Flory-Vrij equation

A plot of Tm l) against 1/7 would give a straight line with an intercept Tra(0, p) given by the Flory-Vrij equation and a gradient having a different interpretation than that from Eq. (2.17). 

The low molecular weight fractions of polymers contain many chain ends, which serve as lattice defects in the large crystal to lower their melting points. The Flory-Vrij equation has also been well verified by the experiments (Mandelkem 2002) and simulations (Hu and Frenkel 2005). 

Fig. 4.9 Melting temperature (T /EJks) of polymers with different chain length. The Solid line is calculated from (4.19) and dashed line is results of Flory-Vrij equation. The Circles are results from Monte Carlo simulations of polymer solutions with a 0.9375 concentration in a 32 cubic lattice for short chains and a 64 cubic lattice for long chains (Hu and Frenkel 2005)...

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. 

This equation can be used to determine ae using the experimentally determined Tm if the other parameters are known. In the Flory-Vrij model, oe is the sum of the free energy of formation from the melt of the amorphous/crystalline interface, (x0, and the free energy of formation from the melt of the amorphous layer, ad (Ashman and Booth 1975 Cooper et al. 1978)... 

The first term in braces represents the surface energy. With Eq. (8.24) we employ bulk PVT properties for evaluation of surface tension. Application of Eq. (8.24) to copolymers also allows for discussion of the composition dependence of surface tension. Patterson and Rastogi [1970] and Slow and Patterson [1971] published a similar approach in terms of the Flory, Orwoll, and Vrij equation of state. 

JAN Janssen, S., Schwahn, D., Mortensen, K., and Springer, T., Pressure dependence of the Flory-Huggins interaction parameter in polymer blends. A SANS study and a comparison to the Floty-Orwoll-Vrij equation of state (experimental data by S. Janssen), Macromolecules,... 

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... 

Jo WH, Lee HS, Lee SC (1998) Temperature and molecular weight dependence of interfacial tension between immiscible polymer pairs by the square gradient theory combined with the Flory-OrwoII-Vrij equation-of-state theory. J Polym Sci B Polym Phys 36 2683-2689... 

The term ln(r) represents a change in conformational entropy upon cutting. Inserting Equations 13.12 and 13.13 into Equation 13.11, 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]. 

Figure 15 Plot of melting temperature as a function of the number of carbon atoms in the chain. Curve A, Flory-Vrij anaysis. Curves B and C, theoretical calculation for equations (18) and (19) with = 1200 and 4600 respectively. , values for...

Table 7. Characteristic Parameters for the Flory, Orwoll and Vrij Equation of State VI-599...

TABLE 7. CHARACTERISTIC PARAMETERS FOR THE FLORY, ORWOLL, AND VRIJ EQUATION OF STATE... 

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. 

Flory-Orwoll-Vrij [1964] The FOV equation of state was developed to correlate properties of macromolecular liquids, sacrificing rigor of mathematical derivation for simplicity. The authors followed Prigogine s derivation up to the configurational partition function and then modified the expression for the intermolecular energy, Eq. The final product is indeed algebraically simple ... 

Reviews of the Equation of State for Polymer Melts There are many publications comparing different equations of state to each other, and a few more ambitious reviews that discuss theoretical fundamentals, derivation, and application, and provide tabulated characteristic reducing parameters and deviation from the experimental volume [Curro, 1974 Zoller, 1989 Rodgers, 1993 Rudolf et al., 1995, 1996 Lambert et al 2000], For example, Zoller [1989] examined the equations of state by Spencer-Gilmore (S-G) [1949, 1950], Flory-Orwoll-Vrij (FOV), Sanchez-Lacombe (S-L), and Simha-Somcynsky (S-S). Large deviations (<0.01 mL/g) were observed for S-G. While the FOV and S-L expressions were useful at low P, the S-S equations of state consistently provided the best representation of data over extended ranges of T and P, with deviations AV< 0.003 mL/g, eomparable to the experimental error for density. 

Flory, P. J., Orwoll, R. A., and Vrij, A., Statistical thennod5mamics of chain molecule liquids 1. An equation of state for normal paraffin hydrocarbons, J. Am. Chem. Soc., 86, 3507-3514 (1964a). 

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 ... 

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. 

For polymer systems, a lot of EOS have been developed (e.g., Flory-Orwoll-Vrij EOS [10], Sanchez-Lacombe EOS (11), Panayiotou-Vera EOS [12], lattice gas EOS [13], group-contribution lattice fluid EOS [14]). In all these equations, the pressure is introduced via empty lattice sites allowing the compressibility of the lattice. 

Redlich-Kwong equation of state and Soave modification Peng-Robinson equation of state Tait equation for polymer liquids Flory, Orwoll, and Vrij models Prigogine square-well cell model Sanchez-Lacombe lattice fluid theory... 

Derive an expression for Hildebrandt solubility parameter as given by Equation (4.7) for the Flory- Orwoll-Vrij EOS summarized as... 

Nine different equations-of-state, EOS theories are described including Flory Orwoll Vrij (FOV) Prigogine Square Well cell model, and the Sanchez Lacombe free volume theory. When the mathematical complexity of the EOS theories increases it is prudent to watch for spurious results such as negative pressure and negative volume expansivity. Although mathematically correct these have little physical meaning in polymer science. The large molecule effects are explicitly accounted for by the lattice fluid EOS theories. The current textbooks on thermodynamics discuss... 

The first equation-of-state theory developed for polymers was due to Flory, Orwoll and Vrij (1964) and Eichinger and Flory (1968), the so-called FOVE model. Each of the components is characterized by p, V and T obtained from data for the pure components. Two interaction terms, denoted Xj2 and Qi2, associated with the enthalpy and entropy of the mixture, are introduced. The equation of state is ... 

Flory PJ, Orwoll RA, Vrij A (1964) Statistical thermodynamics of chain molecule liquids. I. An equation of state for normal paraffin hydrocarbons. J Am Chem Soc 86 3507 3514 Nose T (1976) Theory of liquid liquid interface of polymer systems. Polym J 8 96 113 de Gennes P G (1977) Qualitative features of polymer demixtion. J Phys Lett 38 L441 L443 Ouhadi T, Fayt R, Jerome R, Teyssie Ph (1986) Molecular design of multicomponent polymer systems. 9. Emulsifying effect of poly(alpha methylstyrene b methlyl methacry late) in poly(vinylidene fluoride)/poly(alpha methylstyrene) blends. Polym Commun 27 212 215... 

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