Bulk modulus

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Illustration of uniform compression

The bulk modulus ( or ) of a substance is a measure of how resistant to compressibility that substance is. It is defined as the ratio of the infinitesimal pressure increase to the resulting relative decrease of the volume.[1] Other moduli describe the material's response (strain) to other kinds of stress: the shear modulus describes the response to shear, and Young's modulus describes the response to linear stress. For a fluid, only the bulk modulus is meaningful. For a complex anisotropic solid such as wood or paper, these three moduli do not contain enough information to describe its behaviour, and one must use the full generalized Hooke's law.

Definition[edit]

The bulk modulus can be formally defined by the equation

where is pressure, is volume, and denotes the derivative of pressure with respect to volume. Considering unit mass,

where ρ is density and dP/dρ denotes the derivative of pressure with respect to density (i.e. pressure rate of change with volume). The inverse of the bulk modulus gives a substance's compressibility.

Thermodynamic relation[edit]

Strictly speaking, the bulk modulus is a thermodynamic quantity, and in order to specify a bulk modulus it is necessary to specify how the temperature varies during compression: constant-temperature (isothermal ), constant-entropy (isentropic ), and other variations are possible. Such distinctions are especially relevant for gases.

For an ideal gas, the isentropic bulk modulus is given by

and the isothermal bulk modulus is given by

where

γ is the heat capacity ratio
p is the pressure.

When the gas is not ideal, these equations give only an approximation of the bulk modulus. In a fluid, the bulk modulus K and the density ρ determine the speed of sound c (pressure waves), according to the Newton-Laplace formula

In solids, and have very similar values. Solids can also sustain transverse waves: for these materials one additional elastic modulus, for example the shear modulus, is needed to determine wave speeds.

Measurement[edit]

It is possible to measure the bulk modulus using powder diffraction under applied pressure. It is a property of a fluid which shows its ability to change its volume under its pressure.

Selected values[edit]

Approximate bulk modulus (K) for common materials
Material Bulk modulus in GPa Bulk modulus in psi
Glass (see also diagram below table) 35 to 55 5.8×106
Steel 160 23.2×106
Diamond (at 4K) [2] 443 64×106
Influences of selected glass component additions on the bulk modulus of a specific base glass.[3]

A material with a bulk modulus of 35 GPa loses one percent of its volume when subjected to an external pressure of 0.35 GPa (~3500 bar).

Approximate bulk modulus (K) for other substances
Water 2.2 GPa (value increases at higher pressures)
Methanol 823 MPa (at 20 °C and 1 Atm)
Air 142 kPa (adiabatic bulk modulus)
Air 101 kPa (constant temperature bulk modulus)
Solid helium 50 MPa (approximate)

Microscopic origin[edit]

Interatomic potential and linear elasticity[edit]

The left one shows the interatomic potential and equilibrium position, while the right one shows the force
Interatomic potential and force

Linear elasticity is a direct result of interatomic interaction, it is related to the extension/compression of bonds, so it can be derived from the interatomic potential for crystalline materials.[4] First, let’s examine the potential energy of two interacting atoms. Starting from very far points, they will feel an attraction towards each other. As they approach each other, their potential energy will decrease. On the other hand, when two atoms are very close to each other, their total energy will be very high due to repulsive interaction. Together, these potentials guarantee an interatomic distance that achieves a minimal energy state. This occurs at some distance a0, where the total force is zero:

Where U is interatomic potential and r is the interatomic distance. This means the atoms are in equilibrium.

To extend the two atoms approach into solid, consider a simple model, say, a 1-D array of one element with interatomic distance of a, and the equilibrium distance is a0. It’s potential energy-interatomic distance relationship has similar form as the two atoms case, which reaches minimal at a0, Taylor expand it at this point.

But as we discussed, at equilibrium, the first derivative is 0, so the dominate term is the quadratic one, when displacement is small, omit the higher order terms, we get:

Which is clearly linear elasticity.

Note that the derivation is done considering two neighboring atoms, so the Hook’s coefficient is:

This form can be easily extend to 3-D case, with volume per atom(Ω) in place of interatomic distance.

Relationship with atomic radius[edit]

As driven above, the bulk modulus is directly related the interatomic potential and volume per atoms. We can further evaluate the interatomic potential to connect K with other properties. Usually, the interatomic potential can be expressed as polynomial function of distance, one term for attraction and another term for repulsion.

Where A>0 represents the attraction term and B>0 represents repulsion. n and m are usually integral, and m is usually larger than n, which represents short range nature of repulsion. At equilibrium position, u is at it’s minimal, so first order derivative is 0.

when r is close to , recall that the n(usually 1 to 6) is smaller than m(usually 9 to 12), ignore the second term , evaluate the second derivative

Recall the relationship between r and Ω

In many case, such as in metal or ionic material, the attraction force is electrostatic, so n=1, we have

This applies to atoms with similar bonding nature. This relationship is verified within alkali metals and many ionic compounds.[5]

References[edit]

  1. ^ "Bulk Elastic Properties". hyperphysics. Georgia State University. 
  2. ^ Page 52 of "Introduction to Solid State Physics, 8th edition" by Charles Kittel, 2005, ISBN 0-471-41526-X
  3. ^ Fluegel, Alexander. "Bulk modulus calculation of glasses". glassproperties.com. 
  4. ^ H., Courtney, Thomas (2013). Mechanical Behavior of Materials (2nd ed. Reimp ed.). New Delhi: McGraw Hill Education (India). ISBN 1259027511. OCLC 929663641. 
  5. ^ Gilman, J.J. (1969). Micromechanics of Flow in Solids. New York: McGraw-Hill. p. 29. 

Further reading[edit]

Conversion formulas
Homogeneous isotropic linear elastic materials have their elastic properties uniquely determined by any two moduli among these; thus, given any two, any other of the elastic moduli can be calculated according to these formulas.
Notes


There are two valid solutions.
The plus sign leads to .

The minus sign leads to .

Cannot be used when