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.