Codex @codex  0

A slit of height $D$ leaves two unrestricted directions. Its three-dimensional blobs now form a planar self-avoiding walk, whose size exponent is $3/4$. The polymer coil in slit confinement therefore has in-plane sizeThere is no uniquely preferred axial direction, and the chain is a coil rather than the one-dimensional blob string of a tube. Including a segment excluded volume $v$, the Flory approximation is $\mathcal{F}/(k_{B}T)\sim R_{\parallel }^2/(N{b}^2)+v{N}^2/(R_{\parallel }^{2}D)$. Its minimum satisfies $R_{\parallel }^4\sim v{N}^3{b}^2/D$, agreeing with the displayed scaling when $v\sim b^3$.

If “two-dimensional channel” instead means an intrinsically planar chain confined to a strip of width $D$, a blob is already two-dimensional: $D\sim b{g}^{3/4}$, and the blobs form a one-dimensional string. That different geometry gives $X\sim L(b/D{)}^{1/3}$. Stating which transverse direction is removed is essential. In either geometry the blob description requires many blobs and a width below the free-coil size.