Newest models and calculation schemes for quantitative analysis of physical properties of polymers

New models and calculation schemes have been developed for the quantitative analysis of a number of physical properties of polymers — glass transition temperature, flow temperature of polymer nanocomposites, thermal conductivity, boiling point of polymer solutions, water absorption and water permeability of polymers and nanocomposites, strength, viscosity, storage and losses moduli, refractive index and dielectric constant. All calculation schemes are based on the structure of linear and cross-linked polymers; their degree of crystallinity, free volume, the effect of temperature, the composition of copolymers and homogeneous mixtures of polymers, the concentration of nanoparticles, their shape, size distribution, orientation angles, the structure of polar groups grafted to the surface of nanoparticles, the energy of intermolecular interactions are taken into account. All computational schemes are computerized and allow calculations to be carried out automatically after the introduction of the structure of a repeating unit of polymer unit into the computer, as well as the shape and size of nanofillers.


Introduction
In recent years, we have developed new and modified previous calculation schemes for the quantitative description of a number of physical properties of polymers. Among the properties are the glass transition temperature, the flow temperature of polymer nanocomposites, thermal conductivity, boiling point of polymer solutions, water absorption and water permeability of polymers and nanocomposites, the strength, viscosity, storage and losses moduli, refractive index and dielectric constant. All design schemes are based on the structure of linear and cross-linked polymers; their degree of crystallinity, free volume, temperature effect, composition of copolymers and homogeneous mixtures of polymers are taken into account. The concentration of nanoparticles, their shape, size distribution, orientation angles, the structure of polar groups grafted to the surface of the nanoparticles, the energy of intermolecular interactions are taken into account. Spherical nanoparticles, rectangular plates and nanofibers are considered. The calculation scheme for the refractive index and dielectric constant takes into account the influence of the plasticizing effect of the residues of the synthesis products and the solvent, nonlinearity on the Clausis-Mossoty function, the composition of nanoparticles, and temperature. All computational schemes are computerized and allow calculations to be carried out automatically after the introduction of the structure of a repeating unit of polymer into the computer, as well as the shape and size of nanofillers.

Methods
Used methods for constructing models and design schemes, published in [1][2][3][4][5]. To develop computational schemes, it is necessary to calculate the van der Waals volume of repeating units of polymers, as well as the energy of intermolecular interactions. In the calculations of the refractive index and dielectric constant, molar refraction and polarizability were calculated. In the work actively used a computer program "Cascade", developed in INEOS RAS.

Glass transition temperature of polymers
In order to modify the calculation scheme [1][2][3] to estimate the glass transition temperature, we introduced atomic constants and energies of dipoledipole interactions and hydrogen bonds for atoms and atomic groups located in the main and side chains. The equation used to describe the glass transition temperature of polymers T g : where a i are the atomic constants associated with the energy of weak dispersion interaction; b j are the constants associated with the energy of the dipole-dipole interactions of b d and hydrogen bonds b h ;         i i V is the van der Waals volume of the repeating unit.
The correlation diagram containing over 50 polymers is shown in Figure 1; the correlation coefficient is 0.999.

Flow temperature of polymer nanocomposites
Flow temperature T f of polymer nanocomposites is calculated by the ratio, which includes the concentration and radius of the nanoparticles, the number of polar groups grafted on their surface, the density of the nanoparticles, the molecular weight of the polymer.

Thermal conductivity of polymers
This characteristic in this paper is calculated by the formula where a i are the atomic constants characteristic of each atom; b j are the constants for polar groups leading to dipole-dipole interaction or to hydrogen bonds; N A is the Avogadro number, ΔV i is the van der Waals volume of the i-th atom, m is the number of atoms in the repeating unit, c p is the specific heat capacity, M 0 is the molecular mass of the repeating unit, ρ is the density. The arrangement of atoms in the back bone and side chains, the degree of crystallinity are taken into account. The correlation diagram is shown in Figure 2.
where T 0 is the boiling point of the solvent, T 1 is the boiling point of the solution, δ is the solubility parameter (Hildebrand parameter), R is the universal gas constant, V is the molar liquid volume, M p and M s are the molecular masses of the polymer and solvent, respectively, α w,p is the weight fraction of polymer.
The proposed calculation scheme for quantitative estimation of the boiling point of polymer solutions in an organic solvent allows calculations based on the chemical structure of the polymer and solvent, does not require preliminary experiments, and therefore has predictive power.

Water absorption and water permeability of polymers and copolymers
The correlation is proposed for calculating the activation energy of the permeability process, based on the set of atomic constants describing the contribution of each atom and polar groups to the energy of intermolecular interaction with water, which are summarized to estimate the total energy. The chemical structure of the polymer, the degree of crystallinity, temperature and free (empty) volume are taken into account.
where P is the permeability, P 0 is the constant,   The method is computerized. The computer program allows to solve both the direct problem associated with the evaluation of the properties of polymers based on the chemical structure of the repeating unit, and the inverse problem, which consists in searching for such polymer structures that have a given permeability interval.

Design scheme for assessing the permeability of nanocomposites
The calculation scheme has been developed for predicting water permeability through polymer nanocomposites. The chemical structure of the polymer and its concentration, the chemical structure of the nanoparticle surface, the concentration of nanoparticles in the composite and the concentration of polar functional groups on the surface of the nanoparticles, the size and shape of nanoparticles are taken into account. The concentration of nanoparticles and the number of polar groups on their surface have the greatest effect on permeability. The influence of the size distribution of anisometric nanoparticles and their orientation on permeability is analyzed. The calculation method is computerized and included as the special option in the CASCADE computer program (INEOS RAS), which allows calculations and predictions of the permeability of nanocomposites after the chemical structure and the specified parameters of nanoparticles are introduced into the computer.
The permeability of the composite P, consisting of a polymer and nanoparticles, is described by the ratio For nanocomposites, in which the surface of the nanoparticles is modified by chemical groups of a particular chemical structure, the following relationship applies: