Dear Anna,
Thanks for your email. I have a coarse grained polymer system with FENE
bonds and LJ pair-wise repulsion (self avoiding chain). What I did is to
look at the free energy profile based on thermodynamic integration, in
which case I could look at the free energy profile when lj component is
turned off and on, but turning off FENE bonds cannot really be turned off,
and therefore I am interested whether there is a way to see the free
energy contribution of the FENE bonds?
TI provides differences in free energies e.g. from a reference state, but why
would you want to compare the free energy difference between a bonded
and non-bonded system?
Do not forget that nature does not distinguish between pair and bonded
interactions, in reality bonded interactions are just pair interactions with
such a high barrier, that we implicitly coarse-grain them into permanent
bonds. This is computationally very convenient, but does not necessarily
have anything to do with nature.
Perhaps a better reference state for TI would be to take a random chain
conformation, and replace bond interactions by self-springs fixing the
monomers to their instantaneous position in space and perhaps fix the
self-spring constant to reproduce the same monomer fluctuations as in
the polymer?
Anyways, if you take a step back (ignoring pair interactions (and units)
for simplicity), then in the absence of bonds you have an ideal gas, the
partition function is Z ~ V^N where N is the number of particles, and V
the volume.
If you link identical monomers into a RW polymer the partition function
becomes Z ~ V (4pi)^(N-1). Here I assume you integrate over the position
of the first monomer, and bond lengths are fixed so the remaining (N-1)
integrals are just over the solid angles specifying the chain conformation.
In both cases F= kT ln(Z) ~ kT N * const1 + const2 for large N, with
different constants for the free and bonded cases. For more complicated
polymer models, I would expect the same generic behaviour with
suitably renormalised constants.
In the case where you replace bonded interactions with self-springs, the
partition function should still be Z ~Veff^N, where Veff ~ sqrt( k/kT )^3
is the effective volume of monomer fluctuations and k the spring. This
follows since 0.5 k <X^2> = 0.5 kT by the equipartition theorem. But if
you now gradually switch on pair AND bonded interactions, while letting
the self-spring constant go to zero, that is perhaps an interesting free
energy difference, and I would expect that this can be done with LAMMPS?
Note you can still derive the exact free energy of the reference state
analytically.
Note that the self-springed state is fixed in space, while the final state
is free to move, hence you acquire a -kT Log(V) additional free energy
due to the translational entropy of the final state. I would subtract
this off the TI result, to get a number that is reflects the free energy
contribution due to the specific molecular interactions. Secondly note
that this should be averaged over several initial states.