de Gennes confinement scaling ($X\sim L(b/D{)}^{2/3}$)

For a long self-avoiding polymer in a tube of diameter $D$, represent each confinement blob by $g$ statistical segments of length $b$, with $D\sim b{g}^{3/5}$. Excluded-volume repulsion orders the blobs along the tube, giving $X\sim (N/g)D=L(b/D{)}^{2/3}$. More generally, a segment excluded-volume parameter $v$ gives $X\sim L[v/(b{D}^2){]}^{1/3}$. This requires many blobs and $D$ smaller than the free coil size; $b\ll D\ll L$ alone does not ensure that. An ideal chain without intersegment repulsion is not a one-dimensional string of nonoverlapping blobs.

In a slit, a confinement blob is three-dimensional but the chain of blobs forms a two-dimensional self-avoiding walk. With $g\sim (D/b{)}^{5/3}$ and in-plane exponent $3/4$, $R_{\parallel }\sim D(N/g{)}^{3/4}$. There is no single preferred axial extension as in a tube. A strictly planar chain confined further to a narrow strip is a different geometry and gives $X\sim L(b/D{)}^{1/3}$ under the two-dimensional blob estimate.