Polymer Physics

challenge due to a path memory to avoid any self-intersection. Performing com-
puter simulations on powerful facilities, scientists have figured out a scaling
relationship of SAWs. With fixed step lengthsn, SAW exhibits the total possible
path number as (q1)nng^1 , whereq1isthe connective constant(its practical
value is slightly smaller thanq1) andgisthe critical exponentrelated only to the
dimensionality. The mean-square end-to-end distance of a self-avoiding chain
follows a scaling relationship

<R^2 saw>n^2 n (4.15)

where for 3D,n¼3/5; for 2D,n¼3/4; and for 1D,n¼1 (Madras and Slade 1993 ).
For the three-dimensional self-avoiding walks, the critical exponent of the
polymer coil is 3/5, which is larger than the critical exponent of the ideal chain
(1/2). This implies that the volume exclusion of the polymer chain leads to coil
expansion. Such an expansion makes chain conformation deviate from its most
probable state, causing a recovery force originated from the conformational
entropy. Therefore, the single coil could not expand unlimitedly, and there exists
a thermodynamic balance between the energy gain of volume exclusion and the
entropy loss of chain conformation.
Flory proposed a mean-field treatment to calculate the above energy gain and
entropy loss. The repulsive energyErep is mainly originated from two-body
interactions, and each chain unit in the single coil feels the repulsion from the
other chain units with an internal concentration

Cint

n

Rd

(4.16)

whereRis the coil size of linear polymers, anddis the dimensionality. Since the
repulsive energy for each pair of chain units is comparable with the thermal
fluctuationkT, each chain unit contains the repulsive energykTn/Rd, and the total
repulsive energy fornchain units is scaled as

ErepkT

n^2

Rd

(4.17)

On the other hand, according to the Boltzmann relationship and the Gaussian
distribution of chain conformations, the conformational entropy loss is proportional
tokT(R/R 0 )^2. The total conformation entropy loss of the single coil is thus scaled as

EelkT

R^2

n

(4.18)

Under the thermodynamic balance, the total free energy change is

F¼ErepþEel (4.19)

50 4 Scaling Analysis of Real-Chain Conformations