Polymer Physics

and the coils are assumed unperturbed, asR//¼(nb^2 )1/2, then one obtains

Cint

b

d

(4.90)

which shows no dependence on the chain length. Whend!b,Cint!1. This
implies that with the decrease ofd, the chain will be expelled from the other chains,
and the coils decrease interpenetration with each other. In the extreme case, the
polymer coil is actually expanded a bit from the 3D unperturbed coil into the 2D
non-interpenetrated coil.
For the melt polymer chains confined in a tube,

Cint

nb^3

d^2 R==

(4.91)

and assuming again the unperturbed coils asR//¼(nb^2 )1/2, one obtains

Cintn^1 =^2 ð

b

d

Þ^2 (4.92)

WhenCint~1,

R==

nb^3

d^2

(4.93)

When d!b,

R==nb (4.94)

and the chains are fully stretched.
SinceCint<1, when the tube diameter

d>n^1 =^4 b¼ðR 0 bÞ^1 =^2 (4.95)

polymer chains will maintain the unperturbed conformation states.
The above scaling analysis did not take interacting boundaries into account,
which has recently been studied via a semi-analytic theory by Freed et al.( 2010 ).

4.4.3 Adsorption

Adsorption of polymers on a solid flat substrate is an important physical chemistry
issue in many applications of polymers, such as composites, coatings, adhesion,
lubricates, gel chromatography, wetting, colloidal stability, piping transportation,

4.4 Single-Chain Conformation Under External Forces 69