Property Prediction

2       Polymer Properties

The Polymer Properties module is a tool for analyzing some important structural features of polymers. These analyses can reveal the characteristic nature of polymer structures, thereby playing an important role in helping you understand the behavior of polymer materials in various environments. The results obtained can be directly correlated to the applications of polymer materials.

The structural characteristics analyzed include radius of gyration tensor, end-to-end distance of a polymer chain, distribution of selected dihedrals, orientation function (order parameter), dipole moment and quadrupole tensor, and Voronoi volume distributions. Useful plots can be generated from the analyses.

Sections in this chapter

General Methodology

Selecting the trajectory file

Selecting the trajectory file frames

Calculating physical and chain properties

To analyze the Voronoi volume

Calculating dihedral distributions

Specifying the torsion selection rules

Obtaining orientation function results

Specifying the bond selection rules

Theory

For information about		See
Editing and manipulating graphs		The Cerius2 Modeling Environment book
Calculating the compatibility of polymer mixtures, analyzing binding energies, or calculating thermodynamic mixing variables		The "Blends" chapter in this book
Calculating the pair distribution function and structure factor of polymers		The Cerius2 Simulation Tools book
Calculating the velocity autocorrelation function and predicting vibrational properties, calculating the self-diffusion constant, or performing statistical analysis of properties		The Cerius2 Simulation Tools book

The trajectory file is selected and loaded using the browser box on the Polymer Trajectory control panel (see the online help for more information on control panels).

Note

To select the trajectory file to be analyzed

1.   Select Trajectory File from the POLYMER PROPERTIES card to bring up the Polymer Trajectory control panel.

2.   Use the browser box to select the trajectory file to be analyzed. Trajectory file names use the .trj, .qtrj, or .atrj extensions.

To specify the trajectory frames to be analyzed

1.   Choose Frame Selection from the POLYMER PROPERTIES card to bring up the Trajectory Frames control panel.

2.   Enter the starting and ending frame numbers in the First and Last entry boxes.

3.   Specify the frame interval by entering the number of frames in the Step entry box.

4.   To reset to include all frames, click the Reset to Full Trajectory button.

Note

To calculate physical or chain properties

1.   If using trajectory file data:

a.   Choose Trajectory File from the POLYMER PROPERTIES card and use the browser box to select the trajectory file to be analyzed.

b.   Specify the frames that are to be included in the analysis (see "Selecting the trajectory file frames" on page 23).

Otherwise, read in or build the model that is to be used in the analysis.

Note

2.   Choose Properties from the POLYMER PROPERTIES card, then select Physical & Chain to bring up the Physical & Chain Properties control panel.

3.   Check the box for each physical property to be calculated:

Dipole Moment

Quadrupole Tensor

Molecular Weight

Density

4.   If calculating dipole moment and using all the atoms in the model, choose All from the popup; otherwise, choose Selected and select the atoms to be included from the model window.

5.   Check the box for each chain property to be calculated:

Radius of Gyration

End-to-End Distance

6.   Click the Calculate button.

1.   If using trajectory file data:

a.   Choose Trajectory File from the POLYMER PROPERTIES menu card and use the browser box to select the trajectory file to be analyzed.

b.   Specify the frames that are to be included in the analysis (see "Selecting the trajectory file frames" on page 23).

Otherwise, read in or build the model that is to be used in the analysis. A periodic model must be used.

2.   Choose Properties from the POLYMER PROPERTIES card, then select Volume to bring up the Volume Properties control panel.

3.   To calculate the Voronoi volume:

a.   Check the Voronoi Volume box.

b.   If using all the atoms in the model, choose All from the popup; otherwise, choose Selected and select the atoms to be included.

c.   Enter the cutoff distance to be used.

d.   Enter the number of bins to be used in the distribution histogram.

4.   To calculate the coordination number, check the Coordination Number box.

5.   If you want to list additional data in the text window, check the Use Long Output Format box.

6.   To calculate the cell volume, check the Cell Volume box.

7.   Click the Calculate button.

Calculating dihedral distributions

1D or 2D plots, time
evolution

Polymer Properties can be used to calculate and plot the distribution of selected types of rotatable dihedrals; a 1D plot of frequency versus dihedral angle is generated. The interdependence of two consecutive dihedrals can also be illustrated with a 2D plot of dihedral one versus dihedral two (only the discrete points are plotted). If trajectory file data is being used, averaged values for all the snapshots are used in the plots. The time evolution of a single dihedral in the trajectory file can also be plotted.

Where variables are set

You perform the calculations using options on the Dihedral Distributions control panel (see the online help). The type of plot generated and the dihedrals used are also specified on this control panel.

Input

The calculations can be performed either on a single isolated structure or periodic model, or on data from a trajectory file. The trajectory file and frames are specified using both the Polymer Trajectory control panel and the Trajectory Frames control panel (see the online help). The structure must be a single-chain polymer consisting of well-defined regular repeat units; at least four repeat units are required.

Defining the dihedrals

You define the dihedrals by selecting them from a list. The type of dihedrals that appear in the list are specified using the options on the Torsion Selection Rules control panel (see the online help). You can have all rotatable dihedrals in the polymer listed, or you can list only unique dihedrals. You can specify that only dihedrals with a particular central bond pair type, such as C--O, C--N, or C--C, be shown. You can also determine whether dihedrals with terminal hydrogen bonds are listed and, if so, whether those with one or all terminal hydrogens are included.

You can then use the Find button to find and list all the specified dihedrals (the element label and number are given for the four atoms). You define the dihedrals to be used in the plots by selecting them from those listed.

To obtain a 1D distribution or 2D plot

1.   If using trajectory file data:

a.   Choose Trajectory File from the POLYMER PROPERTIES card and use the browser box to select the trajectory file to be analyzed.

b.   Specify the frames that are to be included in the analysis (see "Selecting the trajectory file frames" on page 23).

Otherwise, read in or build the model that is to be used.

Note

2.   Choose Properties from the POLYMER PROPERTIES card, then select Dihedral Distributions to bring up the Dihedral Distributions control panel.

3.   Specify the type of plot by selecting from the Trajectory Plot Type popup:

1-D Average to plot a dihedral distribution

2-D Average to plot the interdependence of two dihedrals

4.   If doing a distribution plot, enter the bin width to be used.

5.   If you want to list additional data in the text window, check the Use Long Output Format box.

6.   Specify the torsion selection rules (see "Specifying the torsion selection rules" on page 28).

7.   Click the Find button to find and list all specified dihedrals.

8.   Select the dihedrals to be analyzed from the Corresponding Dihedrals of the Repeat Unit list.

9.   To highlight the selected dihedrals on the model, click the Show Torsions button.

10.   To deselect all dihedrals, click the DeSelect All button. To redefine dihedrals, repeat steps 6 through 9 as needed.

11.   Click the Calculate button.

1.   Choose Trajectory File from the POLYMER PROPERTIES card and use the browser box to select the trajectory file to be analyzed.

2.   Specify the frames that are to be included in the analysis (see "Selecting the trajectory file frames" on page 23).

Note

3.   Choose Properties from the POLYMER PROPERTIES card, then select Dihedral Distributions to bring up the Dihedral Distributions control panel.

4.   Select Evolution from the Trajectory Plot Type popup.

5.   Enter the repeat unit to be used in the Repeat ID Number entry box.

6.   If you want to list additional data in the text window, check the Use Long Output Format box.

7.   Define the dihedral to be plotted as described in steps 6 through 10 in "To obtain a 1D distribution or 2D plot" on page 27.

8.   Click the Calculate button.

Only rotatable dihedrals are allowed. You can have all the rotatable dihedrals listed, or you can restrict the listings to a particular type; this often makes the selection process easier. For example, you can specify that only unique dihedrals be listed or that only dihedrals with a particular central bond pair type, such as C--O, C--N, or C--C, be shown. You can also determine whether dihedrals with terminal hydrogen bonds are listed and, if so, whether those with one or all terminal hydrogens are included.

Defaults

The defaults list only unique rotatable dihedrals; all central bond pair types are included, but dihedrals with terminal hydrogens are not allowed.

To specify the torsion selection rules

Note

1.   Choose Properties from the POLYMER PROPERTIES card, then select Dihedral Distributions to bring up the Dihedral Distributions control panel.

2.   Click the Rules... button to bring up the Torsion Selection Rules control panel.

3.   To list only unique dihedrals, check the Unique Torsions Only box; otherwise, uncheck it.

4.   Specify the torsion center bond types to be listed:

a.   Click the Torsion Center Bond action button to update the torsion center bond types for the current model

b.   Select the type desired from the adjacent popup (All or a bond type such as C--C or C--O)

5.   Specify the dihedrals with terminal hydrogens that are to be included by selecting one of the options from the Terminal Hydrogens popup (None, One, or All).

The calculations can be performed either on a single isolated structure or periodic model, or on data from a trajectory file. The trajectory file and frames are specified using both the Polymer Trajectory control panel and the Trajectory Frames control panel (see the online help). The model must be a single-chain polymer consisting of regular repeat units (at least four repeat units are required).

Where variables are set

The reference direction, structural unit vector, and plot and output variables used in the orientation function calculations are specified using the options on the Orientation Function control panel (see the online help).

Reference direction

The reference direction is specified by selecting an axis (x, y, or z), by entering the x, y, and z components of a vector, or by selecting atoms to define a vector.

Structural unit vector

The structural unit vector is defined by selecting a particular bond or set of bonds from those that appear in a list.

You first specify the type of bonds that are to be included in the list. This is done using options on the Bond Selection Rules control panel (see the online help). You can specify that all bonds in the repeat unit be listed, only single, double, or triple bonds be listed, or only particular bond pairs, such as C::O, C::N, or C::C, be shown. Next, you click the Find button to find and list all specified bonds in the repeat unit (the element label and number are given). Finally, you define the structural unit vector by selecting the desired bonds from those listed.

Information obtained

The distribution of , the average angle , and the orientation function P₂ are calculated and displayed in the text window, and a histogram showing the orientation angle distribution is plotted in the graph window. If a trajectory file is being used, averages are calculated and displayed for each snapshot, and the average orientation angle distribution for all the snapshots is plotted. Alternatively, the time evolution of for a single vector in the trajectory file can be plotted.

To calculate the orientation function and angle distribution

1.   If using trajectory file data:

a.   Choose Trajectory File from the POLYMER PROPERTIES card and use the browser box to select the trajectory file to be analyzed.

b.   Specify the frames that are to be included in the analysis (see "Selecting the trajectory file frames" on page 23).

Otherwise, read in or build the model that is to be used.

Note

2.   Choose Properties from the POLYMER PROPERTIES card, then select Orientation Function to bring up the Orientation Function control panel.

3.   Select Average from the Trajectory Plot Type popup.

4.   Enter the bin width to be used in the distribution histogram.

5.   If you want to list additional data in the text window, check the Use Long Output Format box.

6.   Select the reference direction method from the Define By popup (Axis, Atoms, or Components):

• If Axis is chosen, select the coordinate axis to be used from the popup (X-axis, Y-axis, or Z-axis).

• If Atoms is chosen, select the atoms that are to define the vector from the model window, then click the Pick button.

• If Components is chosen, enter the x, y, and z components of the vector in the Cartesian Direction entry boxes.

7.   Specify the bond selection rules (see "Specifying the bond selection rules" on page 32).

8.   Click the Find button to find and list all specified bonds in the repeat unit.

9.   Select the bonds that define the structural unit vector from the Corresponding Bonds of the Repeat Unit list.

10.   To highlight the selected bonds on the model, click the Show Bonds button.

11.   To deselect all bonds, click the DeSelect All button. To redefine the structural unit vector, repeat steps 7 through 10 as needed.

12.   Click the Calculate button.

1.   Choose Trajectory File from the POLYMER PROPERTIES card and use the browser box to select the trajectory file to be analyzed.

2.   Specify the frames that are to be included in the analysis (see "Selecting the trajectory file frames" on page 23).

Note

3.   Choose Properties from the POLYMER PROPERTIES card, then select Orientation Function to bring up the Orientation Function control panel.

4.   Select Evolution from the Trajectory Plot Type popup.

5.   Enter the repeat unit to be used in the Repeat ID Number entry box.

6.   If you want to list additional data in the text window, check the Use Long Output Format box.

7.   Define the reference direction and structural unit vector as described in steps 6 through 11 in "To calculate the orientation function and angle distribution" on page 30.

8.   Click the Calculate button.

Defaults

The defaults specify that all bond pair types and bond orders (single, double, and triple), except bonds with terminal hydrogens, be included in the list.

To specify the bond selection rules

Note

1.   Choose Properties from the POLYMER PROPERTIES card, then select Orientation Function to bring up the Orientation Function control panel.

2.   Click the Rules... button to bring up the Bond Selection Rules control panel.

3.   Click the Bonding Pair button to update the bonding atom pair types listed for the current model, then select the type desired from the adjacent popup.

4.   Select the bond type desired from the Bond Order popup (All, Single, Double, or Triple).

5.   Check the Include Hydrogens box.

Dipole moment

The dipole moment formed by a pair of equivalent point charges with opposite sign is defined as:

Eq. 1            

µ = Dipole moment

q = Charge

l = Vector pointing from the negative charge to the positive one

The components of µ are calculated from the atomic coordinates and atomic charges using the following equations:

Eq. 2            

x_i, y_i, z_i = Coordinates of atom i

Qi = Charge on atom i

Eq. 2 can be used as a more general definition of a dipole when a system is not neutral.

The dipole moment can be calculated for the entire model or just for selected atoms. Values reported to the text window are the magnitude of the dipole moment and the three Cartesian components µ_x, µ_y, and µ_z.

Quadrupole tensor

The components of the quadrupole tensor are calculated using the following equations and their equivalents:

Eq. 3             Diagonal components

Eq. 4             Off-diagonal components

Molecular weight

Molecular weight is calculated from the atom masses. Currently, calculations include the entire model. However, the ability to use multiple-chain structures and to select specific molecules will be available in the future. The value reported in the text window is given in grams.

Density

Density of periodic systems is calculated from both the molecular weight and the unit cell parameters. Values listed in the text window are given in g/cc. If trajectory file data is being used, the time evolution is also plotted in the graph window.

Chain properties -- radius of gyration and end-to-end distance

Radius of gyration
tensor (S)

Eq. 5            

r₀ (x₀, y₀, z₀) = Position of the center of gravity

r_i (x_i, y_i, z_i) = Position of atom i (i = 1,N)

Here the average is over all atoms.

The center of gravity of the polymer can be calculated from weight-averaged coordinates:

Eq. 6             , ,

If every atom (or segment) has the same mass, then , and simplified equations like the following can be used to calculate the elements of the S matrix:

Eq. 7            

Eq. 8            

Principal values
and directions

The diagonalization of the S matrix gives its principal values S₁², S₂², S₃² (eigenvalues) and their orientation (eigenvectors) in the Cartesian frame.

Radius of gyration (s)

The radius of gyration, s, is the square root of the first invariant of S:

Eq. 9            

The end-to-end distance, R, is simply calculated between the first and last skeleton atoms in the polymer chain.

Values reported to the text window are the radius of gyration tensor, its principal values and directions, and the scalar value of the radius of gyration. If trajectory file data is used, plots also appear in the graph window, showing the time evolution for the radius of gyration and end-to-end distance.

Voronoi volume analysis

Voronoi polygon

In the two-dimensional case, the Voronoi polyhedron becomes the Voronoi polygon. The construction of a Voronoi polygon is illustrated in the figure below:

Figure 1 . Voronoi polygon

In three dimensions, the edges shown above become faces. Each face of the polyhedron is a polygon.

Determining Voronoi volume

The volume of a Voronoi polyhedron for an atom is determined as follows:

1.   All the neighboring atoms within a given cutoff distance are found.

2.   The polyhedron for the atom is defined:

a.   For each neighboring atom, a plane is defined whose normal is the vector pointing from the center of the atom to this neighbor. The plane is positioned so that the space between the two atoms is divided equally.

b.   Only the closest planes are used.

3.   The volume of the polyhedron is calculated:

a.   The volume contributed by each face is calculated:

Eq. 10            

Where:

A_face = Area of the face

d = Distance from the face to the center atom

b.   The volume of the polyhedron is determined by summing the volumes of all the faces.

The information obtained from Voronoi volume analysis includes the coordination number (that is, the number of faces) for each atom, the Voronoi volume for each atom, the average Voronoi volume, and the Voronoi volume distribution. These results are reported in the text window, and a histogram of the Voronoi volume distribution is plotted in the graph window. If trajectory file data is used, the time evolution of the Voronoi volume is also plotted.

Voronoi volume analysis can be performed on all atoms in a periodic model or only on selected atoms.

The calculations are performed using options on the Volume Properties control panel (see the online help). The cutoff distance and other variables are also set on this control panel.

Cutoff distance

Specifying an appropriate cutoff distance is often critical. The calculation time for large systems can be rather long (on the order of hours). Unnecessarily large cutoff distances significantly increase the calculation time; this is because the number of neighboring atoms increases with distance . On the other hand, if the cutoff distance is too small, some neighboring atoms may be excluded, leading to incorrect results. The default value, 6.0 Å, is usually appropriate for systems with an even distribution of atoms.

Order versus disorder

For a completely ordered structure, the shape of the Voronoi polyhedron is well-defined. The number of types of polyhedrons is determined by the number of atom types in the model. For a disordered structure, however, the random nature of the local environment can cause the shape of the Voronoi polyhedron to be different, even for the same type of atoms. The distribution of Voronoi volumes is therefore quite different for disordered structures.

External effects

External influences such as temperature, strain, and electric field can change the Voronoi volume distribution. An analysis of these changes can reveal useful information about the structural changes of polymers.

Calculating the orientation function (order parameter)

A measure of alignment can be obtained by calculating the orientation function. Useful information can also be revealed by calculating the average orientation angle, and by analyzing orientation angle distributions.

Defining the terms

Hermans orientation function, P₂

Eq. 11            

= Angle between the structural unit vector and the reference direction.

Order parameter, S

In liquid crystalline science, the term order parameter is often used, designated by the symbol S. The definition of the order parameter is identical to that given for the orientation function in Eq. 11. Therefore, they are completely equivalent.

Average value used, <cos²>

Because a model system can contain a large number of structural units, cos² in Eq. 11 has a particular distribution. As a result, Polymer Properties uses the average value, <cos²>, in calculating P₂:

Eq. 12            

The orientation functions for three typical situations are shown in the table below.

	<cos2>	*	P2
Perfect alignment	1	0	1
Random orientation	1/3	54.73	0
In perpendicular plane	0	90	-0.50
*(0 90)

The values for P₂ vary from 1 (perfect alignment) to -0.50 (perpendicular); the P₂ for random orientation is 0.

Random orientation

In the case of random orientation, it can be shown that the distribution function, f(), is a sine function:

Eq. 13            

The <cos²> is obtained from:

Eq. 14            

Average angle

The average angle is given by:

Eq. 15            

The values for used in the calculations and reported in the text window are restricted to:

Eq. 16