Polymer Physics

Dme 1 ¼

@DHemix

@N 1

T

@DSemix

@N 1

(4.37)

The excess mixing enthalpy

@DHmixe

@N 1

¼kTK 1 f^22 (4.38)

The excess mixing entropy

@DSemix

@N 1

¼kC 1 f^22 (4.39)

whereK 1 andc 2 are the empirical enthalpy and entropy parameters, respectively.
f 2 is the volume fraction of polymers. Therefore, when

Dme 1 ¼kTf^22 ðK 1 C 1 Þ¼ 0 (4.40)

K 1 andc 1 will compensate with each other, leading to the pseudo-ideal state of the
single-chain system. Flory defined the temperateyK 1 /c 1 T. Accordingly,

K 1 C 1 ¼C 1 ð

y

T

 1 Þ (4.41)

To approach the pseudo-ideal state, one can either adjust the temperature to the
theta temperature, or use thetheta solvent.
Flory’s analysis focuses on the thermodynamic interactions between polymers,
and defines the theta point at the critical polymer concentration for phase separation
(equal to the critical concentration of chain units within a single chain upon collapse
transition), similar to the Boyle point of the non-ideal gas. We can perform Virial
expansion on the osmotic pressure of dilute polymer solutions, as

P¼kTðA 1 CþA 2 C^2 þA 3 C^3 þ::::::Þ (4.42)

Here, the first Virial coefficient is

A 1 ¼

1

MN

(4.43)

which reflects the colligative property of ideal solutions. The second Virial coeffi-
cient is

A 2 ¼C 1 ð 1 

y

T

ÞFðxÞ

n^2

v 1

(4.44)

4.2 Single-Chain Conformation in Polymer Solutions 57