## 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}

where *E*_{coh} is the cohesive energy,
*V*_{m}(*T*), *V*_{0}
are the molar volume at the chosen temperature *T* and at zero Kelvin
and *K* is a numerical fitting parameter.
Seitz^{1} and Bondi^{2} suggested following
relationship between the *Van der Waals volume* and* *
the* zero point molar volume* of liquid or rubber like polymers: * V*_{0} ≈ 1.3 *V*_{w}
whereas for polymers in the glassy state Seitz suggested * V*_{0}
≈ 1.4 *V*_{w}.
The molar volume of a glassy and rubbery polymer can be written as * V*_{m}
≈ 1.6 *V*_{w}. Using these relations, the
bulk elastic modulus becomes

*B _{am}* ≈ (9.0 ±
2)

*E*

_{coh}= (9.0 ± 2)

*δ*

^{2}

The bulk elastic modulus of polymer crystals is about twice the
modulus of glassy polymers, *B _{cr}* ≈ 2

*B*and that of rubbery polymers is about half,

_{am}*B*≈ 1/2

_{r}*B*.

_{am}##### References

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