Uneffective bias for Umbrella Sampling, for studying aggregation

[Aden RANA]

Dear PLUMED developers and users,

I hope that all of you are doing well.

I am facing some issues while learning how to study aggregation of Coarse Grain particle. To do so, I am using LAMMPS in NPT, with a toy potential and PLUMED v.2.10. My initial structure has 1000 particles, two types, 500 particles of each.

I am playing with PLUMED commands starting with "CLUSTER_...", to perform Umbrella Sampling, to get the Gibbs Energy of aggregation by using the size of the biggest cluster as a CV and applying a bias on it. I took inspiration from the following paper: http://dx.doi.org/10.1021/acs.jctc.6b01073 .

I don´t know why, but even by using a ridiculously high  "KAPPA" for the bias, the system is not willing to reach a configuration where the size of the biggest cluster is equal to the value given to "AT", not even close. Moreover some particles are dissolving.

I have five question/hypothesis: 

1) The way I define the size of the biggest cluster is wrong and not directly usable as a CV to perform US. Am I misusing the RESTRAINT command ?

2) Should I stick to the paper:  http://dx.doi.org/10.1021/acs.jctc.6b01073 and use metadynamic instead of umbrella sampling ?

3) The problem comes from my LAMMPS input.

4) I should "help" the simulation and start the MD with an initial structure that already has an aggregate with the wanted size.

5) As the bias is: 

V(s)=0.5*KAPPA(s-AT)^2 with s the CV.

I assume that the biased force applied on the i-th particle will be: 

F_i(s)= -KAPPA(s-AT)*(∂s/ ∂r_i)​ ​with r_i the coordinates of the i-th particle

If the system is not evolving to the wanted configuration, it might mean that  (∂s/∂r_i) is low, or that the function s(r_i) isn't differentiable. Indeed, in my case the CV has discrete value and might not be differentiable everywhere.

In this case, is it possible to use a different CV that will be smoother and still allow to control directly or indirectly the size of the biggest cluster ?

Do not hesitate to ask me if you need more information to help me. 

I attached my LAMMPS (lammps.in) and PLUMED input (plumed.dat). Here is a small snippet of the COLVAR file:

#! FIELDS time step_pl CoordCaC_avg_mean c1 us.bias
 0.000000 0.000000 0.388265 5.000000 101250000.000000
 10.000000 1.000000 0.387178 5.000000 101250000.000000
 20.000000 2.000000 0.384761 5.000000 101250000.000000
 30.000000 3.000000 0.380716 5.000000 101250000.000000
 40.000000 4.000000 0.374813 5.000000 101250000.000000
 50.000000 5.000000 0.367228 5.000000 101250000.000000
...
 54070.000000 5407.000000 0.347744 5.000000 101250000.000000
 54080.000000 5408.000000 0.348952 5.000000 101250000.000000

I sincerely apologize if my questions seems clumsy and I thank you for the time you might take to read me.

Respectfully,

RANA Aden