8.1: Entropy of Single Polymer Chain
- Page ID
- 294306
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Entropy of Single Polymer Chain
Calculate the number of ways of placing a single homopolymer chain with \(n\) beads on lattice. Place beads by describing the number of ways of adding a bead to the end of a growing chain:
A random walk would correspond to the case where we allow the chain to walk back on itself. Then the expression is \(\Omega_P = M z^{n - 1}\)
Note the mapping of terms in \(\Omega_P = M z (z - 1)^{n - 2}\) onto \(\Omega_P = \Omega_{trans} \Omega_{rot} \Omega_{conf}\).
\[\text{ For } n \to \infty \ \ \ M \gg N \ \ \ \Omega_P \approx M(z - 1)^{n - 1}\nonumber\]
\[\begin{array} {rcl} {S_p} & = & {k_B \ln \Omega_P} \\ {} & = & {k_B ((n - 1) \ln (z - 1) + \ln M)} \end{array} \nonumber\]
This expression assumes a dilute polymer solution, in which we neglect excluded volume, except for the preceding segment in the continuous chain.