= Solution
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 size
$$
\boxed{R_\parallel\sim D(N/g)^{3/4}\sim bN^{3/4}(b/D)^{1/4}.}
$$
There 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 $\mathcal F/(k_BT)\sim R_\parallel^2/(Nb^2)+vN^2/(R_\parallel^2D)$. Its minimum satisfies $R_\parallel^4\sim vN^3b^2/D$, 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: $D\sim bg^{3/4}$, and the blobs form a one-dimensional string. That different geometry gives \b[$X\sim L(b/D)^{1/3}$]. 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.
Back to article page
