The Kremer-Grest (KG) model is the defacto standard for coarse-grain modeling of polymers using Molecular Dynamics. It represents a polymer as a string of beads. The beads are essentially hard spheres and the springs increase in stiffness to prevent chains from passing through each other. So far there is somewhat more than 1500 papers studying various aspects of polymer physics using the KG model. The KG model is a generic model, and was not designed to match any particular specific polymer chemistry.
Below is a tool for 1) mapping real polymers to a Kuhn representation, and 2) deriving a Kremer-Grest model matching a given Kuhn representation, that we have developed based on our recent papers on Kuhn mapped KG models. Combining the two, we can derive a KG model matching the large scale emergent properties of any commodity polymer species.
Our mapping is based on Univerality rather than explicit coarse-graining. Universality suggests that all physical properties (for concentrated polymer solutions/melts above the glass-transition temperature) depends on just two numbers. The Kuhn number which determines the chemistry of the polymers, we want to describe, and the number of Kuhn segments per chain describes the chain length of the polymer melt we are targetting. The plot above shows how by varying a single parameter - the chain stiffness - we can design KG polymer models that match fairly accurately the entanglement modulus of most commodity polymers (symbols). The entanglement modulus is just one univeral emergent quantity, we expect all other universal emergent quantities to be equally well matched.
Note that if you are interested in studying e.g. glassy dynamics or other non-universal properties which depends critically on the specific monomeric structure and interactions, then the current approach will most probably fail. You might use well equilibrated KG polymer models as a stepping stone to specific coarse-grained models or models based on atomistic force fields.
Please cite the papers and site, if you use the mapping tool:
Below is our online tool for mapping real polymers to a Kuhn description, and mapping a Kuhn description to a KG model. If you already have a Kuhn mapping you can jump to "Kuhn mapped KG model" and enter the numbers to generate a corresponding KG model. If not, then start by entering the single chain properties. Clicking Calculate will derive quantities that you need to propagate the properties.
Values may differ slightly from the tables of the mapping paper. The reason is that here we use eq. 30 of the mapping paper to calculate chain stiffness kappa from a known Kuhn number, whereas in the paper we invert eq. 25 using the numerical expression for the Kuhn length eq. 24.
To derive the Kuhn mapping, we need the spatial extent and length of a polymer. A polymer in a theta solvent or melt state adopts a random walk conformation. To characterize the spatial extent of the conformation we use the mean-square of the chain end-to-end vector <R2>. To characterize the "length" of a polymer, we use Rmax which is the maximal length of a polymer chain assuming its pulled to a straight line conformation. Notice that we respect chemical structure, that is Rmax is in general somewhat shorter than the sum of the lengths of the back bone bonds of the polymer, which defines its contour length L.
To derive entanglement related properties we need to know the bulk density of the polymer, as well as its entanglement modulus Ge. This characterize the elastic contribution due to entanglements in the melt state. The entanglement modulus is related to the experimental plateau modulus as Ge= 1.2 GN. Experimentally 4/5 of the entanglements are lost due to chain contraction post deformation.
Please enters the properties below or choose a polymer from the Polymer Handbook
| Name of polymer | ||
| Mean square extension per unit mass <R2>/M= | (Å2/[g/mol]) | |
| Maximal extension per unit mass Rmax/M= | (Å/[g/mol]) | Use Rmax calculator below to estimate value. |
| Molecular mass of monomer Mm= | (g/mol) | |
| Bulk density rho= | (g/cm3) | |
| Entanglement modulus Ge= | (MPa) | Note Ge= 5/4 GN where GN is the experimental plateau modulus |
| Temperature T= | (K) | Temperature where the experiments above were conducted. |
In the Kuhn description we describe the experimental polymer as a freely jointed chain with the same <R2>=lK2NK and Rmax=lKNK. Where lK and NK are the Kuhn length and the number of Kuhn steps respectively. The second key quantity is the dimensionless density of Kuhn segments nK=rhoK lK3 where rhoK=rhobulk/MK, where MK is the mass of a Kuhn segment.
| Kuhn Number nK= | |
| Kuhn length lK= | (Å) |
| Mass of Kuhn segment MK= | (g/mol) |
| Monomers per Kuhn segment | |
| Kuhn density rhoK= | number/nm3 |
Based on the Kuhn number above, we can derive the stiffness (kappa) of a KG polymer model that reproduces this Kuhn number since nK=rhobeadlbondlK2. Where we assume rhobead=0.85sigma-3 and lbond=0.965sigma as the standard values of the KG model.
Given the entanglement modulus the number of Kuhn segments between entanglements NeK is given by Ge=rhoK kT/NeK, from which we can estimate the tube diameter which we identify with the Kuhn length of the PPA: app2=dT2=lK2NeK as well as the packing length p=(rhoK lK2)-1. alpha=app/p is the number of entanglement strands per entanglement volume.
| Kuhn segments per entanglement NeK= | |
| Tube diameter dT= | (Å) |
| Packing length p= | (Å) |
| alpha= |
| Stiffness kappa/kT= | (number) |
| Kuhn length lK= | (sigma) |
| Beads per Kuhn segment. cb= | (number) |
| Beads per monomer. cm= | (number) |
| Beads between entanglements. Neb= | (number) |
When we know the chain stiffness, we also know lK(kappa) in sigma units. Then cb=lK/lbond provides the number of beads per Kuhn segment, from which we obtain the SI mass of a bead. We can also identify the KG Kuhn length in KG units with the polymer Kuhn length in SI units to obtain a length scale mapping. Finally kT in SI units we can identify with 1epsilon of simulation energy unit.
| KG bead mass mb= | (g/mol) |
| KG length scale sigma= | (Å) |
| KG energy unit epsilon= | (J) |
| KG unit of stress: 1epsilon/sigma3= | (MPa) |
Assuming the mapping relations have been filled out, enter one of the following four parameters:
| Number of beads per polymer Nb= | (number) | |
| Number of monomers per polymer Nm= | (number) | |
| Molecular weight of polymer Mp= | (g/mol) | |
| Number of Kuhn segments NK= | (number) |
When the chain length is defined (NK), you can calculate everything else.
| Mean-square end-to-end extension <R2>= | (Å2) |
| (sigma2) | |
| Maximal length Rmax= | (Å) |
| (sigma) | |
| Number of entanglements per chain Z= | |
| Kuhn time: tauk= | (tau) |
| Entanglement time: taue= | (tau) |
| Rouse time: taur= | (tau) |
| Maximal relaxation time: taumax= | (tau) |
Here we assumed zetabead=24.0+21.0/nK for the effective friction of a single bead as function of stiffness From this we can predict the friction of a Kuhn segment, and hence the time it takes a Kuhn segment to diffuse its own size. This is tauK which is the fundamental clock in the dynamical hierachy of polymer dynamics. The next time scale is the time it takes for a polymer to feel entanglement effects (taue=NeK2tauK). The time it takes to relax inside the tube (the Rouse time tauR)=NK2tauK), and finally the time it takes for a chain to completely escape its tube (the disentanglement time taumax=3 NK3tauK/NeK)
To generate a time mapping, additional experimental input is required to match one of the time scales above to an experimental value.
The contour length of a polymer L is the sum of all the bonds in the backbone. Analogously the contour length of a monomer is the sum of its bond lengths. Rmax and L are proportional, since both are extensive in number of monomers in a polymer. But Rmax is shorter than L due to chemical structure. Typically PE like structure is assumed for most polymers. For PE the bond angle is 68degrees, hence projecting a C-C bond onto the end-to-end direction of the polymer produce a factor of cos(theta/2)=0.83 for PE.
| Contour-length per monomer | (Å) | |
| Maximal extension/chain contour length Rmax/L= |
If you have derived the KG force field above, then
to generate a LAMMPS script for running the force field
In case of questions, bugs, etc. please contact Carsten Svaneborg at zqex(at)sdu(dot)dk
This page is copyright Carsten Svaneborg, 2020