| Property Prediction |
Second derivative
The second derivative method uses a single-point energy calculation to obtain the second derivatives of the lattice energy with respect to the lattice parameters and the atomic coordinates. The following energy expression is used:
Eq. 1
Where: 
+ higher order terms
= Strain
can be used to calculate the components Cij of the stiffness matrix:
Eq. 2
The stiffness matrix computed by this method is always symmetric; that is, 
.
Constant-stress minimization
Constant-stress minimization applies an external stress to a minimized periodic system. The structure is then reminimized, allowing all the lattice parameters to vary, and the resultant strain is measured. This is repeated for a series of stresses. The variation of the measured strain as a function of external stress is used to derive the stiffness matrix. Stresses can be applied in the x, y, or z directions, or shear stresses can be applied. Positive stresses result in expansion; negative stresses result in compression.
In default mode, the stiffness matrix is calculated in the simplest manner. Each of the six independent elements of the stress tensor is varied individually while the other five elements are kept at zero. This results in six sweeps being defined. For each sweep, calculations are performed for up to 20 values of the variable tensor element. Default values for each sweep can be edited, but each sweep must remain within elastic limits in order to produce a true stiffness matrix. The sweep parameters are specified using the options on the Constant Stress Minimization Prefs control panel (see "Specifying the applied stress/strain and mode" on page 9).
The slopes of the stress-strain graphs correspond to the elements of the compliance matrix. Both the compliance matrix and its inverse, the stiffness matrix, are used to derive the other mechanical properties.
In custom mode, only one sweep is used, allowing all six elements of the stress tensor to vary independently. You have complete control over the stress profile; that is, you can specify applied stress values for all the elements of the stress tensor. Custom mode is specified by checking the Use Only One Customized Sweep box on the Constant Stress Minimization Prefs control panel (see the online help for more information).
Constant strain minimization
This technique is the complement of the constant stress minimization described above. Small strains are applied to a periodic structure at an energy minimum. The structure is reminimized keeping the lattice parameters fixed, and the resultant stress in the minimized structure is measured. This is repeated for a series of strains. The variation in the measured stress as a function of applied strain is used to derive the stiffness matrix. Strains can be applied in the x, y, or z directions or shear strains can be applied.
For constant stress minimization, either the default or the custom mode can be used. They work as for constant stress minimization except that the strain tensor is varied (instead of the stress tensor) and the sweep parameters are specified using the options on the Constant Strain Minimization Prefs control panel (see the online help for more information).
Advantages and drawbacks of the three methods
The second derivative method is generally considered the best method of the three; its principal drawback is that for large systems it is both slow and memory-hungry. The constant stress method works for larger systems than the second derivative method but is also slow. The constant strain method is the quickest for large systems but is the least likely to be accurate.
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The shear modulus cannot be obtained using the second derivative method. To calculate the shear modulus, you must either use the constant stress method or the constant strain method.
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Calculating mechanical properties
Module design
The mechanical properties of a periodic model can be calculated by simply clicking the Calculate button on the Mech Props Run control panel (see the online help). The calculation uses the parameters currently specified.
The module is designed so that the occasional user can quickly and easily predict mechanical properties, but features are also available that allow the expert to fine-tune the calculations and to model a custom stress or strain profile.
Three property prediction methods are provided: second derivative, constant-stress minimization, and constant strain minimization. Select the method you want to use from the Method popup.
The minimization methods involve applying a series of stresses or strains to the periodic structure. The direction and value of the external stresses or strains and the mode used are specified on the Constant Stress Minimization Prefs and Constant Strain Minimization Prefs control panels, accessed using a Preferences... button. You can use the defaults or you can set up your own custom stress or strain profile (see "Specifying the applied stress/strain and mode" on page 9).
The results of the mechanical properties calculations are displayed in the text window and saved in a text file. Alternatively, you can save the results in table format. The controls that specify output format and level of output detail are set on the Mech Props Output control panel. Mechanical Properties also outputs files containing the run parameters, model coordinates, and trajectories. You should enter a seed name for all the output files before initiating a new calculation. For details, see "Displaying and saving results" on page 12.
All of the methods rely on minimizing the structure before performing the calculations. Thus, the Minimize Model First switch is preset to on. The structure is minimized under zero stress using the minimization variables currently specified. The default settings use the conjugate gradient algorithm (for both the atoms and unit cell) and termination criteria of 500 steps or rms force of 0.1000 kcal/mol/Å. You can specify different termination values using the options in the Minimizer. For information about when you should do this, see the note on page 8.
An Accumulate Averages switch allows you to calculate cumulative averages from successive calculations or reloads. Averages from different calculation methods cannot be done (for example, only data from a constant stress minimization can be averaged with data from another constant stress minimization). A generation number is inserted into the names of the output files when the Accumulate Averages switch is on; files are simply overwritten when the switch is off.
Analysis functions are provided for plotting and fitting data obtained from constant stress and constant strain minimization. Several different types of plots can be created including stress-strain and pressure-volume curves. You can fit the plotted data to a range of curve types, automatically derive moduli, and calculate average Y values. The derived mechanical properties can also be recalculated using the fitted values. For details, see "Analyzing the data" on page 14.
Options are provided that allow you to investigate the effects of constraints imposed by crystal symmetry on the mechanical properties of a material. When a material belongs to certain crystal classes, relationships exist between elements of the stiffness matrix. For example, when the crystal class is cubic, the following relationships apply within the stiffness matrix C:
The data saved from a mechanical properties calculation can be reloaded and analyzed at a later date (see "Restoring data for analysis" on page 16).
To calculate the mechanical properties of a model
1. Place a periodic structure in the current model space. The structure
can be loaded in from file or created using the Cerius2
Builders. (If the structure's symmetry is not primitive, convert
it to P1 using the Superlattice button on the Crystal Building
control panel of the Crystal Builder.)
3. Select Run on the MECHANICAL PROPERTIES card to bring
up the Mech Props Run control panel.
4. Select the method to be used from the Method popup.
5. If using one of the minimization methods (Const Stress Min or
Const Strain Min), specify the mode and values for applied
stress/strain in the Preferences control panel (see "Specifying
the applied stress/strain and mode" on page 9).
6. Enter the file name seed for the output files in the Files Prefix
entry box.
9. Specify the level of detail and file(s) used to save the results (see
"Displaying and saving results" on page 12).
10. Click the Calculate button.
1. Calculate the mechanical properties of the model given no constraints (the default). Follow the procedure "To calculate the mechanical properties of a model" described on page 7.
3. Click the Redisplay Results button.
Sweep defaults
Default values for applied stress or strain are easily specified using the options under Sweep Defaults. Initial and final values are used to define the range. The total number of points must also be entered (as many as 20 points are allowed for each sweep). The other values are automatically calculated from the initial and final values so that the intervals between points are equal.
You should choose initial and final sweep values that result in only small (elastic) deformations of the initial structure. While these values will depend upon the stiffness of the material, you should aim to keep the change in cell parameters to less than 1%. In some cases, if the material has a soft shear modulus, the maximum elastic deformation should be even smaller -- on the order of 0.1%.
Either the default or custom mode can be used to perform constant stress and constant strain minimization. These modes are described under "Constant-stress minimization" on page 3. Custom mode is specified by checking the Use Only One Customized Sweep box. Default mode is specified by leaving this box unchecked (default).
The values shown in the Applied Stress Table determine the stress profile that will be applied during the mechanical properties calculations. Similarly, the Applied Strain Table reflects the strain profile. The values specified under Sweep Defaults are initially assigned to each sweep. In custom mode, these defaults are assigned to only two components of the stress/strain tensor (xx and yy); the values for the other four components are set to zero.
The stress/strain tensor values are ordered and applied following the standard Voigt notation. For example, an xy stress value of 0.1 corresponds to setting the values of the xy and yx elements of the stress tensor to 0.1.
If you don't want to use the default values, you can easily enter new values in the Applied Stress/Strain Table. When using default mode, values can only be entered for one sweep at a time. The sweep to be edited is selected from a popup menu. Each sweep is referred to by the component of the stress/strain tensor that is allowed to vary. When using custom mode, values for all components of the stress/strain tensor appear in the table and can be edited.
You can view the currently specified stress or strain profile by clicking the Plot Stress/Strain Profile button.
Default or custom mode can be used to perform constant stress and constant strain minimization. Custom mode is specified by checking the Use Only One Customized Sweep box. Default mode is specified by leaving this box unchecked (default). These modes are described under "Constant-stress minimization" on page 3.
To specify variables for constant-stress minimization
1. If necessary, open the Mech Props Run control panel by selecting
Run from the MECHANICAL PROPERTIES card.
2. Make sure the Method popup is set to Const Stress Min.
3. Click the Preferences... button to bring up the Constant Stress
Minimization Prefs control panel.
4. Enter a value for Number of Points per Sweep (2 to 20).
5. Enter a value for Initial Stress (GPa).
6. Enter a value for Final Stress (GPa).
8. If not using the default values for applied stress, edit the
Applied Stress Table:
a. Select the stress tensor element whose values are to be edited from the Stress Components Which Vary During Sweep popup.
b. Enter the new values for this sweep in the entry boxes.
c. Repeat the above two steps for each sweep to be edited.
9. To view a plot of the stress profile:
b. Click the Plot Stress Profile button.
In custom mode, all the tensor elements are displayed on the plot in the graph window. In default mode, only the element selected by the popup is plotted.
To specify the constant strain minimization variables
1. If necessary, open the Mech Props Run control panel (select
Run from the MECHANICAL PROPERTIES menu card).
2. Make sure the Method popup is set to Const Strain Min.
3. Click the Preferences... button to bring up the Constant Strain
Minimization Prefs control panel.
4. Enter a value for Number of Points per Sweep (2 to 20).
5. Enter a value for Initial Strain (%).
6. Enter a value for Final Strain (%).
8. If not using the default values for applied strain, edit the
Applied Strain Table:
a. Select the strain tensor element whose values are to be edited from the Strain Components Which Vary During Sweep popup.
b. Enter the new values for this sweep in the entry boxes.
c. Repeat the above two steps for each sweep to be edited.
9. To view a plot of the strain profile:
b. Click the Plot Strain Profile button.
Displaying and saving results
Each time a mechanical properties calculation is performed, the results of the energy calculations are displayed in the text window. When the energy calculations are complete, the mechanical properties are automatically calculated and displayed. These include:
Output variables
The level of detail and the format used to save the results of the mechanical properties calculations can be specified. This is done using options on the Mech Props Output control panel (see the online help for more information).
The results are written to a text file (.txt) by default. You can turn this off. You can also specify that the results should be saved in table format (.dat file). The two check boxes under Files are provided for this purpose.
The Output Level popup is used to specify the detail level of the results that are displayed in the text window and written to the text file. Three settings are provided: None, Normal (default), and Detailed.
To specify the output variables
1. Select Output from the MECHANICAL PROPERTIES card to
open the Mech Props Output control panel.
2. Select the level of detail (None, Normal, or Detailed) from the
Output Level popup.
| The seed name for the output files is specified using the Files Prefix option on the Mech Props Run control panel (see the online help). |
Analyzing the data
The data obtained from constant stress minimization can be analyzed using the options on the Constant Stress Minimization Analysis control panel (see the online help). An identical control panel is provided for analyzing constant strain minimization data (see the online help for the Constant Strain Minimization Analysis control panel). Two types of analysis are provided: data plotting and data fitting.
The plots are displayed in the graph window. They can be saved and their appearance can be manipulated using the functions in the Graphs module (see Cerius2 Modeling Environment).
Data fitting
Four data fitting algorithms can be used to fit the plotted data: linear, quadratic, cubic, and quartic. The type is selected from the Curve Type popup. Click the Fit __ Curve to Data Points button to initiate the data fitting. The fitted curve is added to the plot and the correlation coefficient is calculated and displayed in the text window. You can exclude certain points from the fit, if appropriate. This is done by selecting a subset of data points from the graph window. Only these are used in the fit. Average Y values can also be calculated.
To analyze constant-stress minimization data
1. Open the Constant Stress Minimization Analysis control panel.
Select Analysis from the MECHANICAL PROPERTIES card,
then choose Constant Stress from the popup menu.
2. If analyzing saved data from a previous session, reload the
appropriate .mp file (see "Restoring data for analysis" on
page 16).
a. Select the trajectory file to be used from the Sweep Trajectory
File popup.
b. Select the type of plot from the Plot Type popup.
d. Click the Plot From Sweep button.
e. To display additional plots, repeat Steps a through d.
4. To fit the most recently plotted data:
a. Select the type of curve from the Curve Type popup (Linear,
Quadratic, Cubic, or Quartic).
b. If not using all the data points, select from the graph window
those to be included in the fit.
c. Click the Fit __ Curve to Data Points button.
5. To calculate the average Y values for the last plotted data:
b. Click the Calculate Average Y Value of Data Points button.
To analyze constant-strain minimization data
1. Open the Constant Strain Minimization Analysis control panel.
(Select Analysis from the MECHANICAL PROPERTIES card,
then choose Constant Strain from the popup menu.)
2. Follow Steps 2 through 8 under "To analyze constant-stress
minimization data" on page 15.
Restoring data for analysis
You can reload a mechanical properties calculation file (.mp) that was created from a previous run. Reloading restores the run parameters, recalculates the derived properties, and sets up the trajectory file(s) for data analysis. This allows you to run the calculation at one time and then reload the results for analysis later.
Reloading is done using the controls on the Mech Props Reload control panel (see the online help). The .mp file is selected using a browser box. Information about the file including the method and type of applied stress/strain can be viewed using the Show Information button. The Reload Sweeps button is used to reload the file.
To reload a mechanical properties calculation file
1. Select Load from the MECHANICAL PROPERTIES menu
card to open the Mech Props Reload control panel.
2. Use the browser box to select the .mp file to be reloaded.
3. To display information about the file, click the Show Information
button.
4. Click the Reload Sweeps button.
Theory
Deriving mechanical properties
Any of the three methods described in this chapter (second derivative, constant-stress minimization, or constant-strain minimization) can be used to calculate the stiffness matrix. Once the stiffness matrix has been obtained, several related mechanical properties can be derived from it.
The compliance matrix, S, is calculated as the inverse of the stiffness matrix:
Eq. 3
Compressibility
Volume compressibility is calculated from the compliance matrix as follows:
Eq. 4
Bulk modulus
Bulk modulus is the inverse of the compressibility:
Eq. 5
Young's modulus
Young's modulus is calculated from the compliance matrix. Values are given for three directions (x, y, and z) as follows:
Eq. 6
Poisson's ratios
Poisson's ratios are calculated from the compliance matrix for six components:
Eq. 7
Velocities of sound
The velocities of sound, V, are calculated from the eigenvalues of the corresponding stiffness submatrix,
sc, and the density, 
:
Eq. 8
Three submatrices can be derived from the stiffness matrix, C. Each of these submatrices has three eigenvalues, giving a total of nine eigenvalues and, therefore, nine components for the velocity of sound.
Lamé constants are calculated from the stiffness matrix and apply only to isotropic materials:
Eq. 9
Eq. 10
Model, forcefield, and energy setup
Models
No limits are set on the unit cell size or on the number of atoms in the unit cell. In practice, however, the amount of available memory and/or the time required for large calculations eventually limits the number of atoms you can use. The second derivative method is more memory-intensive than either of the two minimization methods.
All of the property prediction methods perform energy calculations on the model. The energy is calculated using the current forcefield energy expression. If no energy expression is set up, Mechanical Properties does it automatically using the default settings for the Open Force Field. However, you are more likely to want to exercise control over the forcefield and energy expression setup. Cerius2 provides a wide range of forcefields and parameters that allow you to choose the one best suited to your problem. For information about the forcefields available and how to load them, atom typing of models, choosing energy terms, setting atom exclusions and restraints, and setting up the energy expression, see the Cerius2 Simulation Tools book and the Cerius2 Forcefield-Based Simulations book.
Fan, C. F.; Hsu, S. L. Macromolecules, 25, 266 (1992).
References