) laws. It has been shown, that in both casesincrease of the dipolar particles concentration leads to the shift of the potential prole to the regionof the smaller distances. The latter is associated with the occurrence of the additional screeningof the test ion charge due to the presence of the charged centers of dipolar particles in its vicinity.Besides that, the behavior of the electrostatic potential of the test point ion in the aqueous saltfree solution of the dipolar particles with dipole moments, corresponding to those of the proteinmolecules (p ≈ 1000 D), was investigated.
It has been found, that eects of non-locality becomesignicant at the distances of r ∼ 1 − 2 nm, which are much smaller than dipole length l = 20 nm,but, nevertheless, are big enough in the comparison with the characteristic sizes of the solventmolecules.Figure 6: Ratio of the electrostatic potential ψ(r) of the point charge in the dipolar particlessurrounding, obtained within the non-local theory, to the potential ψloc (r), obtained within thelocal theory, as a function of the distance, at dierent concentrations nd of the dipolar particles.The data is shown for l = 20 nm, ε = 80, T = 300 K .Then, the evaluation of the electrostatic free energy of the ion-dipole mixture within the Gaussianapproximation (GA) was formulated.
Expanding the functional S[ϕ] in (31) into a power series neara mean-eld ϕ(r) = iψ(r) and truncating it at the quadratic term over the potential ϕ, following19expression has been obtained:ZQ ≈ exp {−S[iψ]}Dϕβ−1exp − ϕG ϕ ,C2whereG −1 (r, r0 |ψ) = kB T(50)δ 2 S[iψ]δϕ(r)δϕ(r0 )is the renormalized reciprocal Green function with the mean-eld electrostatic potential ψ(r), satisfying the equation (33). Thus, evaluating the Gaussian functional integral using the standardmethods, the following general relation for the conguration integral within GA has been obtained:1Q ≈ exp −S[iψ] + tr (ln G − ln G0 ) ,(51)2where symbol tr(..) means the trace of the integral operator.
In the absence of the external charges(i.e., ρext (r) = 0), electrostatic potential ψ(r) = 0, so the mean-eld contribution to the electrostatic(M F )free energy Fel= kB T S[0] = 0. Thus, in this case the electrostatic free energy is fully determinedby the thermal uctuations of the electrostatic potential near its zero value and adopts the followingform: Zdkκ 2 (k)κ 2 (k)V kB Tln 1 +−,(52)Fel =2(2π)3k2k2where screening function κ 2 (k) is determined by the expression (36).
For the distribution function,dened by the relation (38), in the absence of the ions in system, the integral in (52) is calculatedanalytically:V kB TFel = − 3 σ(yd ),(53)lwhere the following auxiliary dimensionless function is introduced:√6σ(yd ) =(2(1 + yd )3/2 − 2 − 3yd ).(54)4πElectrostatic free energy of the solution of the dipolar particles has been analysed in two limitingregimes, resulting from (53-54), that is: √ 4 2− 6πe ln2d , yd 1Fel3(εkB T )=(55)− 1 3 , yd 1.V kB T12πrDIt is noted, that in the rst regime the gas of the dipolar particles, interacting via the Keesompotential, is realized, whereas in the second regime the charged groups of the dipolar particles canbe considered as unbound ions, so the electrostatic free energy is described by the DH limiting law.Then the limiting regimes of the electrostatic free energy behavior in the presence of the saltions at yd 1 and yd 1 have been analysed.
At yd 1 the result is√√√Felκ3s3 6ys3 6 1 + 3 ys + ys 2=−−y + O(yd3 ),(56)√ yd −√V kB T12π 2πl3 1 + ys16πl3 (1 + ys )3 dwhere the rst term in the right-hand side describes the contribution of the ionic species to theelectrostatic free energy within the DH approximation. The second and third terms describe thecontributions of the ion-dipole and dipole-dipole pair correlations, respectively. In the oppositeregime, when the yd 1, we arrive to the DH limiting law:1Fel'−V kB T12π8π(nd + I)e2εkB T3/2.(57)It is worth noting, that in this case the charged groups of the polar particles behave as unboundedions, participating in the screening of the charge along with the salt ions.20In the conclusion of the Chapter 3 it is noted, that developed formalism is unclosed, as it containsundened probability distribution function g(r) of the distance between ionic groups of the dipolarparticles.
Thus, for the applications of the obtained relations for the thermodynamic descriptionof the real polar uids, it is necessary to independently determine the distribution function g(r).It is noted, that for this purpose it is necessary to solve a quantum-mechanical problem on thedetermination of the orbitals of the ionic groups of the isolated dipolar particle (for example, usingthe Hartree-Fock method), which, in their turn, will allow one to evaluate desired distributionfunction. Besides that, possibilities of the applications of the developed formalism is discussed. Itis noted, that current formalism could be applied to the description of the phase behavior of thepolar uids with the consideration of the formation of the chain clusters of the polar particles,their dispersion and excluded volume interactions.
Finally, it is mentioned that the theory can beeasily generalized for the description of the polar particles mixtures, and also for the particles witharbitrary electrical structure.Chapter 4It is well known from the experimental research (for reference [Mel'nikov, et al. J. Am. Chem.Soc., 1999]) and molecular dynamic simulations [Gavrilov et al., Macromolecules, 2016], that in thedilute salt-free solutions of the exible polyelectrolytes in the regime of good solvent a coil-globule(CG) transition can take place. Such unconventional conformational transition is usually accompanied by the counterion condensation onto the backbone of the polyelectrolyte macromolecule, whichreects its electrostatic nature.
Besides, considering that transition takes place in the regime ofgood solvent, i.e. when polyelectrolyte is highly soluble, we can not apply classical models of theCG transition, based on the concept of theta-temperature, for its description.Nowadays, there are two possible mechanisms of the CG transition in the dilute polyelectrolytesolutions are proposed. The rst one, ”coulomb” mechanism (ref.
[Brilliantov et al., Phys. Rev.Lett., 1998]) is based on the idea, that this transition is caused by strong electrostatic correlations ofthe counterions. In more details, when characteristic electrostatic energy becomes much larger thanthermal energy kB T , it becomes thermodynamically protably for the counterions to ”adsorb” ontothe polymer ”surface”, neutralizing its charge. Within this mechanism, counterions are not localisedon the monomers, but can freely move along the polymer backbone.
Despite full neutralization ofthe macromolecule charge due to the counterion condensation, the thermal uctuations of the chargedensity near its zero value have to take place due to the thermal motion of the counterions. Theseuctuations of the charge density, in their turn, lead to the cooperative mutual attraction of themonomers (so-called Kirkwood-Shumakher interaction [Kirkwood, Shumakher, Proc. Natl. Acad.Sci. USA., 1952]), resulting in the CG transition. Because of the fact, that this transition takesplace in the region of the strong electrostatic interactions, electrostatic contribution to the total freeenergy of the system must be taken into account beyond the Debye-Huckel (DH) theory, which isvalid, as it is well known, in the region of the weak electrostatic interactions.
A successful attemptto describe such electrostatic CG transition beyond the DH theory for the rst time was made inthe paper [Brilliantov et al., Phys. Rev. Lett., 1998], where a Flory-de Gennes type theory of thestrongly charged polyelectrolyte chain was formulated. It is worth noting, that accounting for theelectrostatic correlations of the counterions within the one-component plasm (OCP) model allowedone to understand the scaling laws for the gyration radius of the polyelectrolyte chain, obtainedrecently within the MD simulation [Tom et al., Phys. Rev.
Lett., 2016].The second, ”dipole” mechanism is based on the assumption, that CG transition happens dueto the attraction of the thermally uctuating dipole moments, appearing along the polymer chainbackbone due to the counterion condensation. This mechanism was for the rst time proposed inthe paper [Schiessel and Pincus, Macromolecules, 1998] and was studied timorously in the followingpapers. However, in all existing theoretical models the dipole correlations are considered at the pairwise approximation level: through the renormalization of the second virial coecient of the volumeinteractions of the monomers due to the existence of the eective Keesom attraction, occurringbetween the polar particles at the long distances, or with the aid of the addition of the eectiveKeesom pair potential to the potential of the volume interactions.
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