Helfand-Tagami theory

Figure 4.3. Verification of the molecular weight dependence of the interfacial tension coefficient, as predicted by the Helfand-Tagami theory (see Eq 4.6)...

K. W. Foreman and K. F. Freed (1998) Lattice cluster theory of multicomponent polymer systems Chain semiflexibility and speciflc interactions. Advances in Chemical Physics 103, pp. 335-390 K. F. Freed and J. Dudowicz (1998) Lattice cluster theory for pedestrians The incompressible limit and the miscibility of polyolefin blends. Macromolecules 31, pp. 6681-6690 E. Helfand and Y. Tagami (1972) Theory of interface between immiscible polymers. 2. J. Chem. Phys. 56, p. 3592 E. Helfand (1975) Theory of inhomogeneous polymers - fundamentals of Gaussian random-walk model. J. Chem. Phys. 62, pp. 999-1005... 

For high molecular weight (M — °o) binary blends, the Helfand and Tagami theory predicts that in binary blends (i) the interfacial thickness, A/ is inversely proportional to the interfacial tension coefficient,v , the product, A/ v being independent of the thermodynamic interaction parameter, X, (ii) the surface free energy is proportional to (iii) the chain-ends of both polymers concentrate at the interface (iv) any low molecular... 

The Helfand-Tagami lattice theory predicts that there is reciprocity between the interfacial tension coefficient and the interfacial thickness, and the product, Al , is independent of the thermodynamic binary interaction parameter, Furthermore, the theory led to the conclusions that (i) the surface free energy is proportional to... 

Interphase theories Immiscible blends density profile compatibilized compatibilized - semi experimental, p vs. ((). Helfand Tagami, 1971, 1972 Helfand Sapse, 1975 Noolandi, 1984 Utracki, 1991 1992... 

The Helfand-Tagami lattice theory predicts that 1. Product, Al Vi2 = const. 2. Surface free energy is proportional to 3. Polymeric chain-ends concentrate at the interface ... 

We have reviewed the theories of polymer-polymer interfaces. We began by presenting the early semiempirical attempts. Then, we discussed in some detail the microscopic theories of polymer interfaces, with emphasis on the theories of Helfand and coworkers as well as on subsequent theories. One should emphasize here the significant influence of the original Helfang-Tagami theory on the field of polymer interfaces. The expression for the interfacial tension in the limit of infinite molecular weights, y = pobk T (40), has been utilized extensively for... 

E. Helfand, Y. Tagami, Theory of the interface between immiscible polymers, J. Polym. Sci. B-Polym. Phys. 34(12) 1947 1952, 1996 (reprinted from J. Polym. Sci., Polym. Lett. 9 741-746, 1971)... 

Helfand E and Tagami Y 1972 Theory of the interfaee between immiseible polymers J. Polym. Sc/. Polym. Lett. 9 741 Helfand E and Tagami Y 1972 J. Chem. Phys. 56 3592... 

Helfand, E., and Y. Tagami. 1972. Theory of the interface between immiscible polymers. J Chem Phys 57 1812. 

Let us consider a molten, immiscible, binary blend of polymers A and B, without compatibilizer. Helfand and Tagami [1971], Helfand [1975], Roe [1975], and Helfand and Sapse [1975] have developed a quantitative lattice theory of the interphase that twenty years later still provides good basis for understanding. 

It is noteworthy that Leibler s (see Eqs 4.20 and 4.22) and Noolandi s theories (see Eq 4.28) predict that the product V Al depends on the binary interaction parameter Thus, the reciprocity between and A1 predicted by Helfand and Tagami for binary systems is not expected to exist in compatibilized binary blends. 

Roe (1975) developed a quasicrystalline lattice model for conditions where Xi2 Xcr (where Xcr is the critical value of the interaction parameter at the phase separation) and for Xn Xcr Xcr- Under the first conditions (high immiscibility), the theory predicted a proportionality between Vj2 and Xn whereas under the secmid (near the phase separation), a proportionality between Vi2 and xn was predicted. By contrast with the previously summarized Helfand and Tagami predictions. Roe s theory indicates that the product V12A/ should be proportional to... 

Lattice theory predicts that the density profile across the interface follows the exponential decay function (Helfand and Tagami 1971, 1972) ... 

Helfand, E. and Tagami,Y. (1971) Theory of the interface between immisdble polymers. J. Potym. Sd., Pdym. Lett.,... 

By definition, the free energy of mixing of a polymer blend interphase is positive, or else molecular mixing would continue to completion, and at equilibrium the interphase would vanish. Helfand and Tagami (10,11) developed a mean-field theory of polymer interfaces, or interphases, as they are now called. They were particularly interested in the equilibrium composition and interphase density across the interphase, the interfacial tension and thickness (see Section 12.3.7.2), and conformation of the polymer chains making up the interphase. 

The theory was originally compared to three polymer pairs, namely PS/PMMA PMMA/poly(n-butyl methacrylate), PnBMA and PnBMA/poly(vinyl acetate), PVA. The calculated interfacial tension agreed exactly with the experimental value for PnBMA/PVA it compared well for PMMA/EhiBMA and differed by 50% for PS/PMMA. Helfand and Tagami suggested that, if z is too large, then the characteristic interfacial thickness is too small for the mean-field theory to be appropriate. The theory has been widely used to estimate the interfacial tensimi in many different polymer-polymer systems with acceptable success. 

However, the theory cannot be used if the asymmetry between A and B is too severe. Helfand and Sapse [29] refined the theory of Helfand and Tagami so as to remove the restrictive approximation of property symmetry of the two polymers. For a Gaussian random walk in a mean field, they obtained ... 

Tagami [200,201] extended the theories of Helfand and coworkers to the case of compressible nonsymmetric polymer mixtures. A slight decrease in the predicted interfacial tension was found, due to the presence of finite compressibility of the polymers. This tendency was particularly apparent in the case of nearly symmetric polymer pairs, when the intersegmental interactions are of ncmlocal nature. The results reduce to the results of Helfand and Sapse in the appropriate limits. However, the resulting equations are much too complicated, although the results do not differ significantly fi-om those predicted by (41). 

For pol3nner-polymer interfaces, the first substantiated theory was developed in a number of works by Helfand and Tagami. " It describes the state of the interphase region between two immiscible phases, polymers A and B, and relies on the existence of repulsive forces acting between dissimilar molecules and related to the thermodynamic interaction parameter, %ab- The theory assumes limited mixing of components of two phases... 

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