Flory-Vrij theory

The isothermal crystallization of PEO in a PEO-PMMA diblock was monitored by observation of the increase in radius of spherulites or the enthalpy of fusion as a function of time by Richardson etal. (1995). Comparative experiments were also made on blends of the two homopolymers. The block copolymer was observed to have a lower melting point and lower spherulitic growth rate compared to the blend with the same composition. The growth rates extracted from optical microscopy were interpreted in terms of the kinetic nucleation theory of Hoffman and co-workers (Hoffman and Miller 1989 Lauritzen and Hoffman 1960) (Section 5.3.3). The fold surface free energy obtained using this model (ere 2.5-3 kJ mol"1) was close to that obtained for PEO/PPO copolymers by Booth and co-workers (Ashman and Booth 1975 Ashman et al. 1975) using the Flory-Vrij theory. 

The number of folds n = t - 1. The melting point in the Flory-Vrij theory, as applied to block copolymers, is (Ashman and Booth 1975) ... 

The term ln(r) represents a change of conformational entropy upon cutting. Inserting (4.16) and (4.17) into (4.15), the equilibrium melting point predicted by the Flory-Vrij theory can be calculated. 

Approaches used for crystallization in homopolymers may be used to calculate the change in melting temperature due to finite crystal thickness (Thompson-Gibbs equation), lamellar crystal surface energies (Flory-Vrij theory), and growth rates (kinetic nucleation theory). Details can be obtained from [1]. 

The cell theory of Prigogine has been applied by Clark and Schmidt and by Hicks and Young to benzene + polyphenyl systems and n-alkane mixtures respectively. The Flory, Orwoll, and Vrij theory has also been applied by Hicks and Young to n-alkane mixtures. 

The first successful theoretical approach of an equation of state model for polymer solutions was the Prigogine-Flory-Patterson theory. It became popular in the version by Flory, Orwoll and Vrij and is a van-der-Waals-like theory based on the corresponding-states principle. Details of its derivation can be found in numerous papers and books and need not be repeated here. The equation of state is usually expressed in reduced form and reads ... 

Figure 13.4 compares the melting temperatures of bulk polymers with different chain lengths derived separately by the lattice theory, the Flory-Vrij equation, and Monte Carlo simulations. The dimension of the temperature unit is reduced as Eglik T ). Their agreements are satisfying, although they hold various assumptions. 

Casassa (S3) has modified a theory of the configuration of polymer molecules in solution due to Vrij (84) and applied it to branched molecules. If certain terms neglected in Flory s derivation are retained, Eq. (6.10) becomes ... 

It is not surprising that attempts have been made to derive equations of state along purely theoretical lines. This was done by Flory, Orwoll and Vrij (1964) using a lattice model, Simha and Somcynsky (1969) (hole model) and Sanchez and Lacombe (1976) (Ising fluid lattice model). These theories have a statistical-mechanical nature they all express the state parameters in a reduced dimensionless form. The reducing parameters contain the molecular characteristics of the system, but these have to be partly adapted in order to be in agreement with the experimental data. The final equations of state are accurate, but their usefulness is limited because of their mathematical complexity. 

On the other hand, Sauer and Dee (1994) have demonstrated that, by refining the equation of state used in the gradient theory, by using better experimental data and a different equation of state model, the Flory, Orwoll and Vrij model (the FOV model) (Flory et al. 1964), excellent agreement, at the level of better than 1%, between theory and experiment is obtained. The square gradient coefficient still has to be adjusted somewhat from the mean-field value, but it is still treated as entirely enthalpic in origin, with no temperature dependence. 

Here, the term A includes the contribution of interaction entropy (Floiy 1970). Such a contribution can be miderstood from the concept of compressible free volume in the fluids. When two fluids are mixed with each other, part of molecules of one species enters the free volume of another species, and then the total volume is not a simple addition of the two individual components. Yamakawa made an approximate estimation from the expansion theory (Yamakawa 1971). Prigogine attributed this contribution to a combinatorial contribution of molecular geometry and a non-combinatorial contribution of molecular stmctures, and proposed an equation-of-state theory (Prigogine 1957b). Hory, Orwell and Vrij further considered the contribution of free volume, and employed separate parameters to describe the hard-core volume and surface contacts of chain units (Flory et al. 1964 Flory 1965 OrwaU and Rory 1967). This work makes the equation of state fit better to the e erimental results, and derives the so-called Flory-Orwell-Vrij equation of state for pure polymers, as given by... 

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