Setting up Calculations on
Polymers using Periodic Boundary Conditions
As an example
of how to work with a polymer in periodic boundary conditions we choose to work
with polyethylene. Although there is a
known crystal structure for polyethylene we will load it into a box with
dimensions appropriate to the length of the molecule in order to demonstrate
the general procedure for chain packing of polymers. At a later stage we will examine the
crystalline form of polythylene.
Procedure
for Packing an All-trans Polymer Chain into Periodic Boundary Conditions
First, we read in the coordinate file (car file). Use the Molecule/Get command to read in polyethylene.car.
The image of
the molecule looks like this on screen.
Use the Measure/Distance command to determine the distance between two carbon atoms.
For example,
if we wish to use PBC with an six carbon chain we
would click on one of the carbons and then on the seventh carbon away from that
carbon. The distance we measure in this
case is 7.52 Angstroms. Note that the
choice of a six carbon unit is for illustrative purposes to demonstrate how the
chain packs in the unit cell. We could
have chosen four or two. Clearly a
two-carbon unit will be the least expensive for computations. This is considered further below when we
introduce the unit cell parameters of crystalline polyethylene.
Use the Transform/Axes command to define the x-axis as being along the length of alkane chain.
Choose any
carbon not on a line with the chosen x-axis to be the y-axis.
Then use the Transform/Rotate command to rotate the molecule. Make sure to use Absolute and not Relative.
Leave the angles as the default (0.0 0.0 0.0).
You may also
wish to center the molecule. Use the Transform/Center command. Center_of_Mass is a good choice for our purposes. Then use the Transform/Move
command. Here again Absolute should be the choice rather than Relative and
the center is x = 0.0, y = 0.0, z = 0.0.
Finally, use
the Transform/Apply command to apply these angles. Now rewrite the coordinates with in the new
coordinate system. Use the Molecule/Put command. Check the box Transformed to
write out the transformed coordinates.
Give the new file a different name from the original polyethylene.car
file. For example, I would call the new
file polyethylene_x.car. The molecule has the following appearance at
this point.
The molecule
needs to be converted into a fragment that will be replicated using periodic
boundaries. This is done by deleting all
of the atoms except six. Ideally you would
choose the six central atoms, but this is not essential. To delete atoms use the Builder. The Builder is called from the upper
left-hand corner of the InsightII window using a left mouse click. The Builder submenu (underneath the main
menu) will appear. Then use the Atom/Delete command. If you delete a
carbon atom, the hydrogens bonded to it are
automatically deleted. Once you have
deleted all but six of the carbons your fragment should have the following
appearance.
Now to place
the fragment in a periodic coordinate system use the Assembly/Cell command. Input the cell
dimensions as x = 7.52, y = 5.0, z = 5.0.
The x value was obtained from the measurement of the distance between
atom 1 and atom 7. The other two values
are arbitrary. The most difficult part
of the procedure will be the refinement of those values and the angles to
obtain the most accurate chain packing.
You will also want to check the Center_in_Cell box in this menu before executing the
command. Once you have executed the
command the fragment and cell will have the appearance shown in the figure
below.
Then use the Assembly/Cell_Display menu.
Check the Border_and_Pack option and then Execute. The periodic boundaries are now evident for
the first time. Note
in the figure below that the fragment is replicated once in both x and y. Note that you can choose to include z if
needed in the menu.
Now, you can
write the molecule including the periodic boundaries using the Molecule/Put command. Be sure to
check the box labeled PBC_File. When you examine the car file that you have
created (e.g. using vi editor) you will find that it
has the the PBC=ON specification and the there is an
extra line in the file that gives the dimensions and angles for the periodic
box that contains the molecule.
Initially we
chose a distance of 5 Å.
Perhaps a more reasonable distance between polyethylene chains would be
4 Å.
You could either repeat the procedure for creating the
unit cell starting with the 6 carbon fragment or you could copy the car file
and edit a new version.
> cp polyethylene_pbc1.car polyethylene_pbc2.car
> vi polyethylene_pbc2.car
Once in the vi editor replace
the 5.0000 Angstrom dimensions by 4.0000 Å.
How do we know whether 4.0 Å is a better value than 3.9 Å or 4.1 Å. Well, that it is
point of DFT calculations. Now we can
perform DFT calculations on a different unit cells in order to determine the
effect of distortions of the geometry on the energy of the system. In this way we can calculate the bulk modulus
and elastic properties of the polymer as well as vibrational frequencies if
these are of interest.
Construction
of a Model Starting with Unit Cell Parameters of a
Polyethylene
has been crystallized in the two unit cells.
The high density form has an orthorhombic unit cell and the low density
form has a triclinic unit cell.
The unit cell
parameters for the orthorhombic (high density form) are:
a = 2.534, b
= 4.930, c = 7.400, a =
90.0, b =
90.0, g = 90.0
The unit cell
parameters for the trirhombic (low density form) are:
a = 2.506, b
= 4.285, c = 4.285, a =
108.0, b =
90.0, g = 110.0
Note that the
parameters we discovered for the chain packing procedure above are not far from
the triclinic unit cell. If we had used
two carbons instead of six our x value would have been a = 2.507.
The high
density form has two molecules per unit cell.
This is important because it gives rise to chain packing the looks
hexagonal close packed. The low density
form has one molecule per unit cell and has a square arrangement of polymer
chains when viewed along the x-axis.
This is shown in the figure below where the side and top view of the two
crystalline forms of polyethylene are illustrated.
|
Orthorhombic
Cell |
Triclinic
Cell |
|
Top view |
Top view |
|
Side view |
Side view |
To pack the
orthorhombic unit cell we first copy the file above and then delete four of the
carbon atoms. It really does not matter
which ones are deleted, however, we can choose to keep the two carbons that are
closest to the point (0,0,0). To create the second polyethylene molecule
we use the program cellstagger. To use this program alter the input file so
that the unit cell parameters have the final desired values. So for the high density orthorhombic form the
input file would have the PBC parameters.
a = 2.534, b
= 2.465, c = 3.700, a =
90.0, b =
90.0, g = 90.0
The program
will prompt you for the displacements of the duplicate of the molecule. In the orthorhombic cell you will want to
place a duplicate of the original chain at the position 0,1/2,1/2
in the unit cell. So the displacements
will be:
X = 0.000
Y = 2.465
Z = 3.700
These values
are added to the unit cell dimensions so that the final values correspond to the
orthorhombic unit cell given above.
Once you have
generated the new .car file using cellstagger you can
examine the output using insightII to observe the
chain packing shown in the Figure above.
The high
density form of polyethylene can now be used to calculation of molecule
properties:
Elastic
constants: Young's modulus and Bulk modulus
Vibrational
Frequencies
Charge
Carrier Density
Band Gap
Density
References
1. J. Tadokoro "Structure of Crystalline
Polymers" Wiley,
2. J. Brandrup, E.
H. Immergut and E. A. Grulke
"Polymer Handbook" Wiley,