| Property Prediction |
1. The Bravais Friedel Donnay Harker (BFDH) method.
2. The Attachment Energy (AE) method for growth morphologies.
3. The Surface Energy (SE) method for equilibrium morphologies.
4. The Hartman-Perdok (HP) method.
The BFDH method is an approximation based on geometrical considerations. The AE method relies on a calculation of the energy released when a growth slice is added to a growing plane. Both predict the relative growth rates of possible growth faces. In contrast, the SE method predicts the morphology which minimizes the total surface energy of the crystal, by determining the surface energy of the relevant faces. Finally, the HP method uses a powerful algorithm for identifying stable growth planes by generating connected chains and nets of strong bonds in the crystal. Attachment Energies of these growth planes are then calculated and used to predict the growth morphology.
Step 1: Setup
Before you start, you must ensure that the current model space contains a crystalline model, loaded from a file or constructed using the Crystal Builder. For all calculations except BFDH (step 2), you will also need to load and configure an appropriate Cerius2 force field, unless you wish to use the provided default force field.
You can make predictions using a number of different methods.
The calculated morphology is displayed in the model window. Visualization options allow you to display the external morphology with or without the atomistic structure, to control the transparency of the crystal faces and to label the faces with Miller indices or surface area. You can set these variables using the options on the Morphology Display control panel (see Displaying the morphology).
Step 4: Listing, editing, adding and removing growth faces
A conventional way to describe crystal morphology is to list the Miller indices of all present faces and their perpendicular distance from the center of the crystal; that is, the center-to-face distance. After a morphology calculation, such a list is available on the Edit/Add Faces control panel. This control panel also allows you to edit the properties of the growth faces, add new faces or remove undesired faces (see Editing, adding and removing crystal faces).
You can compute several properties of the calculated morphology using the options on the Morphology Analysis control panel. These include interplanar angles, aspect ratio, and the percentage surface area accounted for by certain forms (see Analyzing the morphology).
You can save the morphology of the crystal structures to a file and access it later (see Storing morphologies).
If you use the Hartman-Perdok method, further analysis capabilities for the bonds and connected nets within the crystal are provided (see Generating and editing the Crystal Graph and Generating and editing Periodic Bond Chains and Connected Nets).
General Methodology
Applications
Morphology allows researchers both to study particle shape and to consider the effects of altering the growth rate of particular faces on crystal morphology. In particular, Morphology can look at the effect of tailor-made additives in modifying growth. Knowledge of shape and aspect ratio is essential to understanding packing, flow problems, clogging of filters, and other problems. The Morphology module can also provide insight into other properties, such as the texture of powders and polymorphism. In addition, the Hartman-Perdok module enables you to understand the role of bonded networks in stabilizing particular faces and to study the effect of altering interactions between molecules ("bonds") on the morphology. Morphology's application areas include pharmaceuticals, agrochemicals, food sciences, petrochemicals, cements, and commodity and specialty chemicals.
Morphology is an ideal complement to other Cerius2 modules:
Growth planes and growth rate list
The prediction generates a list of possible growth planes that satisfy the Donnay Harker rules for the current symmetry. The calculation loops through all values of h, k, and l choosing faces. Any face forbidden by the symmetry (Donnay Harker) has its indices increased as necessary (for example, 2 0 0 may replace 1 0 0). Next, a center-to-face distance is assigned according to Bravais Friedel as being proportional to the reciprocal of the lattice spacing. For theoretical details, see Bravais Friedel Donnay Harker method.
The list of generated planes is displayed in the Edit/Add Faces control panel. The listed center-to-face distances indicate the relative growth rates of the planes. The actual values assigned are arbitrary and are in fact equal to 100/d, where d is the inter plane spacing or slice thickness.
The 1000 planes with the smallest center-to-face distances are listed; these planes must all have a d value greater than the minimum slice thickness. This value controls the number of faces considered. If the number of planes exceeds 1000, a larger value for the minimum slice thickness is automatically used.
morphology
From this list of relative growth rates, Cerius2 deduces the crystal morphology and displays it in the model window. By default, the morphology is shown with the molecular model, and the planes making up the displayed morphology are transparent. You can change this and other visualization variables using the options on the Morphology Display control panel. You should consider the shape you obtain to be a good first approximation to the morphology of any crystalline system and an ideal starting point for attachment energy calculations in molecular crystals.
To calculate morphology with the BFDH method
1. Place the crystal structure in the currently active model space.
You can either load the crystal structure from a file or build it
using the Crystal Builder.
3. Click the Calculate button in the Bravais Friedel Donnnay
Harker subpanel.
Calculating morphology with the Attachment or Surface Energy methods
As with the BFDH method, you do morphology calculations using the options on the Calculate Morphology control panel (see the online help). Once you set up the energy calculation, you can predict the morphology of the current structure by simply clicking either the Calculate button in the Growth Morphology subpanel (for attachment energy calculations) or the Equilibrium Morphology subpanel (for surface energy calculations).
It is important that you build the crystal with correct bonding, allowing the calculation to deduce the nature of each isolated molecule present in the crystal.
The attachment or surface energy is calculated for all crystal faces listed in the Edit/Add Faces control panel. This list could come from the BFDH calculation, which is frequently used as a screening method to identify possible growth planes. The Do BFDH First option ensures that the BFDH method automatically generates the list before running the calculation.
Calculating attachment energies
The morphology prediction using the Attachment Energy method is controlled within the Growth Morphology subpanel. For a given slice the program evaluates the attachment energy using Eq. 11. Two different algorithms for the attachment energy calculation are available.
2. The second mode of calculation (Bond-Energy-List) computes
the lattice energy and the energy of the growth slice by applying
a force field to directly sum the interactions of pairs of molecules
having centers that lie within the specified interaction
radius. Since these interactions are finite in range, they are precalculated
and tabulated for each pair of molecules, making the
calculation of slice energies faster. However, this method is less
accurate than a calculation using Ewald summation (see the
first Note below).
The positioning of growth slices affects the calculated energies. If you select Full OFF Support for the evaluation of the attachment energies (and thus use the default method above for attachment energy calculations) or if you calculate surface energies, all theoretically possible slice positions leading to non-identical growth slices are found automatically. If you select the Bond-Energy-List mode for the attachment energy calculation, an additional Preferences ... button appears which allows you to set variables affecting the Slice positioning.
The morphology prediction using the Surface Energy method is controlled within the Equilibrium Morphology subpanel. The surface energy of a crystal is calculated by simulating the semi-infinite crystal by a slab of finite thickness. The substrate thickness is a user-defined parameter. By increasing the substrate thickness, the accuracy of surface energies is systematically improved, at the expense of larger computation times. The calculation is based on the algorithm used for the attachment energy calculation and is always performed by creating a temporary two-dimensional model for each surface. (please see The Equilibrium Morphology for more information on surface energy calculation and see Calculating attachment energies for more information about the attachment energy algorithm)
Each growth face in the list is assigned a center-to-face distance that is proportional to the attachment or surface energy calculated. The Edit/Add Faces list is updated with the new center-to-face distances, and the calculated morphology is displayed in the model window.
To calculate morphology with the Attachment or Surface Energy methods
1. Place the crystal structure in the currently active model space.
You can either load the crystal structure from a file or build it
using the Crystal Builder.
4. If you want to automatically generate the list of growth planes
with the BFDH method, check the Do BFDH First box (default
setting). Otherwise, either enter your own list or do the BFDH
calculation separately, then edit the list (see Calculating Morphology
with the BFDH method and Editing, adding and removing
crystal faces).
6. If you wish to calculate the growth morphology using the
Bond-Energy-List mode, specify how the growth slices are considered
when calculating their energies (see Slice positioning).
Please see the online help for more detailed information on the Calculate Morphology control panel.
Setting up the energy calculations
The procedure for setting up the energy calculations depends on the method used for morphology prediction. Except for the case where the Bond-Energy-List option for the calculation of attachment energies has been selected, variables affecting the attachment energy calculations are specified using the OPEN FORCE FIELD card on the OFF SETUP card deck (see the Cerius2 Simulation Tools manual). There, you can specify the non bond interaction terms included in the energy calculation, the method (EWALD, DIRECT, or SPLINE) to evaluate Coulomb interactions, the cutoff radius for direct non bond interactions, and a large number of more advanced parameters affecting your energy calculations.
On the other hand, if you use the Bond-Energy-List option to calculate attachment energies, you specify variables affecting the energy calculation using the options on the Bond-Energy-List Preferences control panel. This panel is accessible via the additional Preferences... button which appears as soon as you specify Bond-Energy-List mode on the Growth Morphology subpanel. These options specify the nonbond interaction terms included in the energy calculation and the interaction radius. A button for checking the lattice energy is also on this control panel.
To calculate attachment or surface energies, you must load a force field and assign atom types to the model. The choice of force field is an essential variable in the morphology prediction. Most force fields are parametrized for particular types of systems, thus two force fields can give widely diverging results for the same model. On the other hand, you can use suitably chosen force fields to validate each other. Roberts and Docherty (1988) have shown the effects of using different force fields for predicting the morphologies of a number of systems.
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By default, Cerius2 uses the Universal force field. Note that you still need to assign QEq charges (using the CHARGES card on the OFF SETUP card deck) before using this force field.
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You can use the OFF to load a different force field and to assign atom types and charges; you can use the Force Field Editor to change parameters (see the chapters titled "Open Force Field" and "Force Field Editor" in the Cerius2 Simulation Tools manual).
The attachment and surface energy calculations use only the non bonded terms from the chosen force field. They use all these terms by default (van der Waals, Coulomb, and hydrogen bonding), but you can exclude some of these if you wish using either the Energy Terms/Selection option on the OPEN FORCE FIELD card (all methods except Bond-Energy-List mode) or the Bond-Energy-List Preferences control panel (Bond-Energy-List mode).
For neutral molecules, you do not always require Ewald summations to perform the Coulomb summations--direct treatment of the interactions may be sufficiently accurate and significantly reduce the time required to perform the calculation. However, you then need to ensure that the cutoff distance is large enough for the lattice energy to be converged. If you want to calculate the growth morphology and you do not wish to use Ewald summation techniques, we recommend that you switch to Bond-Energy-List mode. In this mode, the cutoff distance is set using the Bond-Energy-List Preferences control panel which is accessible via the Preferences... button. The Check Lattice Energy button allows you to plot the lattice energy and determine the convergence distance. This option is also useful in verifying that the value of the lattice energy is as expected for a stable crystal structure (it should be negative).
A switch is provided that allows you to save essential information to an output file named interactions.dat. For the format of this file, see "Morphology File Formats" in Appendix B of the Cerius2 Computational Instruments online documentation.
To set up the energy calculations
1. Go to the OPEN FORCE FIELD card, load a force field and
select options for the various non-bonded terms on the Energy
Terms ... menu. In particular, decide whether to use Ewald
summations for the Coulomb term, and set the cutoff distance
for van der Waals and Coulomb interactions to a converged
value.
5. Click Preferences... next to the Calculate button to bring up the Attachment Energy Preferences panel. Check the Auto Force Field box to generate automatic atom typing for the structure. To do this manually, uncheck the box and use the options in the OFF; to edit force field parameters, use the Force Field Editor. To save the calculated energy data, check the Save Interactions File box.
To check the lattice energy
1. Go to the MORPHOLOGY card and select the Calculate item
to display the Calculate Morphology control panel (see the
online help for more control panel info).
2. Select the Bond-Energy-List mode on the Growth Morphology
subpanel.
3. Click the Preferences... button to bring up the Bond-Energy-
List Preferences control panel.
4. Enter the number of data points to use in the plot.
5. Click the Check Lattice Energy button.
Slice positioning
The way that growth slices are positioned affects the calculated energies. If Full OFF Support has been selected for the evaluation of the attachment energies or if surface energies are calculated, all theoretically possible slice positions leading to non-identical growth slices are found automatically, and you do not need to set any parameters (you can skip this section). For the Bond-Energy-List mode for attachment energy calculations, you can set variables affecting the slice positioning using the options on the Slice Positioning control panel (see the online help).
In calculating slice energies with the Attachment Energy method, it is important to find the most stable slice (that is, the one with the most negative energy), since this is the slice most likely to be involved in the growth process.
Eq. 9 Step size = d(hkl) / (Number of slice steps + 1)
Center slice on all
molecules
When working with mixed crystals, you can obtain more accurate results by repeating the slice energy calculation with each of the molecules positioned at the center of the slice. The Center Slice on All Molecules option is provided for this purpose.
Positioning a slice so that the molecule is exactly at its center is likely to place other molecules on the edge of the slice. This can lead to difficulties in determining which interactions contribute to the stability of the slice. You should therefore position the slice so that the molecule is slightly offset from the center. You can use the Slice Offset option to specify the offset value (the default value is 0.01 Angstroms).
To specify the slice positioning variables
1. Go to the MORPHOLOGY card and select the Calculate item
to display the Calculate Morphology control panel.
2. Select the Bond-Energy-List mode on the Growth Morphology
subpanel.
3. Click the Preferences... button to bring up the Bond-Energy-
List Preferences control panel.
5. Specify the Slice Offset (in Angstroms).
6. Enter a value for the number of slice steps to be used.
Please see the online help for more detailed information on the control panels.
Editing, adding and removing crystal faces
Face list
You can choose whether or not invisible faces will appear in the list by checking or unchecking the Include Invisible Faces in List button.
Selecting faces
To edit or remove faces in the list, you need to select these faces. You can select a face in the list by clicking the corresponding row with the left mouse button. Multiple faces can be selected by holding down the < Ctrl > key while clicking additional rows. Holding down the < Shift > key during a second selection selects the whole range of faces between the first and second selection. Double-clicking the list selects all faces in the list.
You can edit faces in the list using the three input fields immediately below the face list. Selecting a single face in the list will fill these input fields with the Miller indices, center-to-face distance and color of the selected face. You can then modify the center-to-face distance (second) and color (third) fields for this selected face. You can also select a face by typing its Miller indices into the first of the input fields. Changes are reflected in the displayed morphology. If multiple faces are selected you can set the center-to-face distance or the color of all selected faces to a common value. This provides an efficient way, for example, to change the display color of the habit in the model window.
If you type into the first input field below the list of faces the Miller indices of a face which is not symmetry related to any of the faces currently in the list, this form is added to the list of faces. Once the face is added, you need to set its center-to-face distance and color attributes using the second and third input fields. If you do not enter a center-to-face distance, a default of 10 Angstroms is assigned. You can also compute the true value by running a Growth or Equilibrium morphology calculation. If visible, the added face is shown in the displayed morphology.
You can remove one or more faces permanently from the list by first selecting them and then hitting the Remove Selected Faces button. You can also permanently remove all faces from the list that are not present in the current morphology (that is, all those that are listed as not visible), using the Remove Invisible Faces button. This is often useful in decreasing the number of faces considered in an Attachment or Surface Energy calculation. However, you should make sure that significant growth faces are not removed accidentally. Note that in order to regenerate faces which have been removed, you either have to reenter their Miller indices by hand or redo your morphology calculation.
You can use the List All Faces to Text Window option to print a more detailed listing of all current faces in the text window. The listing is grouped into faces that are in the same form (that is, symmetry related), and is ordered according to center-to-face distance. In addition to the Miller indices, the center-to-face distance and the color of each face, the d-spacing, surface area and number of corners are also given.
To edit, add, and remove crystal faces
1. Go to the MORPHOLOGY card and select the Edit/Add Faces
item to display the Edit/ Add Faces control panel.
4. Remove faces by selecting them and executing Remove
Selected Faces.
5. Remove all invisible faces using the Remove Invisible Faces
option.
Please see the online help for more detailed information about the control panels.
Calculating morphology using the Hartman-Perdok Method
Morphology calculations applying the Hartman-Perdok method use the options on the Crystal Graph and Hartman-Perdok panels, both accessible via the Calculate Morphology control panel. The program implements the Hartman-Perdok method for building stable growth planes from an analysis of strong intermolecular interactions ("bonds") in the system. The Attachment Energy method is then used to simulate the relative growth rate of these growth planes.
The Crystal Graph menu is used to generate a set (graph) of strong crystal bonds for the current structure. The Crystal Graph panel is accessible via the Crystal Graph ... button on the Calculate Morphology menu. After you specify a spatial range, the program generates a list of bonds between molecules in the crystal and calculates bond energies. This bond list may be edited, and an energy window can be specified determining those bonds within the bond list which are to be included in the subsequent identification of strong chains of bonds (periodic bond chains) and networks of bonds (connected nets) (see Generating and editing the Crystal Graph).
After a crystal graph has been defined, you may generate all possible connected nets associated with that crystal graph by simply clicking Calculate Morphology on the Hartman Perdok panel (accessible by clicking the Hartman Perdok ... button on the Calculate Morphology menu). The program selects only connected nets which have the correct stoichiometry, i.e., which include the same ratio of different types of molecules as the full crystal. In addition, for a set of forms {nh nk nl}, n=1, 2, ... only those connected nets are shown which correspond to the smallest n for which the given form is not excluded due to symmetry constraints. The connected nets generated are listed in the Hartman Perdok panel for further analysis (see Generating and editing Periodic Bond Chains and Connected Nets).
After the connected nets resulting from a given crystal graph have been identified, the attachment energy of each net is calculated, and the energy of the most stable net for each generated form is used to assign a center-to-face distance for the corresponding face. You may select from two different methods to calculate the attachment energy of a connected net. In the default mode (E(Att)) the attachment energy is calculated using total energies for the evaluation of lattice and slice energies. Alternatively, the E(Bond) mode evaluates sums over bond energies as defined in the crystal graph. The list of planes and the predicted center-to-plane distances used to calculate morphology is accessible via the Edit/Add Faces menu on the MORPHOLOGY card (see Editing, adding and removing crystal faces).
Creating a model of a connected net
The program allows you to generate and visualize any net in the list of connected nets within a new model space, which can then be used for further analysis. In addition to the molecules forming the net, the model contains dummy atoms at the center of geometry of each molecule, connected by bonds, illustrating the two one-dimensional periodic bond chains defining the connected net.
Both the crystal graph as well as the structure of the connected nets are saved to the data model. You may permanently save this information by saving the Model to disk.
To calculate morphology with the Hartman-Perdok Method
1. Place the crystal structure in the current active model space.
The crystal structure can either be loaded from a file or built
using the Crystal Builder.
Generating and editing the Crystal Graph
The first step of a Hartman-Perdok morphology prediction is the identification of strong interactions ("bonds") between molecules forming the crystal. The program generates a crystal graph describing the interactions between the molecules in the crystal structure in terms of those bonds. Generating crystal bonds is done using the Crystal Graph panel accessible via the Calculate Morphology control panel.
Before generating the crystal graph, you need to specify the spatial range of the bonds which you wish to be included in the crystal graph. You define the spatial range by specifying a maximum distance along each lattice vector using the entry fields below the Generate Bonds button.
If you click the Generate Bonds button, the program automatically generates all bonds that fall into the specified spatial range, calculates their energies and displays a list of representatives for each family of symmetry related bonds in the Crystal Graph panel. The program also prints a list of all bonds into the text window.
The visualization of the crystal graph is controlled by the Display Crystal Graph button. If checked bonds included in the crystal graph are displayed in the model window as green lines connecting the centers of geometry of the molecules forming that bond. Selecting a bond in the bond list (left-clicking with the mouse on the corresponding line in the list box) highlights the corresponding bond (and all its symmetry related copies) in the model window.
To Generate and edit the Crystal Graph
1. Place the crystal structure in the current active model space.
The crystal structure can either be loaded from a file or built
using the Crystal Builder.
5. Click the Generate Bonds button and the program automatically
generates a list of bonds.
8. Modify the bond energy of a selected bond using the Energy
entry field.
Generating and editing Periodic Bond Chains and Connected Nets
After a set of strong crystal bonds has been identified, you can generate all two-dimensional connected nets of strong bonds. These nets define the stable growth planes of a crystal. Generating and editing connected nets is done using the Hartman-Perdok panel accessible via the Calculate Morphology control panel.
If you click the Calculate Morphology button, the program automatically constructs all one-dimensional periodic bond chains (PBCs), then examines all combinations of pairs of PBCs to find all unique two-dimensional connected nets that are flat and stoichiometric. For a set of forms {nh nk nl}, n=1, 2, ... only those connected nets are selected which correspond to the smallest n for which the given form is not excluded due to symmetry constraints. In addition, the program calculates the attachment energy of the connected nets using the force field settings specified in the OFF. In particular, Ewald summations can be used to perform these attachment energy calculations.
From the attachment energies of the energetically most stable connected nets, the program determines the morphology. Two different methods may be used to calculate the attachment energy of a connected net. In the default mode (E(Att)) the attachment energy is calculated using total energies for the evaluation of lattice and slice energies. This method is equivalent to the one applied if Full OFF Support is selected for the calculation of the growth morphology (see Calculating morphology with the Attachment or Surface Energy methods). Alternatively, the E(Bond) mode evaluates sums over bond energies as defined in the crystal graph. From the generated list of connected nets the most stable one for each form is selected and used to predict the crystal morphology. These connected nets considered in the morphology calculation appear in the face list on the Edit/Add Faces control panel (see Editing, adding and removing crystal faces).
All generated connected nets are listed in the Hartman Perdok panel. Since the Hartman-Perdok analysis may generate several connected nets for a form (hkl) all connected nets of a given form are grouped together and ordered according to their stability. In the list of connected nets the forms are ordered according to the stability of their most stable connected net. Both the attachment energy based on the total energy of the connected net E(Att) and the attachment energy calculated using the bond energies of the crystal graph E(Bond) is given for each connected net.
The Hartman-Perdok analysis may generate a large number of different connected nets for a given form (hkl) some of which are considerably more unstable than the most stable connected net of this form (appearing as Type 1). Those relatively unstable nets may usually be ignored for the analysis of crystal morphology. In addition, since the connected nets are saved to the data model a large number of connected nets may affect the performance of Cerius2. You may delete all nets within each form which are less stable than the most stable connected net of this form by a user-specified energy value by clicking the Delete Types less stable by button. The energy value is specified using the entry field beside the Delete Types less stable by control. Note, that the energy criterion is applied to each form separately and that no forms are deleted from the list.
Selecting a connected net from the list on the Hartman Perdok panel (left-clicking with the mouse on the corresponding line in the list box) gives you access to two modes of analysis:
1. You can create a model of the connected net.
Create a model of a connected net
If you have selected a connected net from the list on the Hartman-Perdok panel (left-clicking with the mouse on the corresponding line in the list box), you may create an atomic model of this connected net using the Create Selected Face option. In addition to the molecular arrangement, the model illustrates the PBCs forming the connected nets by dummy atoms at the centers of geometry of the molecules forming the PBC and bonds between them.
The Display Style pull-down menu allows you to change the display style for a model of a connected net. You can show the atoms only (Atoms), the PBCs only (PBCs) or both the atoms and the PBCs together (Atoms + PBCs).
To Generate and edit Periodic Bond Chains and Connected Nets
1. Generate a crystal graph (see Generating and editing the Crystal
Graph).
6. Generate a model of the selected connected net by clicking the
Create Selected Face button.
Displaying the morphology
The calculated morphology is displayed in the model window. You can specify the way the morphology is displayed using the options on the Morphology Display control panel.
With visualization options you can choose to display the external morphology either with or without the internal molecular structure, or display the molecular structure alone. By default, both are displayed.
The morphology is displayed with the center-to-face distance for each plane represented in Angstrom units multiplied by a scale factor. The units are used to enable easy comparison of molecular structure and morphology and should not be regarded as representing the true size of the crystal. The larger the scale factor, the larger the size of the morphology relative to the molecular model. A scale factor of 1.000 is used by default.
You can change the transparency of the crystal faces, allowing the molecular structure to be viewed inside the morphology model. An entry box or slider bar may be used to specify the value (0 is opaque, 1 is transparent). The default setting is 0.400.
The face label options allow you to label the faces of the model showing their Miller indices (Indices) or their surface area as a percentage of the total area (Area). Use the None option if you do not want labels.
Usually, any change to the list of faces on the Edit/Add Faces panel results in an automatic update of the morphology display in the model window. However, in some cases, such as a manual change of lattice parameters, you may wish to update the morphology display manually. You can do this using the Redisplay Morphology button.
To specify the display controls
1. Go to the MORPHOLOGY card and select the Display item to
bring up the Morphology Display control panel.
3. Enter the scale factor to apply.
Please see the online help for more information on the control panels.
Analyzing the morphology
Several properties of the crystal morphology can be calculated using the options on the Morphology Analysis control panel (see the online help). These include inter planar angles, aspect ratio and the percentage area accounted for by certain forms.
Buttons are provided that allow you to calculate the inter planar angle between two faces or the surface area of a face. Faces are identified by entering their Miller indices. Results are reported in the text window.
You can use the List Areas by Form option to list all forms present in the current morphology and to display the percentage of the total surface area accounted for by certain forms.
You can also calculate the aspect ratio. The ratio is defined as D/d, where D is the maximum center-to-corner distance and d is the minimum center-to-face distance. Knowledge of shape and aspect ratio is essential to understanding packing, flow problems, clogging of filters, and related questions.
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If you perform a Hartman-Perdok analysis the Cleave Selected Face control is not active and is replaced by the Create Selected Face command on the Hartman-Perdok panel.
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To analyze the crystal morphology
1. Go to the MORPHOLOGY card and select the Analysis item to
bring up the Morphology Analysis control panel.
4. To list the forms and their percentage areas, click the List Areas
by Form button.
5. To calculate the aspect ratio of the crystal, click the Aspect Ratio
button.
Please see the online help for more detailed information about the control panels.
Storing morphologies
Saving
You can save the current morphology in two different ways. The usual way is to use the File/Save Model option of the main Cerius2 menu. If you choose the .msi format, the morphology is saved together with the underlying molecular structure, the crystal graph and any connected nets which may have been generated.
If a morphology has been saved as part of an .msi file, it can be loaded using the File/Load Model option of the main Cerius2 menu. You can also read morphologies saved in CIF-formatted files using options on the Load Morphology control panel (see the online help). You can add a morphology to the current model space or put it into a new one. However, you can add a morphology to the current structure only if both have the same cell and space group symmetry.
To save the current morphology
1. Go to the File/Save Model item on the main menu to bring up
the Save Model control panel if you wish to save a morphology
as part of an .msi file, or go to the MORPHOLOGY card and
select the File/Save item to bring up the Save Morphology control
panel.
To load a morphology
1. Go to the File/Load Model item on the main menu to bring up
the Load Model control panel if you wish to load a morphology
saved as part of an .msi file, or go to the MORPHOLOGY card
and select the File/Load item to bring up the Load Morphology
control panel.
2. Select the model store to be used (click either the Current or
New button).
3. Use the browser to specify the file name and load the file.
This section provides detailed theoretical information for the four methods used by Morphology: the Bravais Friedel Donnay Harker method, the Attachment Energy method, the Surface Energy method, and the Hartman-Perdok method.
Theory
Bravais Friedel Donnay Harker method
The Bravais Friedel Donnay Harker (BFDH) method is a geometrical calculation that uses the crystal lattice and symmetry to generate a list of possible growth faces and their relative growth rates. From this, crystal morphology can be deduced.
Generally, observed crystal morphology is found to be dominated by lower-order faces. Bravais (1913) and Friedel (1907) observed that the center-to-face distance for a given plane tended to be related to the inverse-plane spacing,
Eq. 10 D ~ 1/d
where:
Donnay and Harker (1937) refined this approach by developing rules that related the crystal symmetry to the possible growth planes. These rules account for the effect of translational symmetry operators, meaning that higher-order planes can grow in preference to lower-order ones. For example, in a body-centered cell, a molecule at the origin repeats in the center of the cell. This implies that growth in the [1 0 0] direction could occur by adding the (2 0 0) slice, rather than the thicker (1 0 0) slice.
The Attachment Energy method
The Attachment Energy method can predict the shape of the crystal more accurately because it takes the energetics of the system into account.
The attachment energy, Eatt, is defined as the energy release on the attachment of a growth slice to a growing crystal surface (Docherty et al. 1991). Eatt is computed as (Berkovitch-Yellin 1985):
Eq. 11 Eatt = Elatt - Eslice
where:
The attachment energy is calculated for a series of suitable slices (hkl) that are chosen either by performing a Donnay-Harker prediction, or by entering your own data. From the energy calculation and, hence, the growth rate, a center-to-face distance is assigned to each face. This information is used to deduce the morphology using a Wulff plot (Wulff 1901).
The attachment energy model assumes that the surface is a perfect termination of the bulk and that no surface relaxation takes place. This has been shown to be significant in the case of inorganic systems such as a-Al2O3 and a-Fe2O3. The attachment energy model works well in the case of many organic molecular systems.
The Equilibrium Morphology
The equilibrium morphology of a crystal is determined by the minimum of the surface free energy for a given volume and temperature (Gibbs 1928). If the surface free energies, or, at zero temperature, the surface energies are known for all relevant crystal faces the morphology of a crystal in equilibrium with its surrounding can be visualized using a Wulff plot (Wulff 1901).
In the program, the surface energy is calculated from the energy of a slab of finite thickness:
Eq. 12
where:
It is readily seen that Eq. 12and Eq. 11 are closely related. In the limit of a slab thickness of one growth layer, a very rough approximation of the surface energy may be obtained using the attachment energy:
Eq. 13 Esurf(1) = 1/2 Eatt / Ahkl = 1/2 dhkl Eatt/V
where
The equilibrium morphology is calculated at zero temperature. It is assumed that the surface is a perfect termination of the bulk and that no surface relaxation takes place. In the slab method, the calculated surface energy is an average between the surfaces with Miller indices {h k l} and {-h -k -l}. The latter restriction is important for crystal structures not having a center of inversion.
The Hartman-Perdok method
The Hartman-Perdok method (Hartman-Perdok 1955, Bennema 1996, Grimbergen et al. 1998) provides a systematic way to generate the stable growth planes of a crystal. It removes one of the assumptions of the common Attachment Energy Method, namely that the growth planes are always ideal, flat crystal planes. Once the stable growth planes of a crystal have been identified, the attachment energy is used to simulate the growth morphology of the crystal (see The Attachment Energy method). In addition, the Hartman-Perdok method gives you critical insight into why surfaces are stable and thus provides you with a powerful tool (for example) for designing growth additives.
The implementation of the Hartman-Perdok theory is based on the program FACELIFT (Grimbergen and Meekes 1997) developed at the University of Nijmegen.
The Hartman-Perdok theory starts from the concept that crystals are stable because of the ability of the growth units to form strong attractive bonds between each other. The network of these bonds in the crystal defines the crystal graph.
According to the Hartman-Perdok theory, the stable growth planes are those planes which consist of two-dimensional connected nets of strong bonds in the surface plane. Generally, those planes will have a non-zero step energy hindering nucleation and a low attachment energy (Grimbergen et al. 1998).
The attachment energy is calculated for all connected nets (see The Attachment Energy method). From the attachment energy calculation and, hence, the growth rate, a center-to-face distance is assigned to each face. This information is used to deduce the morphology using a Wulff plot (Wulff 1901). If more than one connected net exists for given Miller indices {hkl}, the most stable (the lowest absolute value of the attachment energy) is defined as the growth plane.
The program assumes that the growth units are single molecules (or single atoms if not part of a molecule).
Bravais, A., Etudes Crystallographiques, Paris (1913).
References