Polymer Physics

The total free energy change becomes

F¼EelþEads¼kTnð

b

d

Þ^1 =nen

b

d

(4.99)

By minimizingFwith respect tod,∂F/∂d¼0, one obtains

dbð

e

kT

Þn=ð^1 nÞ (4.100)

Clearly, the absorbing thickness is irrelevant to the chain lengthn, and decreases
when the temperature decreases or with the absorbing-force increases.
The above free energy calculation considers only a local entropy loss of the single
coil due to the confinement of surface absorption equilibrated by the thermal energy,
but neglects the entropy loss of the global-chain conformation demonstrated in
Fig.4.16. In addition, the contact to a flat substrate makes parallel deformation of
chain segments (called the train segments). The loops and tails connecting train
segments lose some of their conformation entropy as well, due to their ends anchoring
temporarily to the substrate surface. Therefore, there exists a critical energyec> 0
for the desorption-adsorption transition to get over all the above-mentioned entropy
losses (Rubin 1965 ). With a high affinity of the surface (but not so high as in a
chemisorption), the entropy loss must be compensated by more enthalpy gains with
the surface contacts, leading to a richness of monomers at the contact layer in
accompany with a decay of monomer concentrations along the perpendicular direc-
tion of the surface. According to the consistent results from Monte Carlo simulations
(Eisenriegler et al. 1982 ) and from the theoretical series expansions (Ishinabe 1982 ),
the monomer concentrations scale with the distancexfrom the substrate surface, as

Cð

b

x

Þm (4.101)

withm1/3 in a good solvent. With this so-called proximity effect on a highly
affinitive substrate, the adsorption thickness for a single chain in a good solvent will
be updated as well (De Gennes 1983 ), as given by

dbð

e

kT

Þn=½^1 nð^1 mފ¼bð

e

kT

Þ^1 (4.102)

which coincides to the roughly estimated result from (4.100) in a theta solvent (the
ideal-chain model discussed in the early history (De Gennes 1979 )).
The self-consistent mean field model could elucidate the relative importance of
loops and tails along their distances from the substrate surface (Scheutjens and
Fleer 1980 ; Fleer et al. 1993 ). The chain ends are expelled away from the adsorbing
surface, probably due to their relatively high mobility to other monomers. There-
fore, the loops are rich in the region close to the adsorbing surface, and the tails are
mainly in the region away from there. Quantitative description can also be obtained
from the scaling arguments (Semenov and Joanny 1995 ).

4.4 Single-Chain Conformation Under External Forces 71