Theta condition in simulations
The three types of solvent environments (good, bad, Theta) can be easily demonstrated in simulations of a very simple chain on a cubic lattice, as demonstrated in Tanaka, G.; Mattice, W. L. "Chain Collapse by Lattice Simulation" Macromol. Theory Simul. 1996, 6, 499-523.
The simulations uses a single chain, which consists of N+1 beads, connected by N steps of the same length on a cubic lattice.
There are two components to the energy of the system.
- No two beads can occupy the same site simultaneously.
- An energy epsilon, made dimensionless through division by kT, is applied to each unoccupied nearest neighbor site of every bead.
Since the cubic lattice has a coordination number of six, the fully extended conformation has an energy of [(4(N - 1) + 10} epsilon, with 4 epsilon contributed by each internal bead, and 5 epsilon contributed by the two end beads.
Another conformation may have a different energy, if that conformation places two or more nonbonded beads as nearest neighbors on the cubic lattice, thereby reducing the contribution by epsilon.
The simulation proceeds by randomly changing the position of one or more beads, calculating the difference between the energies of the old conformation and the proposed new conformation, Delta E, and employing the Metropolis rule to produce an equilibrated ensemble:
- Accept the new conformation if Delta E < 0.
- Otherwise form the Boltmann factor, exp (- Delta E) ... recall that epsilon is dimensionless ,.. and compare the result with a random number in the range 0 to 1. Accept the new conformation if the random number is smaller than the Boltzmann factor. Otherwise replace the proposed new conformation with another copy of the previous conformation.
After a large number of iterations, the average of the squared end-to-end distance gives <r2>.
Repeat with chains having N ranging from 40 to 1000, each studied with a variety of values of epsilon.
- If epsilon = 0, the chains feel excluded volume, due to the requirement that no two beads can occupy the same site. The chains in the simulation have mean square dimensions that obey the relationship
<r2> = 0.163 N1.2
The exponent, 1.2, is the one expected for chains in a good solvent (nu = 3/5).
- If epsilon is very large and positive (meaning that the chains are strongly repelled by their environment), the chains contract, and they now obey the relationship
<r2> = (N2/3 - 1)/4
The exponent, 2/3, is the one expected for collapsed chains in a poor solvent (nu = 1/3).
- The Theta conditions must lie somewhere between the above two limits.
Thus epsilon must be positive (but not too large) at the Theta condition. When epsilon = 0.154, the simulation yields the very simple result
<r2> = N/4
The exponent, 1, is the value expected in a Theta solvent (nu = 1/2).
And the numerical value of <r2> reproduces exactly the result expected from application of the Rotational Isomeric State model to this system.
The Theta state for this lattice chain requires epsilon = 0.154.
Good solvents have smaller values of epsilon.
And poor solvents have larger values of epsilon.
In the Theta state, the weak repulsive interaction with the solvent (epsilon = 0.154 in the simulation) is exactly the right size to compensate for the hard core exclusion, caused in the simulation by the absolute prohibition for double occupany of any site.
In a real system, the chain has a slightly repulsive interaction with the Theta solvent.
This repulsion is just enough to compensate for the self-exclusion of the chain, causing it to have the same <r2> as it would have had if self-intersections were no problem (if it had no excluded volume).
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July 6, 1999
Wayne L. Mattice: wlm@polymer.uakron.edu