Polymer Physics

If a single polymer chain is confined between two parallel plates with a distance
d, we can assumeN/gblobs, each with the linear sized¼bg3/5, as illustrated in
Fig.4.15a. Similar to Flory’s mean-field treatment, each blob has the volume
repulsive energykTd^2 (N/g)/R//^2 , and the overall volume repulsive energykTd^2 (N/
g)^2 /R//^2. On the other hand, the entropic elasticity of chain conformations formed by
blobs still follows the Gaussian distribution, askTR//^2 /(d^2 N/g). In the thermody-
namic equilibrium, the total free energy changeF¼Erep+Eelis minimized with
respect toR//, and we obtain the equilibrium size

R==dð

n

g

Þ^3 =^4 ¼n^3 =^4 bð

b

d

Þ^1 =^4 (4.85)

When the plate spacing approaches the monomer size, i.e.d!b, we have

R==n^3 =^4 b (4.86)

In this case, the extremely confined polymer chain follows the scaling relation-
ship of 2D SAWs, and exhibits the scaling law of the coil size for a 2D real chain.
If the single polymer chain is confined in a cylindrical tube with diameterd, the
calculation is straightforward, as

R==dð

n

g

Þ¼nbð

b

d

Þ^2 =^3 (4.87)

Whend!b,

R==nb (4.88)

and the chain becomes fully stretched under the extreme 2D confinement.
The above cases are for isolated single chains. For polymer melt confined
between two parallel plates, the internal concentration of each coil

Cint

nb^3

dR==^2

(4.89)

Fig. 4.15 Illustration of the blob models for the single polymer chain confined in (a) two parallel
plates and (b) a cylindrical tube

68 4 Scaling Analysis of Real-Chain Conformations