Polymer Physics

whereI 1 andI 2 are the ionization energies,Ris the distance,a 1 anda 2 are the
polarizabilities. Normally,I 1 andI 2 are close to each other, andRdoes not change
significantly; therefore, they can be expressed collectively as a constantk. Then we
have

B 12 ¼ka 1 a 2 (4.5)

In polymer solutions, one can split a polymer chain into small monomers with
their sizes comparable to that of the solvent molecules. If we consider only the
contact pairs among monomers and solvent molecules, the dissolution process can
be expressed by

½Šþ 1  1 ½Š! 2  2 21 ½Š 2 (4.6)

where 1 represents the solvent, and 2 represents the monomer. Upon dissolution, the
pair interactions of monomers (Guggenheim 1952 ) and the pair interactions of
solvent molecules (Mandelbrot 1983 ) are disassembled, to form two pairs of mixing
interactions (Mandelbrot 1983 ; Guggenheim 1952 ) between the monomers and the
solvent. This process is similar to the reaction between H 2 and Cl 2 that one hydrogen
molecule reacts with one chlorine molecule to form two hydrogen chloride
molecules. The energy change in this process defines the mixing energy. In the
theory of solution thermodynamics, such a way to treat the mixing interactions is
known asquasi-chemical approximation(Guggenheim 1952 ). A mixing interaction
parameterBis used to describe the dissolution process, as given by

BB 12 

B 11 þB 22

2

(4.7)

Mainly concerning the contributions of dispersion forces, one can have

B¼kða 1 a 2 

a^21 þa^22

2

Þ¼

k

2

ða 1 a 2 Þ^2 ¼

1

2

ðB^111 =^2 B^122 =^2 Þ^2  0 (4.8)

Corresponding to (4.8), the macroscopic change of mixing energy is

DUmix

Vmix

¼f 1 f 2

DE 1

V 1

 1 = 2



DE 2

V 2

"# 1 = 2 2

(4.9)

whereV 1 ,V 2 , andVmixare the molar volumes of the solvent, monomer and the
mixtures, respectively;f 1 andf 2 are the volume fractions of the solvent and
monomer, respectively;DE 1 andDE 2 are the corresponding molar evaporation
heats. Assuming that the total volume remains constant during the mixing, i.e.,
DVmix¼0, (4.9) is called theScatchard-Hildebrand equation(Hildebrand 1936 ).
DE 2 /V 2 represents the evaporation heat of unit volume of the liquid, and is defined

4.2 Single-Chain Conformation in Polymer Solutions 47