8.2: Self-Avoiding Walks
- Page ID
- 294307
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To account for excluded volumes, one can enumerate polymer configurations in which no two beads occupy the same site. Such configurations are called self-avoiding walks (SAWs). Theoretically it is predicted that the number of configurations for a random walk on a cubic lattice should scale with the number of beads as \(\Omega_p (n) \propto z^n n^{\gamma - 1}\), where \(\gamma\) is a constant which is equal to 1 for a random walk. By explicitly evaluating self-avoiding walks (SAWs) on a cubic lattice it can be shown that
\[\Omega_p (n) = 0.2 \alpha^n n^{\gamma - 1}\nonumber\]
where \(\alpha = 4.68\) and \(\gamma = 1.16\), and the chain entropy is
\[S_p (n) = k_B [n \ln \alpha + (\gamma - 1) \ln n - 1.6].\nonumber\]
Comparing this expression with our first result \(\Omega_P = Mz(z - 1)^{n - 2}\) we note that in the limit of a random walk on a cubic lattice, α=z=6, when we exclude only the back step for placing the next bead atop the preceeding one \(\alpha = (z - 1) =5\), and the numerically determined value is \(\alpha = 4.68\).
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2. C. Vanderzande, Lattice Models of Polymers (Cambridge University Press, Cambridge, UK, 1998).