Einstein-DEbye Function and Tarasov Model

The temperature dependence of the solid heat capacity of polymers can be explained with a microscopic model of thermal motion. The equation that links the vibrational spectrum to the heat capacity is the Einstein function for independent oscillators. It describes the atomic group vibrations in the polymer.

E(θ/T) = [(θ/T)²exp(θ/T)] / [exp(θ/T) - 1]²

C_E = Nk∑N_E E(θ/T)

and

N_E = 3N_Atoms - N

where N_Atoms is the number of atoms in the characteristic mer unit and N is the number of skeletal modes of vibration in the same structural unit. In the case of polyethylene, the structural unit is CH₂ so that 3N_Atoms = 9 and N = 2.

The distribution of the skeletal vibrations are coupled in a way that they stretch to zero frequency, that is to the acoustical vibrations. In the lower part of the frequency spectrum, the vibrations will couple intermolecularly, so that the wavelengths of the vibrations become larger than the molecular anisotropy of the chain structure. As a result, the detailed molecular arrangement is of little consequence to these frequencies. Therefore, a three-dimensional Debye function applies

D₃(θ/T₃) = 3 (θ/T₃)³∫₀^θ/T₃ {[(θ/T)⁴exp(θ/T)] / [exp(θ/T) - 1]²} d(θ/T)

and

C_v / 3Nk = D₃ (θ/T₃)

where θ₃ = h ν₃/ k is the reference temperature of skeletal mode vibrations normal to the chain axis. The function above describes only the weak oscillations normal the the chain axis that takes place in the three dimensional force field created by the intermolecular van der Waals forces that act mainly normal to the chain axis. The vibrations along the chain axis, i.e. the one-dimensional oscillation of the strong one-dimensional force field created by the chemically bonded polymer groups is described by a Debye function in one dimension:

D₁(θ/T₁) = (θ/T₁)∫₀^θ/T₁ {[(θ/T)²exp(θ/T)] / [exp(θ/T) - 1]²} d(θ/T)

and

C_v / 3Nk = D₁(θ/T₁)

where θ/T = h ν₁ / k is the reference temperature of skeletal mode vibrations in the chain axis. These vibration are responsible for the nearly linear increase of the heat capacity between 0 and 200 K.

To approximate all skeletal vibrations of linear polymers, one should start out at low frequency with a three-dimensional Debye function and then switch to a one-dimensional Debye function. Such an approach was derived by Tarasov (1950-1965). The skeletal vibration frequencies are, thus, separated into two groups; the intermolecular group between zero and ν₃, and the intramolecular group between ν₃ and ν₁:

C_v(Tarasov) = NR/3·{D₁(θ₁/T) - (θ₃/θ₁)[D₁(θ₃/T) - D₃(θ₃/T)] }

Knowing the group vibration frequencies (the frequenzy spectrum), the two θ temperatures and the number of skeletal vibrators, N, it is possible to calculate C_v by applying the Tarasov and Einstein equation. Specific heat capacities are usually measured at constant pressure. The thermodynamic relation between C_v and C_p is:

C_p = C_v + T V α² B 

where B is the bulk modulus and α the volumetric coefficient of thermal expansion. The calculation is quite cumbersome and usually done with a computer program. To understand how the method works, go to Heat Capacity of Polystyrene

References