Rheological properties of the polymeric blends

Abstract. The calculation scheme for storage modulus and loss modulus for compatible polymers as a function of the temperature, stress state, and composition has been suggested. The relations for calculation of the storage modulus at high frequency for polymer of any chemical structure containing carbon, hydrogen, and oxygen atoms are suggested. Calculation scheme allows evaluation both of the storage modulus and loss modulus for compatible blends of polymers. The calculations are performed for mixtures of polystyrene, poly(methyl methacrylate), polycarbonate, and polyethylene oxide used for the manufacture of building materials.


Introduction
Within the frame of the papers [1] it was shown that the storage modulus at high frequencies is described by the following equation where ; the values of g i characterize an average contribution of each atom into the value of i i i l k S ; f i is the selection of constants which characterize the influence of strong intermolecular interaction (dipole-dipole interaction, hydrogen bonds, etc.); S i is the Vander-Waals surface of the i-th atom, through which the intermolecular interaction occurs; κ i is the elasticity coefficient of the i-th atom bond; l i is the characteristic size of the bond; V is the Van-der-Waals volume of i-th atom entering into repeating unit of polymer.
For compatible blend of polymers the equation ( 1) is transformed to the form: where α 1 and α 2 are the molar parts of polymer 1 and 2, respectively; are the selections of constants for the polymer 1 and 2, respectively.Taking into account that in accordance with the equation ( 1) the values of we can rewrite the equation (2) as: where G 1 and G 2 are the moduli of the component 1 and 2, respectively.Because α 1 +α 2 =1, obtain: The equation (4) allows description of the shear modulus G depending on molar concentration of Polymer 2. Let us examine the example of calculation using the polymers used in the building industry.It is well known that fully compatible blend of polymers is as following: "polystyrene + polycarbonate".The Table 1 of the values g i and f j is borrowed from the paper [1]: Using these parameters as well as the Van-der-Waals volumes of the atoms [2][3][4][5] we obtain for polystyrene  3 (calculations are produced by the software Cascade).So, the value of storage modulus is G PS = 3.15 kg/cm 2 (it should be noted that the experimental value of G = 3.16 kg/cm 2 ).

For polycarbonate
So, the value of storage modulus is G PC = 0.24 kg/cm 2 .The dependence of storage modulus of the blend under consideration on the molar part of polycarbonate is shown on Figure 1.The major curvature of the dependence is due to the great difference between the Van-der-Waals volumes of the components.The other compatible blend is "poly(ethylene oxide) + poly(methyl methacrylate)".For poly(ethylene oxide) =391.26 cm 2 /kg•Å 3 .Van-der-Waals volume of poly(ethylene oxide) So, the value of storage modulus is G PEO = 0.112 kg/cm 2 .For poly(methyl methacrylate) The dependence of storage modulus of this blend on the molar part of poly(methyl methacrylate) is shown on Figure 2. It is well known that molar α m and weight α w fractions are related by where M 1 and M 2 are the molecular weights of the repeating units of polymer 1 and polymer 2, respectively.The dependence of the storage modulus on the weight part α w,2 of the polymer 2 is described by the following formula The dependence of storage modulus on the weight part of polycarbonate is shown on Fig. 4.

The storage modulus and loss modulus of the polymer blends
Now we consider the calculation of storage and loss moduli of the polymer blends.It is well-known that in the Maxwell regime (i.e., when there is a single typical relaxation time W) the storage an loss moduli are given by the following formulae where G is the high-frequency limit of the storage modulus and the relaxation time W is given by where x * is the Euler gamma function, and the parameter k is due to the non-exponential relaxation in polymer systems.
As an example of calculations we consider a mixture of poly(ethylene oxide) and poly(methyl methacrylate).Using values , G PMMA , and G PEO for these polymers, the resulting dependencies on the concentration of PEO are given in Table 2.  To study the concentration dependence of the moduli, assume T = 445 K, and, once again mol g M w / 40000 . Then, depending on the composition of the blend one has the following results for the high frequency storage modulus, the viscosity, and the relaxation time are given in Table 4.The corresponding frequency dependences of the storage and loss moduli are presented in Figures 6 and 7. Storage moduli are plotted in full lines, and loss moduli are in dashed lines.

Conclusions
The possibility of calculation of the modulus of elasticity at high frequency has been demonstrated.All the physical parameters we need for calculations can be predicted by the software Cascade (INEOS RAS).The dependencies of modulus of elasticity on both the molar and weight part possess various forms depending on the Van-der-Waals volume of the components.The proposed calculation scheme may be useful for predicting the rheological properties of polymeric materials and the search for optimal process conditions of production of building materials.

Fig. 1 .
Fig.1.Dependence of storage modulus on the molar fraction of polycarbonate.

2 Fig. 2 .
Fig.2.Dependence of storage modulus on the molar part of poly(methyl methacrylate)

2 Fig. 4 .
Fig.4.Dependence of storage modulus on the weight part of polycarbonate α w,PC .

Table 1 .
Values of constants g i and f j * Parameter f d is introduced for each branching in the back bone or side chain.This parameter is introduced also in the presence of a polar group of any type.

Table 2 .
[6] values of modulus G for the compatible blends of PEO and PMMA.Viscosity of the mixtures calculated by Bicerano method[6]as a function of temperature is given in

Table 3 .
Consider first the temperature dependence of the storage and loss moduli.Assume, for definiteness, that % 40

Table 3 .Table 3 .
The values of η and τ at various temperatures.The corresponding frequency dependences both of the storage and loss moduli are presented in Figure5.Storage moduli are plotted in full lines, and loss moduli are in dashed lines.

Table 4 .
The values of G, η and τ at various concentrations of the components.