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However, as is well known from21the theory of polar molecules, the dipole correlations could be described at the Keesom attractionlevel only when the average distance between the dipoles are much bigger than the dipole length(the case of the dipole gas or dilute solution of the polar molecules).

For the polymer solutions thiscondition, obviously, can be realized only for the case of the coil conformation of the macromolecule.Though, approaching the globule conformation of the polymer, it is crucial to consider the manybody dipole correlations, leading to, strictly speaking, renormalization of each virial coecient.Beyond that, in all of the available theoretical works, related to the dipole mechanism of thecollapse, a presence of the constant dipole moments on the monomers has been taken into accountonly in the electrostatic free energy of the polymer chain, although the dipole structure of themonomers, formally, must also inuence on the excluded volume contribution. Indeed, for example,varying the counterion diameter in the solutions of weak polyelectrolytes, we will aect not only theelectrostatic interactions of the monomers (due to the change of the dipole moment), but also thevalue of the eective excluded volume of the monomer.

Hereby, the equilibrium conformation of thedipolar polymer chain must be dened mostly by the competition between the contributions of theelectrostatic interactions and excluded volume interactions. Physically, in the real polyelectrolytesolutions both mechanisms of collapse may be realized.Which mechanism will be implemented in the real polyelectrolyte solutions is dependent onthe chemical specics of the monomers and counterions. From the general considerations it isclear, that coulomb mechanism can take place in the strong polyelectrolyte solutions, where thecomplete dissociation of the monomers takes place, whereas the dipole mechanism in the weakpolyelectrolyte solutions, where the counterions are localized on the monomers, forming solventseparated ionic pairs (dipolar particles).

Thus, both mechanisms must be closely investigated usingthe methods of the statistical physics. However, statistical theory of the CG transition of thepolymer chain, induced by the dipole-dipole interactions of the monomers, with regard to themany-body dipole correlations, has not been developed until recently in contrast to the theory ofcoulomb collapse. As such, in the Chapter 4 the statistical theory of the conformational behaviorof the exible polymer chain, carrying on its monomers constant dipole moments, with accountingfor the many-body dipole correlations, has been developed.A exible polymer chain is considered, consisting of N monomers, representing a hard sphereof the diameter σp and charge e, connected with the spherical counterion of the diameter σc andcharge −e.

Herewith, counterion can freely rotate around the polymer backbone, keeping the xeddistance d = (σp + σc )/2 from the monomer. Thus, all monomers of the polymer chain containfreely rotating dipoles with constant dipole moments ed. It was assumed, that polymer chain islocated in the low-molecular solvent, which is modeled as a continuous dielectric medium witha constant dielectric permittivity ε. To describe conformational behavior of the dipolar polymerchain, following the methodology, proposed in the papers by Flory and de Gennes, the free energyof the polymer chain as a function of its gyration radius Rg was constructed:F (Rg ) = Fconf (Rg ) + Fev (Rg ) + Fel (Rg ),where9Fconf (Rg ) = kB T46Rg2N b2+N b26Rg2(58)!(59)is interpolation formula for the conformational free energy of the gaussian polymer chain with thexed gyration radius (ref.

[Fixman, J. Chem. Phys., 1962; Budkov, Kolesnikov, J. Stat. Mech.,2016]). The chain bond length is xed and set to b = σp . The contribution of the excluded volumeinteractions is estimated by the virial expansion, truncated at the third term:N B N 2CFev (Rg ) = N kB T+,(60)VgVg2where Vg = 4πRg3 /3 is the gyration volume, B and C are the second and third virial coecientsof the excluded volume interactions, respectively. The last ones are estimated as the coecients ofthe hard dumbbells [Boublik et al., J. Chem. Phys., 1990]:B = v(1 + 3αc ), C = v 2 (3αc2 + 6αc + 1)/2,22(61)where v = π(σc3 + σp3 )/6 is the dumbbell volume and αc = Rc Sc /3Vc is the non-sphericity parameterof the convex body.

Geometrical parameters Rc , Sc and Vc , characterising every convex body, forthe dumbbells take the following form [Boublik et al., J. Chem. Phys., 1990]:Rc =σp2 + σc2 + σp σcππ, Sc = (σp + σc )2 , Vc =(σp + σc )3 .2(σp + σc )224(62)For estimation of the electrostatic interactions contribution to the total free energy the mutualinuence of the connectivity eect of the monomers and their electrostatic correlations is neglected.Electrostatic free energy Fel (Rg ) is calculated within the modied gaussian approximation (MGA)[Gordievskaya, et al., Soft Matter, 2018] for the unbound dipole particles:Vg kB TFel (Rg ) '2Zdk(2π)3 κ 2 (k)κ 2 (k)ln 1 +−,k2k2(63)|k|<Λwhere the screening function takes the form (ref. to description of the Chapter 3)κ 2 (k) = κ2D (1 − g(k)) ,(64)where the square of the reciprocal Debye screening length κ2D = 8πlB ρp is introduced; lB = e2 /εkB Tis the Bjerrum length; kB is the Boltzmann constant; T is the absolute temperature, ε is the solventdielectric permittivity.

For the exception of the nonphysical modes from the summation (integration)over the wave vectors k, corresponding to the small distances between the mass centers of the dipolarparticles, the ultraviolet cut-o is introduced, i.e. maximal wave vector Λ = 2π/rs = (6π 2 ρp )1/3 ,where rs = (3/4πρp )1/3 is the radius of the sphere, containing exactly one monomer (Wigner-Seitzradius) and ρp = N/Vg is the average concentration of the monomers. It is noted, that introducingthe ultraviolet cut-o, associated with the average distance between particles, substantially widensthe region of the GA application, including region of the strong electrostatic interactions [Brilliantov,Contrib.

Plasma Phys., 1998; Gordievskaya, et al., Soft Matter, 2018]. Then, using the modelcharacteristic function of the distribution over the distance between ionic groups of the dipolarmonomers, such as in the Chapter 3, i.e. g(k) = 1/(1 + k 2 d2 /6) and taking into account, thatΛ = (6π 2 ρp )1/3 , the following analytical expression has been obtained:Fel =kB T Vgσ(yd , θ),d3(65)where√ √ 6 3yd6θ3/2σ(yd , θ) = 2 θ ln 1 ++ 2 (2 + 3yd ) arctan θ − 2(1 + yd ) arctan √− θyd ,2π1 + θ22π1 + yd(66)√√with yd = κ2D d2 /6 = lB d2 N/Rg3 and θ = Λd/ 6 = (9π/2)1/3 N 1/3 d/( 6Rg ).Then, the limiting regimes, following from the interpolation formula (65), are discussed:1/34/3lB ρp , 1 θ2 yd− 48πFel12' − 12πr(67)3 , 1 yd θDVg kB T√− 6π l2 dρ2 , y 1.3BpdIt is noted that in the rst regime (rD rs d) the amorphous structure of the densely packedpolar groups, which act as unbounded ions, is realized.

In the second regime (rs rD d), as inthe rst one, the polar groups also can be considered as the free ions. However, here polar groupsmust be organized as in the dilute electrolyte solution, so the electrostatic free energy is described bythe DH limiting law. Finally, the third regime occurs at the condition rD d, at which there is noscreening of the charge at all in the system, and the electrostatic correlations of the monomers arereduced to the eective pairwise-additive Keesom attraction of the dipolar particles, which leads to23the renormalization (decrease) of the second virial coecient of the monomer-monomer interaction.Then, it is shown that for the globular conformation the rst two regimes, mentioned in (67), canbe implemented.

For the case, when rD rs d, the following scaling law for the gyration radiustakes place−1/2Rg ∼ B 1/2 lB N 1/3 .(68)In the second regime (rs rD d) the result is:−1 1/3Rg ∼ B 2/3 lBN .(69)For the coil conformation following relation, up to the numerical prefactor, takes place:Rg ∼ b2/5 Br1/5 N 3/5 ,where(70)√Br = B −6π 2dlB3(71)is renormalized second virial coecient.Figure 7: The dependencies of the gyration radius of the polymer chain with N = 256 on the dipolelength d = (σp + σc )/2 at xed monomer diameter σp = 1 and dierent counterion diameters σc ,plotted at dierent λ = lB /σp . Theoretical curves are plotted as solid lines, and the MD simulationdata as points. There are also screenshots of the globular (on the left) and coil (on the right)conformations, obtained by the MD simulation [Gordievskaya, et al., Soft Matter, 2018].For the investigation of the conformational behavior of the polymer chain it is considered, thatall lengths are expressed in the units of the monomer diameter, i.e.

σp = 1. Besides that, thedimensionless parameter λ = lB /σp is introduced, dening how much the energy of the electrostaticinteractions diers from the thermal energy kB T . To show how the counterion size aect on theconformational behavior of the dipolar polymer chain, the dependencies of the gyration radius ofthe dipolar polymer chain on the dipole length d (g. 7), varying by the change of the counteriondiameter σc , have been plotted at the dierent values of the parameter λ, obtained from the minimization of the total free energy and from the MD simulation [Gordievskaya, et al., Soft Matter,2018].

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