in Flory-Huggins Lattice Theory Tejraj M. Aminabhavi' and Ramachandra H. Baiundgi Karnatak University, Dharwad-5.30 003, India While teaching the thermodynamics of polymer solutions to first-year polymer-chemistry students, i t occurred to us that most of the polymer textbooks (1-6) do not deal with the derivation for the function, AH,i., in a more convincing way. Instead, we suggest the following simple treatment. In calculating the enthalpy of mixing, it is assumed that the distribution of molecules or polymer segments is not influenced by the enthalpy of mixing. This quantity is thus given by the difference between the enthalpies of solution, H12, and the enthalpies Hall and H022 of the pure components Since the size of the macromolecule is several thousand times that of solvent molecules, it is divided into a number of equivalent segments so that the properties of solution can be treated by the usual thermodynamics. Furthermore, an interaction energy tij exists between every interacting species i and j ; each unit thus contributes 0.5cij, and every unit is surrounded by the average number of nearest neighhors, Z, which is the coordination number of a lattice. However, it should he mentioned here that Z , which expresses the number of contacts hetween a segment and its neighhors, is not the same for both polymer segments and solvent molecules. T o derive relevant relations, let us now assume that the term ~j is weighted by a factor A;? denoting the number of contacts between the indicated pairs of segments. The values assigned to them depend, of course, on the model and procedure adopted. With this modification, the enthalpy of the solution Hlz is given as (2) H12 = .,,All + ( 2 2 4 2 + ( ~ 4 1 2 If N1 and N2 represent the number of sites occupied by the solvent molecules and polymer segments, respectively, then we may define the enthalpies of pure states as
and
definitions of Aij's in terms of coordination number. ?bus,
ZN,= 2A,, +A,,
(5)
ZN,= 2A,, +A,,
(6)
and Substitution of eqs 2-6 into eq 1leads to
Here, Ar is the average energy gain per contact. Furthermore, recalling Flory-Huggins interaction parameter x = ZArlkT (where k T represents the thermal energy term), it can be shown that At is actually a measure of Gihbs' free energy and not of the enthalpy. Consequently, x also contains an entropy contribution, which is often found to depend on concentration. The parameter A12in eq 7 may now be defined on similar grounds as that of All and A22 Here, ni and pi (= niVilZ:=l niVJ represent, respectively, the number of moles and volume fractions of the ith component in the solution; Viand NA, are, respectively, the molar volume and Avogadro's number. I t may he noted that this rendition rests on the assumption of random intermixing of polymer segments and solvent molecules. Inserting eq 8 into eq 7 leads to Finally, using the statistical expression for ASmi., the traditional Flory-Huggins relation for Gihhs' free energy of mixing would be
Acknowledgment TMA is extremely grateful to Paul J. Flory, Stanford University, for the constructive comments on this topic. Literature Cited (1) Flory, P. J. "Principle8 of Polymer Chemistry": Corndl: Ithaea, NY, 1967. 12) Billmover. F. W. "Textbook of Polvmer Science". 2nd ed.: Wilev: New York. 1971.
Occupation of lattice sites by equal-size solvent molecules and polymer segments leads us to the phenomenological .".m,..,".
I
Author to whom correspondence should be addressed.
16) Allcock, H. R.; Lampe, F. W. "Contemmrari Polymer Chemistry": Prentiee~Hall:
Englewood Cliff8, NJ, 1981.
Volume 63 Number 7 July 1986
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