Summary_eng (Статистическое описание многочастичных эффектов в классических ион-молекулярных системах), страница 10

Thus, in the second part of the Chapter 5 thetheoretical research of the concentration behavior of the polarizable TC molecules in the volume ofthe polarizable polymer in the constant uniform electric eld within the mean-eld theory has beenperformed.The isolated exible polymer chain is considered with N isotropically polarizable monomers,situated in the solvent mediumm with a small addition of the isotropically polarizable moleculesof the TC.

As in the rst part of the Chapter 5, the solvent is modeled as a continuous dielectricmedium with the constant dielectric permittivity ε. It is considered that solution is conned betweentwo oppositely charged membranes with the surface charge densities ±σ , generating the externalelectric eld, impenetrable for the TC molecules, but permeable for the solvent. It is established,that the system is in the thermodynamical equilibrium at the temperature T and osmotic pressure27Π, produced by the TC molecules.

The problem consists in the exploration of the number of thesorbed TC molecules by the polymer chain volume alongside its conformational behavior dependingon the membranes surface charge density. It is noted that presence of the TC molecules in the bulksolution does not allow one to apply a simple Flory-de Gennes type theory, presented in the previouspart of the Chapter 5. To solve assigned problem, it is necessary to construct a thermodynamicpotential of the system, taking into account that not only the temperature is xed in the system,but also the osmotic pressure. For simplicity it is assumed that total volume of the system isdivided into two parts: internal polymer volume (estimated, as above, by the gyration volume ofthe polymer chain) and the bulk phase.Using the mean-eld approximation (80) for the dielectric permittivityεb = ε + 4πγc ρ,(80)standard thermodynamic relations, the expressions for the free energy φ per particle, chemicalpotential µ, and osmotic pressure Π18π 2 γc σ 2φ = kB T (ln(Λ3 ρ) − 1) + kB T Bc ρ −,2εεb(81) 8π 2 γc σ 2∂(ρφ)= kB T ln(ρΛ3 ) + 2Bc ρ −,∂ρε2b(82) 32π 3 γc2 ρ2 σ 2,Π(ρ, T ) = ρ(µ − φ) = kB T ρ + Bc ρ2 +εε2b(83)µ(ρ, T ) =where γc is the polarizability of the TC molecules, ρ is the TC concentration, T is the temperature,kB is the Boltzmann constant.

The rst term in the right-hand side of the expression (81) denesthe free energy of ideal gas, the second term is the contribution of the TC molecules interactionsto the free energy at the level of the second term of the virial expansion (considered only paircorrelations of the particles) and the third is the free energy of the dielectric body, conned betweentwo semipermeable membranes with the xed surface charge densities ±σ ; Λ is the thermal deBroglie wavelength, Bc is the second virial coecient of the interactions between TC molecules.Then, model of the solution in the presence of the polarizable polymer chain was formulated.

Itis mentioned, that, as against the situation described in the rst part, in this case it is insucientto just write the free energy of the polymer chain as a function of its gyration radius. Instead, itis necessary to construct the thermodynamic potential, considering the equilibrium of the targetcompound in the bulk phase and within the polymer volume at the temperature T and osmoticpressure Π.

Thus, two order parameters are chosen: the gyration radius Rg of the polymer chainand number of TC molecules Nc , located in the polymer volume. In such manner, thermodynamicpotential of the solution is written in the following formwhereΦ(Nc , Rg ) = Fid (Rg , Nc ) + Fvol (Rg , Nc ) + Fel + ΠVg − µNc ,(84) 9Nc Λ32−2Fid = kB T α + α+ Nc kB T ln−14Vg(85)is the free energy of the ideal Gaussian chain, surrounded by the gas of the TC molecules, locatedin the gyration volume Vg = 4πRg3 /3; α = Rg /R0g is the expansion factor of the polymer chain2 = N b2 /6 is the mean-square gyration radius of the ideal polymer chain, b is the Kuhn length of(R0gthe polymer chain). Free energy of the ideal polymer chain is estimated in the same manner as in theprevious part with the use of the Fixman interpolation formula.

Chemical potential µ and osmoticpressure Π of the TC are dened by the relations (82) and (83), respectively. Contribution Fvolof the polymer-TC, TC-TC and polymer-polymer interactions to the total free energy is describedwithin the virial expansion at the level of the rst nonzero terms over the TC and monomersconcentrationsBp N 2 Bc Nc2 2Bpc Nc N++,(86)Fvol = kB TVgVgVg28where Bp , Bc and Bpc are corresponding second virial coecients. Electrostatic free energy of themixture, conned in the gyration volume, can be presented as(87)Fel = Fex,p + Fex,c ,where Fex,p is the excess electrostatic free energy of the mixture, conned in the gyration volume,in comparison with the bulk phase of the solution, and Fex,c is the excess electrostatic free energyof the TC solution, conned in the gyration volume, as relative to the pure solvent.

Within themean-led approximation these contributions can be estimated in the following way:Fex,p = −6πVg σ 2 εp − εbεb 2εb + εpandFex,c = 2πσ 2 Vg(88)ε − εb.εb(89)It is noted, that polymer coil together with the sorbed TC molecules is described by the dielectricsphere of the radius Rg with the dielectric permittivity(90)εp = ε + 4πγc ρc + 4πγp ρp(γp is the polarizability of the monomer, ρc = Nc /Vg is the average concentration of the TC in thegyration volume, ρp = N/Vg is the average concentration of monomers), immersed in the dielectricmedium with the dielectric permittivity εb . Thus, total electrostatic free energy is determinedas the excess electrostatic free energy of the polymer chain and TC molecules, located in thegyration volume, as compared to the pure solvent, occupying the same volume.

The minimizationof the thermodynamic potential over the order parameters leads to the system of the transcendentequations in expansion parameter α and average TC concentration within the polymer volume.Before the analysis of the numerical results, the following dimensionless parameters have beenintroduced:B̃p = Bp /b3 , B̃c = Bc /b3 , B̃pc = Bpc /b3 , γ̃p = γp /(b3 ε), γ̃c = γc /(b3 ε), σ̃ =p34πσ b /εkB T . It is noted, that all calculations are accomplished for the polymer chain withthe polymer-polymer and polymer-TC interactions, being repulsive, so that B̃pc > 0.

Then, thenumerical results for two qualitatively dierent regimes were analysed: γ̃p > γ̃c and γ̃p < γ̃c .2Regime I: γ̃p > γ̃c . The parameters values are xed: N = 10 , B̃p = 0.25, B̃c = 0.25, B̃pc =0.75, ρ̃ = 0.15, which allow one to obtain the conformation of the condensed coil at σ → 0 due tothe ”solvophobic” properties of the polymer chain relative to the TC molecules.0.15γ˜c = 0γ˜c = 0.2γ˜c = 0.40.14ρ̃c0.130.120246810σ̃Figure 9: Dependence of the TC concentration ρ̃c within the polymer volume on the surface chargedensity of the membranes σ̃ at xed polarizability of the monomers γ̃p = 0.8 and dierent polarizabilities of the TC molecules.29At zero electric eld polymer coil is suciently shrunk due to its solvophobicity in relationto the TC molecules, whereas the inclusion of the eld leads to the coil expansion due to theelectrostriction eect.

If the TC molecules are not polarizable (γ̃c = 0), then with the increase ofthe surface charge density of the membranes, the local concentration ρ̃c tends to the concentrationρ̃, settled in the bulk phase. However, in the case of the polarizable TC molecules γ̃c 6= 0) at theincrease in γ̃c , the TC concentration in the gyration volume monotonically grows. In fact, at thenon-zero TC polarizability, the dependence of the local concentration ρ̃c on the eld strength Ẽ hasa pronounced maximum (g. 9). Then, the behavior of the dependence ρ̃c (σ̃) at the increase inthe polymerization degree N of the polymer coil has been examined.

It is shown that going of thepolymerization degree to the innity (N → ∞) leads to the fact that ρc → ρ. It is established, thatat the increase in the polymerization degree the mentioned above non-monotonic dependence ρ̃c (σ̃)becomes less pronounced and disappears in the limit of the innitely long coil.Regime II: γ̃p < γ̃c . In the second regime the dependencies of the expansion factor α andlocal concentration of the TC molecules ρ̃c on the monomers polarizability at the xed surfacecharge densities σ̃ and other parameters of the model have been examined. An interesting feature graphics of the functions, corresponding to the dierent values of σ̃ , have a common intersectionpoint. The common intersection point corresponds to the equality of the dielectric permittivitieswithin the polymer volume and in the bulk phase.0.151.8σ̃ = 4σ̃ = 6σ̃ = 81.7σ̃ = 4σ̃ = 6σ̃ = 80.141.6αρ̃c0.131.50.121.41.30.1100.10.20.30.40γ̃p0.10.20.30.4γ̃pFigure 10: Expansion factor α (on the left) and TC concentration ρ̃c in the polymer coil volume(on the right) as a function of the polarizability γ̃p of monomers for the solvophobic polymer at thexed surface charge densities of the membranes σ̃ .

Polarizability of the TC molecules is γ̃c = 0.5.Finally, within the second regime the case of the non-polarizable (γ̃p = 0), solvophobic, relativelyto the polarizable (γ̃c 6= 0) TC molecules, polymer chain has been examined. The following valuesof the second virial coecients are adopted: B̃p = 0.125, B̃c = 0.125, B̃pc = 0.75. The dependenceof the expansion factor and the local TC molecules concentration on the surface charge density ofthe membranes σ̃ at the dierent TC polarizabilities γ̃c has been investigated. It is shown that atsuciently small surface charge densities of the membranes, the expansion factor remains almostconstant and, further, monotonically increases. It is mentioned, that due to the large enoughsolvophobicity of the polymer at zero charge of the membranes, the TC molecules concentrationwithin the gyration volume is much smaller, than in the bulk phase of the solution (ρ̃c ρ̃), thatleads to the shrinkage of the polymer coil (due to the osmotic pressure dierence).

However, atthe large enough charge density on the membranes, its increase will lead to the increase in the TCconcentration within the polymer volume, that, in turn, will lead to the expansion of the polymercoil.In the conclusion of the Chapter 5 it is noted that both presented theoretical models haveobvious restrictions. They could be applied to the description of the conformational behavior of thepolymer chains with isotropically polarizable monomers. In other words, non-diagonal componentsof the polarizability tensor of the monomer unit must be much smaller than diagonal ones.