| Property Prediction |
Sections in this chapter
References
In contrast to previous approaches aiming at classifying morphologies by means of equilibrium theories, the present approach recognizes the fact that by their very nature these patterns are irregular, and hence can only be characterized via the dynamic properties of the systems. From an industrial perspective this approach is much more realistic, since typical processing times are orders of magnitude shorter than the thermodynamic relaxation time, and thus such non-perfect states contribute substantially to the behavior of the final material.
Introduction
This reveals a menu consisting of Run, Build, Analysis and Job Control items.
Using MesoDyn
Start with a new session of Cerius2.
From the Cerius2·Visualizer card deck menu go to the MESOSCALE card deck and choose the MESODYN card.
| First, make sure that you have a molecular ensemble defined for the MesoDyn simulation. From the Build submenu choose Beads. |
| Check that three beads, named A, B and W with diffusion coefficients are defined here. |
| Next, from the Build submenu, choose Molecules. |
| Check that one "triblock" molecule consisting of the two beads A and B is set (e.g. A 3 B 9 A 3). |
| Press the right mouse button on the Architecture field to look at the help available for the chain architecture input. Note the way in which branched architectures can be defined. |
Next you conclude the definition of the molecular ensemble.
| Close all open panels. |
| Go to the MESODYN card and select the Run item. |
There you find the main parameters for a particular run.
This is all you need to do for now. The subpanels accessible from the Run panel provide further more detailed control which does not concern us for now.
| Go to the MESODYN card and select the Job Control item. |
| Finally, go to the MesoDyn Run panel and press the RUN button. |
You should get the following message in the textport:
MESODYN job FirstTest has been started.On the Job Control panel the job is listed in the Job Status box.
You can monitor the progress of the job by clicking the Monitor status file button. This opens a window displaying the bottom of the status file.
| When the number of steps you had selected has been reached, press the UPDATE Job status button. |
The Status indication in the Job Control Box changes to complete, if the job has finished.
| Click the Collect & Transfer button and then click the COLLECT button on the subpanel. |
An appropriate message tells you what happened.
| Close all panels. |
You are now ready to analyses the results of the run.
| Go to the MESODYN card and select the Analysis item, then select the System item from the submenu. |
This opens a file browser from which you select the MesoDyn_stat file from your run (that is, FirstTest.MesoDyn_stat). This shows the name, title, and date. This is from now on the selected system for any of the other analysis functionalities.
| Go to the MesoDyn card and Select the Analysis/Morphology/Profiles item. |
This displays the name of the selected system, the time step and bead name.
| Click the Create New Profile button. |
| You may not see much evidence of phase separation for this short run, but you can enhance the contrast by clicking the More Editing Options... button and clicking Reset Map to Full Range. |
The density boxes show the lowest and highest concentration of A in the slice.
| Move the slice by using the sliders. |
You can do further analysis by looking at the isodensity surfaces.
| Select the Analysis/ Morphology/Isodensities item. Set the isodensity of A to 0.4. Make sure that you are in an empty model space, and then click the Create Isodensity Display button. |
The isodensity value must fall between the minimum and maximum values you found in the Profiles analysis of that bead.
General methodology
Overview
A graphical user interface module to the MesoDyn simulation has been provided within the Cerius2 molecular modelling environment. This interface handles the input generation, (i.e. the definition of the molecular ensemble, setting of the simulation run parameters, control of the numerical algorithm parameters), the job control (choice of machine for run, parallel nodes and output paths, job monitoring), the output file handling from the parallel processors, and the analysis of the simulation output (3D density fields and scalar thermodynamic functions).
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The first action in many cases will be to build a molecular ensemble. This is done from the Build sub-menu. The Beads, Molecules and Interactions are defined here, as are Geometrical Constraints, i.e. regions representing an organic pigment or inorganic filler particle (see Figure 6 and Figure 7).
The MesoDyn Run control panel (Figure 8) allows you to edit parameters governing the simulation run, such as run time and output frequency.
The Job Control control panel (Figure 9 and Figure 10) allows you to set the run host, associated parameters, the job to be monitored, and the output to be handled and prepared for further analysis.
In the Analysis section, a particular system for which output is available can be selected, its output files inspected (Figure 11), output functions plotted (Figure 12), and the three-dimensional density profiles of the beads analyzed graphically both by slicing (Figure 13) and isodensity surfaces (Figure 14).
Building a molecular ensemble
MesoDyn works on an ensemble of Gaussian chains, which are defined in terms of the constituent beads (covalently bonded groups of atoms, e.g. monomers), their respective diffusion coefficients, the way the beads are joined up in molecules (the molecular architecture), the bond length (Gaussian chain parameter), the concentrations of the molecules and the Flory-Huggins interaction energies between the beads.
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Beads, as stated before, represent the individual chemical species in the system, and for a homopolymer or block copolymer molecule are identical with a statistical (or Kuhn) segment. For a solvent, a bead represents the collective degrees of freedom of a group of solvent molecules. At the moment, all beads are restricted to having the same volume and diffusion coefficient.
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Geometrical constraints are a new feature in this version of MesoDyn. They provide one of the first opportunities to study adsorption and adhesion phenomena in realistic complex liquids. As such, the number of validation studies is still small, however, early results are encouraging.
In addition to the above two geometries, it is possible to use an arbitrary constraint. This is accomplished by editing the mask file (with the file extension MesoDyn_ascii) which describes the constraint (as a grid of ones and zeros). The mask file is created in the working directory when a job is submitted which uses a constraint; this is a simple ASCII file which may be edited with a text editor. The ones represent regions accessible to the beads and the zeros regions denied to the beads. The mask file may be saved and used in a later simulation. The Constraints Panel also allows for an arbitrary mask file to be visualized in the Cerius2 Model Window.
Specifying the Run parameters for a MesoDyn simulation
Having defined a molecular ensemble, a number of parameters need to be set that govern the particular simulation to be done. These parameters are accessed from the MesoDyn Run panel (Figure 8). In particular, each run is identified by a certain name and given as file prefix to all input and output files generated. The model system is further defined by specifying its temperature and the size of the simulation box (with periodic boundary conditions), in terms of the side length of its cubic cells and the number of cells in each cartesian dimension. Typically, the cell size is of the same order as the bead size (i.e. the bond length). Furthermore, the step size of the numerical integration and the length of the run, either in terms of its total time or in terms of the number of steps, must be set.
| The implementation of shear for parallel machines requires the system to be decomposed in the y-direction only. Please see the section below on the Job Control Options Panel for a further explanation. |
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Further "expert" parameters are accessible via the Noise... and Solver... buttons. The output level of the run can be set from the Output... control panel. In particular, the frequency in terms of time steps, with which the status information, density files, potentials files and restart files are written to disk is set here. The amount of disk space required for frequent output of three-dimensional fields, such as contained in density, potentials and restart files, may be very large.
Setting the host machine and parallel nodes, job monitoring and output handling
The Job Control panel (Figure 9) (accessed by clicking Job Control on the MESODYN card) shows the name of the host running MesoDyn and the mode of execution (INTERACTIVE or BACKGROUND). Access information to the remote host (user name and password) may also be set there, if required.
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Also on the Job Control panel, you can set the Base directory on the remote host. The Job Status box shows previous and current jobs and related information such as process ID and status -- information which is required by Cerius2·MesoDyn to collect the results of a MesoDyn run.
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On parallel machines with distributed memory, such as the IBM sp2, correct use of the Options control panel syntax is much more important. Here the default run directory is typically the user's login directory, which is mounted on the control workstation of the sp2, and usually has little disk space. Instead, the output paths should correspond to the local disks on each processor, which are typically very large. The Options control panel displayed in Figure 10 above is correctly set to run on two nodes of the sp2 at MSI, where the directory /msidata has been mounted on each of the local disks.
For sheared systems, as stated before, the system must be decomposed in the y-direction only. Thus, for running the above job on two nodes of the sp2 with shear, the number of processors would be specified as 1 2 1. The node and path definitions would remain unchanged. For four nodes, the processors would be 1 4 1, and so on.
Analyzing the results
Having completed a simulation run, collected and transferred the output files, you are ready to analyze the results. This functionality is accessed via the Analysis submenu on the MESODYN card.
The first step is always to open the System Analysis panel (Figure 12).
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This provides a file browser with a filter for the MesoDyn status output files. Selecting the status file from the desired run sets the name of the system to be analyzed further. The name and title of the system selected will be shown in the box below, and you may examine the log and status files by pressing the relevant pushbutton.
As a next step you can see the thermodynamics of the system as it evolved from its initial to its final state. Open the Thermodynamics control panel (Figure 12) and plot the free energy, potential energy and entropy over the whole or a selected time interval.
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Finally, and often most importantly, the morphology of the molecular ensemble can be investigated and analyzed in two ways.
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Furthermore, although the current MesoDyn interface does not provide an animation facility directly, a sequence of models can be saved in a Log file via the Utilities/Record commands from the menu bar, and then replayed via the Utilities/Playback Script facility.
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Open the Isodensities control panel from the Morphology pullright of the Analysis submenu. For each bead type you can create an isodensity surface at a number of time steps and for up to four different values of the concentration at once.
Finally, the main MESODYN card provides a Reset button, which reinitializes all the panels and sets all values back to their default.
The reason for this is that with the above theorem an external potential can always be found such that the distribution of the ideal system equals that of the real system at the same density. This theory can be used to great effect in the description of polymer fluids.
We take the polymer chain as the fundamental building blocks of the model. In this description, the intrachain correlations can in principle be treated by any suitable model. In practice, a Gaussian chain model is used
of bead positions in space, resulting in three-dimensional concentration fields 
I(r). The evolution of these fields is the result of the dynamics outlined in the following section, in combination with the thermodynamic driving force described in the section thereafter. For simplicity of presentation, in the following we are going to limit the number of bead types to two, named A and B, but the theory will equally apply to any number of bead types. The following description of the theory is based on a paper by Fraaije et al (1996) to which the interested reader is referred also for further references.
It should be noted that some of the approximations imposed in this paper (e.g. the assumption of perfect incompressibility) have now been lifted. The interested reader should consult the references below.
Dynamics
The derivation of the diffusive dynamics of the molecular ensemble is based on the assumption that for each type of bead I the local flux is proportional to the local bead concentration and the local thermodynamic driving force:
Eq. 7
where J~I is a stochastic flux (related to thermal noise). Together with the continuity equation
Eq. 8
this leads to simple diagonal functional Langevin equations (stochastic diffusion equations) in the density fields: 
Eq. 9
with a Gaussian distribution of the noise.
Eq. 10
where 
B is the average bead volume. This condition then leads to "exchange" Langevin equations:
Eq. 11
Eq. 12
Here M is a bead mobility parameter. The kinetic coefficient M
AB models a local exchange mechanism. Hence the model is strictly valid only for Rouse dynamics. (Effects such as reptation lead to kinetic coefficients which extend over a range of roughly the coil size. They lead to computationally expensive non-local operators which in addition, are very complex in the non-linear regime.)
Eq. 13
Eq. 14
and ensures that the time-integration of the Langevin equations generates an ensemble of density fields with Boltzmann distributions.
Thermodynamics
The above Langevin equations contain the bead chemical potential as the thermodynamic driving force of the diffusive dynamics. These chemical potentials can be derived from the thermodynamics of the molecular ensemble. . Since the positions of the beads are correlated to each other this amounts to a multi-dimensional many-body problem. To overcome this, any interchain correlations are neglected, and the system is approximated by a set of independent Gaussian chains embedded in a mean-field.
Eq. 15
The first term is the average value of the Hamiltonian for the ideal system, comprising the internal Gaussian chain interactions:
Eq. 16
where H
G is the Gaussian chain Hamiltonian of chain:
Eq. 17
here a is the Gaussian bond length parameter and the index s goes over all N segments of the chain. The second term in the free energy functional stems from the Gibbs entropy of the distribution. The third term is the non-ideal contribution related to the interchain interactions. 
. In the spirit of the particular application of density functional theory taken here, namely treating the chains as the ideal system, the correlations between the chains are neglected, and the density functional method applies to the correlations within the Gaussian chain only. is such that the free energy functional F[] is minimized. Hence is independent of the history of the system, and is fully characterized by the constraints that it represents the density distribution and minimizes the free energy functional. This constraint on the density fields is realized by means of an external potential UI .
Eq. 18
Finally, a Flory-Huggins type interaction is introduced for the non-ideal (inter-chain) interactions:
Eq. 19
where 
IJ(|r-r'|) is a mean-field energetic interaction between beads type I at r and J at r', defined by the same Gaussian kernel as in the ideal chain Hamiltonian:
Eq. 20
Parameterization: Mapping of the atomistic level to the mesoscale
As a result of the model of the mesoscopic molecular ensemble dynamics outlined above, the following parameters arise as characterizing the system:
At this point it is important to emphasize, that this Gaussian chain might differ considerably from the real chain. For example, the Gaussian chain may be branched while the real chain is not. This, however, does not need to concern us. We only need to know what the mapping is, and in particular what each bead represents in terms of bonded atoms.
On this basis the next steps are to derive in some way, either experimentally, by Molecular Dynamics or other methods, the diffusion coefficients of the beads. Furthermore, by the Cerius2 Amorphous Builder or Blends module or by group contribution methods the interaction coefficients can be determined.
In conclusion, MesoDyn offers a complete path from the atomistic level through to the simulation of mesoscale phase morphologies.
Together these equations form a closed set, which can be integrated efficiently on a cubic mesh by a Crank-Nicholson scheme.
Since a very large number of equations (½about 106 nested Fredhol integrals per time step) has to be solved and memory requirements for systems of up to 10 beads on meshes of the order of 1003 is also very high, a domain decomposition method has been used and implemented with MPI (Message Passing Interface) for parallel platforms with distributed memory.
Briefly, the cubic grid is divided into subgrids, each associated with a processor. Communications scale only linearly with the total mesh length, and thereby an efficiency of more than 75% is achieved on an 8-processor IBM SP2.
J.G.E.M. Fraaije, B.A.C. van Vlimmeren, N.M. Maurits, M. Postma, O.A. Evers, C. Hoffman, P. Altevogt, and G. Goldbeck-Wood, "The dynamic mean-field density functional method and its application to the mesoscopic dynamics of quenched block copolymer melts," Journal of Chemical Physics, 106, 4260 (1997).
Although most of the method description in this paper has been adapted for this manual, the reader may wish to consult the original paper for some early application work.
In addition, five of the more recent applications from the University of Groningen group are also available on MSI's Web site, in the Mesoscale Products area. These include several which explain the implementation of shear, now available in the present release. These papers are available in electronic form here.
A.V.Zvelindovsky, G.J.A.Sevink, B.A.C.van Vlimmeren, N.M.Maurits, J.G.E.M.Fraaije, "Lammelar phase of diblock copolymer melt under shear: kinetics and conformational analysis," Accepted by Progress in Colloid and Interface Science. Title: "Trends in Colloid and Interface Science XII."
N.M.Maurits, A.V.Zvelindovsky, G.J.A.Sevink, B.A.C.van Vlimmeren and J.G.E.M.Fraaije, "Hydrodynamic effects in 3d microphase separation of block copolymers: dynamic mean-field density functional approach," Journal of Chemical Physics, 108, 91,500 (1998).
A.V.Zvelindovsky, B.A.C.van Vlimmeren, G.J.A. Sevink, N.M.Maurits and J.G.E.M.Fraaije, "3D simulation of hexagonal phase of a specific polymer system under shear: the dynamic density functional approach," Submitted to the Journal of Chemical Physics (rapid communications).
B.A.C. van Vlimmeren, N.M.Maurits, A.V. Zvelindovsky, G.J.A. Sevink, J.G.E.M.Fraaije, "Micro-phase separation kinetics in concentrated aqueous solution of the triblock polymer surfactant (EO)13(PO)30(EO)13: an application of dynamic mean-field density functional theory," Submitted to Phys. Rev. Lett.
A.V. Zvelindovsky, G.J.A. Sevink, B.A.C. van Vlimmeren, N.M. Maurits and J.G.E.M.Fraaije, "3D mesoscale dynamics of block copolymers under shear: the dynamic density functional approach," Phys. Rev. E 57 4699 (1998).
Also available in the above-mentioned MesoDyn Product area is the following recent review of the MesoDyn ESPRIT project:
P. Altevogt, O.A. Evers, J.G.E.M.Fraaije, N.M.Maurits and B.A.C.van Vlimmeren, "The MesoDyn project: software for mesoscale chemical engineering," Published in Theochem.
A complete list of the papers of the University of Groningen group is available at the website of the MesoDyn ESPRIT project, which is found here. These include the following three papers, the first two of which explain the numerical method behind MesoDyn and the last of which explains the modeling of compressible systems, which is implemented in the current Cerius2 MesoDyn release.
N.M. Maurits, P. Altevogt, O.A. Evers and J.G.E.M. Fraaije, "Simple numerical quadrature rules for gaussian chain polymer density functional calculations in 3d and implementation of parallel platforms" Comp. Polymer Sci. 6, 1 (1996).
B.A.C. Van Vlimmeren and Fraaije, "Calculation of Noise Distribution in mesoscopic dynamics models for phase separation of multicomponent complex fluids" Comput. Phys. Commun. 99, 21 (1996).
N.M. Maurits, B.A.C. van Vlimeren and J.G.E.M. Fraaije, "Mesoscopic phase separation dynamics of compressible copolymer melts" Accepted by Phys. Rev. E.