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Molar Volume and Density of polymers

The molar volume (V_m) of a polymer is defined as the volume occupied by the polymer molecules in units of volume per mole at a given temperature and pressure, and the density (ρ) is defined as the ratio of molar mass and molar volume:

ρ = M / V_m

Because of the sequential nature of polymers we may also use the molar mass and volume of the structural repeat unit for sufficiently long polymers.

Polymers can be in a crystalline, amorphous or liquid (rubber) state. The crystalline state is associated with a regular arrangement of the chains. The chains might either align parallel to each other over a long distance or they may twist into the form of a helix, or may stack into lamellae, where chains fold back on themselves to form crystallites. A rule of thumbs says, the volume of a crystalline polymer is approximately 1.43 times its van der Waals volume,

V_cr ≈ 1.43 V_w.

Amorphous polymers, on the other hand, show only short range order much like a frozen liquid. They can either be in the rubbery or in the glassy state depending if the temperature is above or below the glass transition point. The amorphous polymers have a noticeably lower packing density than polymers in the crystalline state:

V_am ≈ 1.60 V_w

Many properties are directly connected with mass and packing, that is, they depend on the density (or its reciprocal specific volume), the thermal expansibility and isothermal compressibility. For example, the mechanical properties, such as moduli, and the Poisson ratio depend on mass and packing.

Molar volumes are relative easy to measure, and therefore, have been reported for many common polymers at standard conditions and as a function of temperature. These data can be found in many handbooks for physical properties of polymers. In case, the volume or density is only known for standard conditions, densities at other temperatures can be estimated by using exponential (fitting) functions of the thermal expansion coefficient:

ρ(T) = ρ₀ · exp(-α₀ T)

where α₀ and ρ₀ are fitting parameters which have been tabulated for many polymers. The measured densities can be also fitted to a power series:

ρ(T) = a₀ + a₁ T + a₂ T² + ...

Sometimes no density data are available for the polymer in question. In that case, the molar volume or density can be relative easily estimated with group contribution methods. In fact, the molar volume at room temperature was one of the first physical properties for which group contribution methods had been proposed. However, the majority of the group contribution methods have been developed for low molecular weight compounds and are less accurate when applied to polymers. We recommend the methods of Krevelen and Hoy that have been developed specifically for polymers.¹

The figure below illustrates the increase in molar volume in the glassy anrd rubbery state for polystyrene. The experimental values have been measured by Quach and Simha² and the predicted values have been calculated with the method of Krevelen.¹

 

Molar Volume of Polystyrene

References

1. D.W. van Krevelen and K. Nijenhuis, Properties of Polymers, 4th Edition, Amsterdam (2009)

2. A. Quach and R. Simha, J. Appl. Phys. 42, 4592 (1971)

3. Robert A. Orwoll, Chapter 7: Densities, Coefficients of Thermal Expansion, and Compressibilities of Amorphous Polymers in James E. Mark, Physical Properties of Polymers Handbook, 2nd Ed., Springer 2007