**Elastic
Properties**

**The Bulk Modulus**

The
bulk elastic properties of a material determine how much it will compress under
a given amount of external pressure. The ratio of the change in pressure to the
fractional volume compression is called the bulk modulus of the material.

A
representative value for the bulk modulus for steel is 160 x 10^{9} N/m^{2}

and
that for water is 2.2 x 10^{9} N/m^{2}. The reciprocal of the bulk modulus is called
the compressibility of the substance. The amount of compression of solids and liquids
is seen to be very small.

The
bulk modulus, B, is a material property that relates the change in volume with
a change in pressure:

B =
-V(¶P/¶V)_{T} = -(¶P/¶ln(V))_{ T}

A
typical bulk modulus of medium density polyethylene is about 0.14 GPa (= 10^{9} N/m^{2}). This is for a polyethylene that has a density
of 0.93 g/cm^{3}.

The
bulk modulus can be calculated by distorting all of the dimensions of the unit
cell using the program cellvec_xyz and calculating
the energy as a function of the change in volume. This procedure uses the fact that the
pressure is:

P =
-(¶E/¶V)_{T}

and
so

B =
-V(¶^{2}E/¶V^{2})_{T}

If
we distort along all three dimensions of the unit cell by the same fractional
amount we uniformly change the volume of the unit cell. If we plot the calculated energy as a
function of the volume we will obtain a curve whose curvature (¶^{2}E/¶V^{2}) and unit cell volume V can
be used to calculate the bulk modulus.

** **

**Young's modulus**

For
the description of the elastic properties of linear objects like wires, rods,
columns, which are either stretched or compressed, a convenient parameter is
the ratio of the stress to the strain, a parameter called the Young's modulus
of the material. Young's modulus can be used to predict the elongation or
compression of an object as long as the stress is less than the yield strength
of the material.

Figure 1. Definition of terms used the calculation of elastic constants.

The relation between
the external diagonal stress, P_{i}, and the diagonal components of the
strain tensor, h_{i}, can be expressed as the following matrix
equation:

To
obtain the Young's modulus along the chain (assumed here to be in the
x-direction according to our set-up of the unit cell above one needs to solve
for h_{1} at the condition P_{1}
= 1, P_{2} = P_{3} = 0 GPa.

The
elastic constants are obtained from:

Where D_{i} and D_{j} are dimensionless
displacements along unit cell vectors, V is the unit cell volume and U(D) and U(0) are the energies of the deformed and undeformed
unit cell, respectively.

Example: Calculation of the Young's Modulus of Polyethylene

Using the program elastic_xyz.f
we can create .car files with distortions along each of the unit directions, x,
y and z. The program will prompt you for
a four-letter root name and will then add "_x.car",
"_y.car" and "_z.car",
respectively, to create files:

root_x.car

root_y.car

root_z.car

with the same fractional distortion.
We choose to sample the range from 0.96 to 1.04. We calculate the energy at each of the
distorted unit cells using DFT methods. First,
we perform a single point energy calculation. Plots are given for each of the
dimensions.

Figure 2. The binding energy of the high density (orthorhombic) unit cell with
displacements along the given unit cell directions for a single point energy
calculation. The displacements are given
in fractional coordinates along each unit cell direction.

Note that the minimum energy is not necessarily at
the dimension given the x-ray crystal structure. A deviation likely arises from inaccuracy in
the DFT calculation. The best model is
obtained along the length of the carbon chains (x-direction). Here there is a clear minimum and Young's
modulus could be calculated using these data.
The deviation from experiment is only 1% along the x dimension.

Next
we reoptimize the geometry for each deformed
structure as shown in the Figure below.

Figure 3. The binding energy of the high density (orthorhombic) unit cell with
displacements along the given unit cell directions for a reoptimized
geometry at each deformation geometry.
The displacements are given in fractional coordinates along each unit
cell direction.

Along y and z there are clearly problems. The chain interactions along these dimensions
are weak. DFT theory does not capture dispersion forces and so the weak chain
interactions at larger distances may be an artifact.

Based
on these calculated deformation energies we obtain the following deformation
energies for D = 0.01 (using only one of the dimensions)

U(D_{x}) - U(0) = 1.0 ± 0.15, D = 0.01

U(D_{y})
- U(0) = 0.5 ± 0.25, D = 0.01

U(D_{z})
- U(0) = 0.25 ± 0.15 , D = 0.01

Ignoring off-diagonal terms
in this simple example and substituting into the above formula (V = 2.534 x
4.93 x 7.4 = 92.4 Å^{3}) we find:

Y_{x} ~ 350 ± 54 , Y_{x}
~ 180 ± 90, Y_{x} ~ 90 ± 54 GPa

Experimental
measurements of Y_{x}
range from 230 to 360 depending on the method used. See reference 1.

There
are at least two ways to improve on this result.

1.
Sample smaller changes to determine whether there are well-defined minima. There could be more than one minimum given
that there a number of interactions as the chains pushed together or pulled
apart.

2.
Better DFT calculations can be carried out by employing larger basis sets.

References

1. Miao et al. J. Chem. Phys. (2001), 115, 11317-11324