The exponent of two originates from the dimensionality of the geometries, which has
been defined as log 10 (S)/log 10 (L). Such a scaling law reflects the self-similarity of the
geometrical shape. Therefore, on the basis of the self-similarity and the scaling law,
scientists could estimate the size of the irregular geometry, such as the length of
British coast, or the volume of the floating cloud. In those cases, the dimensionality is
often fractional, and the corresponding subject is called the fractal (Mandelbrot 1983 ).
Similarly, the scaling law of the coil size with respect to the chain length reflects
the self-similarity nature of a random polymer coil constituted by the Kuhn segments.
In fact, irrespective of the length of the sub-molecules beyond the Kuhn segment, we
always have the scaling relationship of the coil size to the number of sub-molecules
asR 0 /n0.5. Such a scaling analysis of chain conformations is crucial for us to
understand the real-chain conformation statistics.
In the following sections, some examples will be introduced, which apply
scaling analysis in the conformation statistics of more realistic polymer chains. In
these chains, the inter-chain interactions (as well as the long-range monomer-
monomer interactions in the single chain) or the external restrictions are consid-
ered. In practice, on the basis of the ideal-chain model, we first consider the single
chain with the interactions of volume repulsion, followed by discussing its inter-
penetration into other chains in a concentrated solution. Subsequently, we will
consider the single chain with the effect of volume exclusion and the inter-chain
attraction. The effect of Columbic interactions and multi-chain interpenetration will
be discussed later on. Finally, we will introduce the conformation statistics of
polymer chains deformed under the external restriction.
4.2 Single-Chain Conformation in Polymer Solutions
4.2.1 An Introduction of Polymer Solutions
Polymer solutions are normally homogeneous mixtures of polymers and small
molecules. Most practically useful polymers are in a certain sort of mixtures, and
polymer solutions are the basic prototype to understand the polymer mixture.
Fig. 4.1 Illustration of the
measurement of the square
with an areaSand a
corresponding lengthLvia a
square ruler with an area
pand a lengthr
44 4 Scaling Analysis of Real-Chain Conformations