Polymer Chain Adsorption on a Solid Surface: Scaling Arguments and Computer Simulations - Nanophenomena at Surfaces - page 189

increases beyond a critical value k B T (where T stands for the temperature of

the system). The adsorption transition can be interpreted as a second-order phase
transition at the critical point (CAP) of adsorption

c ≈

= c in the thermodynamical

limit, i.e., N →∞ . Close to the CAP the number of surface contacts N s scales

N φ . The numerical value of as N s ( = c ) ∼ φ is somewhat controversial and

lies in a range between φ = 0 . 59 [ 7 ] and φ = 0 . 484 [ 14 ]; we adopt, however, the

value 02 which has been suggested as the most satisfactory [ 13 ]by

comparison with comprehensive simulation results.
How does polymer structure vary with adsorption strength? Consider a chain
tethered to the surface at the one end. The fraction of monomers on the surface n

φ = 0 . 50 ± 0 .

=

N may be viewed as an order parameter measuring the degree of adsorption.
In the thermodynamic limit N

N s /

→∞ , the fraction n goes to zero ( ≈ O( 1 / N ) )

N φ − 1 whereas for for c (in the strong coupling

limit) n is independent of N . Let us measure the distance from the CAP by the
dimensionless quantity

c , then near c , n ∼

κ = ( − c )/ c and also introduce the scaling variable

N φ . The corresponding scaling ansatz [ 34 ]isthen n N φ − 1 G

(η) =

( η )

η ≡ κ with

the scaling function

const

for η → 0

G (η) = (8.4)

η ( 1 − φ)/φ for η 1

The resulting scaling behavior of n follows as,

⎧
⎨

1 / N for κ 0

N φ − 1 n ∝ for κ → 0 (8.5)

⎩

κ ( 1 − φ)/φ

κ

for 1

The gyration radius in direction perpendicular to the surface, R g ⊥ (η) , has the form

aN ν G g ⊥ (η) R g ⊥ (η) = . One may determine the form of the scaling function G g ⊥ (η)

aN ν so that from the fact that for κ< 0 one has R g ⊥ ∼ G g ⊥ = const. In the opposite

G g ⊥ (η) ∼ η − ν/φ . In result

η

limit, 0the N -dependence drops out and

aN ν

η ≤

for 0

R g ⊥ (η) ∝ (8.6)

κ − ν/φ for η 0

The gyration radius in direction parallel to the surface has similar scaling represen-
tation, R g (η) = aN ν G g ( η ) aN ν and

κ<

0 the gyration radius R g ∼

.Againat

G g = η

const. At 0 the chain behaves as a two-dimensional self-avoiding walk

aN ν 2 , where

(SAW), i.e., R g ∼ 4 denotes the Flory exponent in two dimen-

sions. In result, the scaling function behaves as

ν 2 = 3 /

G g (η) = η (ν 2 − ν)/φ ,

at η 0.

Thus