Update README for tutorial 1

tut1_polymer/README

+# Tutorial 1 - simulating a simple polymer chain
+
 In this second tutorial, we use LAMMPS to simulate the dynamics of a simple
 polymer (a chain of beads) within a solvent (modelled implicitly by simulating
 the system using Langevin/Brownian dynamics). As in the previous tutorial,
-beads interact with one another via the truncated Lennard-Jones (LJ)
-potential, with the default parameters setting the interactions to be purely
-repulsive.
+beads interact with one another via a truncated and shifted Lennard-Jones
+(LJ) potential, with the default parameters setting the interactions to be
+purely repulsive.
 
 To link consecutive beads of the polymer together, we add finite extensible
 nonlinear elastic (FENE) springs/bonds between them. In formula, the FENE
 potential is given by
 
-   U(r) = -K_f*R_0^2*ln(1-(r/R_0)^2),
+   U(r) = -0.5*K_f*R_0^2*ln(1-(r/R_0)^2) if r < R_0
 
-where r is the distance between two consecutive beads, and K_f is the spring
-constant. The FENE bond differs from the harmonic bond in that it can only
-extend to a finite limit (R_0), and this can help reduce the chance of the
-polymer getting entangled.
+and infty otherwise. Here, r is the distance between two consecutive beads and
+K_f is the spring constant. The FENE bond differs from the harmonic bond in
+that it can only extend to a finite limit (R_0), and this can help reduce the
+chance of the polymer getting entangled.
 
 We also model the polymer to be semi-flexible by adding angle bonds - bonds
 which act between three consecutive beads. Here we use the cosine potential
@@ -27,68 +29,90 @@ persistence length l_p (K_b ~ k_B*T*l_p/sigma), the characteristic length over
 which the polymer loses memory of its direction. This kind of semi-flexible
 polymer models is also known as a Kratky-Porod model.
 
+###############################################################################
+
 What to do:
 
-0. Plot the bond and angle potentials
+0. Familarise yourself with the bond and angle potentials
 
    Using your favourite plotting package, plot both the FENE and cosine
    potentials. Make sure you are happy with how these potentials do what they
-   are meant to do in the simulations. Explore different parameter values and
-   see how that changes the shape of the potential.
+   are meant to do in the simulations! Explore different parameter values and
+   see how that changes the shape of the potential as well as th structural
+   and dynamical properties of the system.
 
 1. Run the simulation with the default parameters
 
    a. Generate the required input scripts/files
 
+      As before, this is done by running the provided bash script
+
          bash init.sh
 
-      As in the first tutorial, this generates two files that are necessary
-      for running LAMMPS:
-      -  run.lam file - a 'driver' script that is read by LAMMPS
-      -  init.in file - a trajectory file containing the initial configuration
-                        of the system (i.e., the atoms' positions, bonds, etc)
-			
-      Have a look at these files individually and make sure you understand
-      what each command and each section does! In particular:
+      which creates two files (run.lam and init.in) that are needed for
+      running LAMMPS. Like before, have a look at these files individually
+      and make sure you understand what each command and each section does!
+      In particular:
       -  What are some new commands added in the run.lam file?
       -  What is the initial configuration of the polymer? And how did we
       	 equilibrate the system?
       -  What additional info have we provided in the init.in file?
 
-      As always, if in doubt, check out LAMMPS documentation on
+      If in doubt, check out LAMMPS documentation on
 
          docs.lammps.org/Manual.html
 
    b. Run the simulation
 
-      This can be done using LAMMPS installed on the school computer
+      This can be done using the LAMMPS installed on the school computer. We
+      start the simulation using the same command as before
 
          lmp -in run.lam
 
-      pos*.lammpstrj files will be generated, storing the beads' positions
-      and velocities
+      The simulated trajectory of the polymer is stored in the
+      pos*.lammpstrj files.
 
    c. Visualise the trajectory of the polymer
 
-      This can be done using VMD as in the first tutorial. Note that because
-      LAMMPS does not output the bead positions in order (i.e., by the
-      bead index) while VMD thinks that is the case, beads are linked together
-      wrongly in the visualisation. We can fix this by typing the following
-      set of commands in the terminal after loading the molecule:
-
-      topo clearbonds
-      set N {nbeads}
-      for {set i 0} {$i < [expr $N-1]} {incr i} {
-      	  set j [expr $i+1]
-	  topo addbond $i $j
-      }
-
-      [You will need to replace the variable nbeads with the actual number of
-       beads in the polymer]
+      This can be done using VMD as in the first tutorial.
       
-2. Explore the effect of changing the
-
-   
-
-3. 
-
+2. Explore the effect of changing different system parameters
+
+   a. Change the interaction between polymer beads
+
+      Modify the pairwise interaction between beads so that they become
+      attractive to one another (i.e., change the cutoff r_c to 1.8 and vary
+      epsilon).
+      -  Can you pinpoint the critical value for epsilon where the polymer
+      	 changes from having an extended/open structure to a compact/
+      	 globule-like one? This transition is also known as the
+	 'coil-to-globule' transition in polymer physics, and can be seen in
+	 experiment, for example, by modifying the salt concentration
+	 of the solvent.
+      -  Can you come up with an observable that allows you to quantify the
+      	 'size' of the polymer?
+
+   b. Change the stiffness/flexibility of the polymer
+
+      -  How does changing the persistence length parameter in init.sh affect
+      	 the overall structure of the polymer?
+      -  In the case where the attraction between beads is strong enough such
+         that the polymer collapses, how does changing the flexibility of the
+	 fibre affect the way how it is folded?
+
+3. (More advanced) Model a heteropolymer chain
+
+   The provided script simulates a polymer chain where all beads are identical
+   (e.g., they interact the same with one another), and this type of chain is
+   known as a homopolymer. A heteropolymer is one where beads/monomers with
+   different properties are chained together. Try modifying the code to
+   simulate a chain with two types of beads (type A and B, and say they are
+   randomly ordered along the polymer, e.g., ABAAABBAB). Set the interaction
+   strength to be different between the two kinds of beads (i.e.,
+   epsilon_AA != epsilon_AB != epsilon_BB).
+      -  At what parameter regime does the polymer form a single globule with
+      	 A and B type beads mixed together?
+      -  At what parameter regime does the polymer 'phase separate' and form
+      	 distinct globules, where A beads are demixed with B beads?
+      -  How is the dynamics here different from the case of the LJ fluid,
+      	 where the beads/colloid particles are not connected together?
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