Bulk Modulus

The bulk modulus (also called compression modulus) of a substance describes ist resistance to compression. It is defined as the ratio of an infinitesimal pressure increase to the resulting relative decrease of the volume,

B = - V · (∂P / ∂V)T

where P denotes te hydrostatic pressure, V volume, and ∂P/∂V is the derivative of pressure with respect to volume at constant temperature. The bulk modulus can also be written in terms of density change,

B = ρ · (∂P / ∂ρ)T

where ρ is the density of the polymer and ∂P/∂ρ is the derivative of pressure with respect to density. The inverse of the bulk modulus is called the compressibility, β,

β = 1 / B

The bulk compression involves only short-range conformational changes whereas shear and tensile forces can cause strong (time-dependent) long-range conformational changes. Hence, the bulk modulus is the only time-independent modulus or, in other words, it is not a viscoelastic but linear elastic property. 

if neighbouring molecules interact only by London dispersion forces (as it is the case for many polymers) then the bulk modulus and the compressibility of a solid are directly related to the Lennard-Jones potential and cohesive energy. For example, for glassy polymers, following relation between the bulk modulus, cohesive energy and molar volume can be derived:1

Seitz

where Ecoh is the cohesive energy, Vm(T), V0 are the molar volume at the chosen temperature T and at zero Kelvin and K is a numerical fitting parameter. Seitz1 and Bondi2 suggested following relationship between the Van der Waals volume and the zero point molar volume of liquid or rubber like polymers: V0 ≈ 1.3 Vw whereas for polymers in the glassy state Seitz suggested V0 ≈ 1.4 Vw. The molar volume of a glassy and rubbery polymer can be written as Vm ≈ 1.6 Vw. Using these relations, the bulk elastic modulus becomes

Bam ≈ (9.0 ± 2) Ecoh = (9.0 ± 2) δ2

The bulk elastic modulus of polymer crystals is about twice the modulus of glassy polymers, Bcr ≈ 2 Bam and that of rubbery polymers is about half, Br ≈ 1/2 Bam.

References
  1. J.T. Seitz, J. of Appl. Poly. Sci., Vol. 49, 1331-1351 (1993)
  2. A. Bondi, Physical Properties of Molecular Crystals, Liquids and Glasses, Wiley, New York, 1968
  • Summary

    Bulk Modulus

    The bulk (compression) modulus  of a material is a measure of its resistance to compression.

  • The inverse of the bulk modulus is called the compressibility.

  • The bulk modulus is  temperature and pressure dependent but nearly time-independent.

  • Unlike shear and tensile modulus, the bulk modulus is not a viscoelastic but only a temperature dependent linear elastic property.

  • The cohesive energy determines the magnitude of the bulk modulus.

Polymer Properties Database

Theromophysical Data

Key data on over two hundred
and fifty polymers.

Polymers Index

Typical Performance

Properties of commercial commodity
and engineering polymers.

Plastics  Index

Physics of Polymers

Physical and mechanical properties
of polymers

Phys. Contents

Chemistry of Polymers

Chemical properties and synthesis
of organic polymers.

Chem. Contents