Supplementary Material

Ultra-slow self-similar coarsening of physical fibrillar gels formed by semiflexible polymers

Martin Kröger, Clarisse Luap, Patrick Ilg

Preprint DOI: 10.20944/preprints202412.1254.v1 (Open Access)

Movies

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Movie	κ (kappa)	realization	tw	duration	content
movie A	50	1	1.28× 106	1.5× 105	filament rupture event
movie B	50	1	105	2× 106	displacements, δt=5× 10^3
movie C	50	1	105	2× 106	displacements, δt=2.5× 104
movie D	50	1	105	2× 106	displacements, δt=2.5× 105
movie E	5	11	-103	2× 103	droplet formation
movie F	10	11	-103	2× 103	network formation
movie G	50	11	-103	2× 103	network formation
movie E+	5	11	103	2× 104	coarsening dynamics
movie F+	10	11	103	2× 104	coarsening dynamics
movie G+	50	11	103	2× 104	coarsening dynamics
movie H	5	1	0	105	droplets and short cylinders
movie I	5	6	0	5× 104	percolated cylinder
movie J	20	1	0	5× 104	coarsening dynamics

All movies have been created by C.L. using Ovito.

Logarithmic and power-law fits

We fitted the ensemble-averaged time series using the logarithmic and power-law expressions mentioned in the manuscript. Both of them involve four parameters for each quantity. The scripts FENE_CB_functions.m (matlab) and FENE_CB_functions.py (python) offer inline functions that allow to evaluate or plot the various quantities (described within the script) versus waiting time (t_w) and/or bending stiffness (kappa). Below are rudimentary examples on how to edit and make use of the scripts.

Using matlab, plot the radius of gyration versus waiting time for 6 different kappa values, using the logarithmic fits:

  figure;
  tw = 10.^linspace(0,5,200); 
  for kappa = [10 20 30 50 75 100]; semilogx(tw,Rg_power(kappa,tw),'k.-'); hold on; end

Using python, plot the radius of gyration versus waiting time for 6 different kappa values, using the logarithmic fits:

  plt.figure()
  tw = np.power(10,np.linspace(0,5,200))
  for kappa in [10,20,30,50,75,100]:
      plt.semilogx(t,Rg_power(kappa,tw),'k.-')
  plt.show()

Using matlab, plot the radius of gyration versus waiting time for 6 different kappa values, using the power-law fits:

  figure;
  tw = 10.^linspace(0,5,200); 
  for kappa = [10 20 30 50 75 100]; semilogx(tw,Rg_logarithmic(kappa,tw),'k.-'); hold on; end

Using matlab, plot the radius of gyration versus kappa at three different waiting times using the power-law fits:

  figure;
  kappa = linspace(10,100,100);
  for tw = [1e2 1e4 1e5]; plot(kappa,Rg_power(kappa,t),'k.-'); hold on; end

Calculating averages, extrapolations

With the functions at hand, you can calculate time-averaged mean values to rate the effect of a chosen averaging interval on the mean values, or to extrapolate quantities to a later waiting time. As the fits are based on data for t_w ∈ [10⁴,10⁶] for κ ∈ {0,10,50,75} and t_w ∈ [10⁴,10⁵] for κ ∈ {2,5,15,20,30,40,100} the predictions far outside these regimes must be regarded as crude estimates. For example, extrapolating the number of edges (E) to unity makes sense, while extrapolating them to zero is certainly over-stressing the fit function.

LAMMPS implementation

The FENE-CB model (manuscript section 4.1.1) studied in this work is coveniently implemented in LAMMPS via

  variable rc equal 1.359  # cutoff distance for nonbonded pairs of beads (corresponds to E_coh = 1.4)
  variable kappa equal 10  # bending stiffness
  
  atom_style angle
  read_data myfile.data nocoeff
  mass * 1.0
  pair_style lj/cut ${rc}
  pair_modify shift yes
  pair_coeff * * 3 1.0 ${rc}
  bond_style fene
  bond_coeff * 30 1.5 1.0 1.0
  special_bonds fene
  angle_style cosine
  angle_coeff * ${kappa}

  timestep 0.005
  comm_modify cutoff 4.0
  neighbor 0.3 bin
  
  fix NVE all nve
  fix LANGEVIN all langevin 1 1 2 123456 zero yes

  # run etc .. 

where it is assumed that you have an initial configuration myfile.data at hand (if not, contact M.K.). The entries within the Atoms section of the data file are: id mol type x y z. The Bonds section lists the permanent intramolecular bonds. The above can serve as the header of your LAMMPS input file.