Lattice-cluster theory

Figure 3a shows the mean-field predictions for the polymer phase diagram for a range of values for Ep/Ec and B/Ec. The corresponding simulation results are shown in Fig. 3b. As can be seen from the figure, the mean-field theory captures the essential features of the polymer phase diagram and provides even fair quantitative agreement with the numerical results. A qualitative flaw of the mean-field model is that it fails to reproduce the crossing of the melting curves at 0 = 0.73. It is likely that this discrepancy is due to the neglect of the concentration dependence of XeS Improved estimates for Xeff at high densities can be obtained from series expansions based on the lattice-cluster theory [68,69].

Lue L, Prausnitz JM. Structure and thermodynamics of homogeneous-dendritic-polymer solutions computer simulation, integral-equation, and lattice-cluster theory. Macromolecules 1997 30 6650-6657. 

A. Lattice Cluster Theory of Glass Formation in Polymer Melts... 

The lattice cluster theory (LCT) for glass formation in polymers focuses on the evaluation of the system s configurational entropy Sc T). Following Gibbs-DiMarzio theory [47, 60], Sc is defined in terms of the logarithm of the microcanonical ensemble (fixed N, V, and U) density of states 0( 7),... 

A wide variety of theories have been developed for polymer solutions over the later half of the last century. Among them, lattice model is still a convenient starting point. The most widely used and best known is the Flory-Huggins lattice theory (Flory, 1941 Huggins, 1941) based on a mean-field approach. However, it is known that a mean-field approximation cannot correctly describe the coexistence curves near the critical point (Fisher, 1967 Heller, 1967 Sengers and Sengers, 1978). The lattice cluster theory (LCT) developed by Freed and coworkers (Freed, 1985 Pesci and Freed, 1989 Madden et al., 1990 Dudowicz and Freed, 1990 Dudowicz et al., 1990 Dudowicz and Freed, 1992) in 1990s was a landmark. 

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... 

Foreman K W and Freed K F 1998 Lattice cluster theory of multicomponent polymer systems chain semiflexibility and specific interactions Adv. Chem. Phys. 103 335... 

Recent theoretical efforts to address the phase behavior of associating polymer blends have involved self-consistent field theory (SCFT) and lattice cluster theory (LCT). The SCFT framework was applied to enumerate all possible linear reaction products for blends of self-complementary and heterocomplementary telechelics. Mesophase regions were identified. Dudowicz and Freed reformulated the LCT to model solvent-telechelic polymer blends and identified several trends, including an enhancement of miscibility as the molar mass of the associating polymer was increased [92, 93]. This trend was explained by unbalanced entropy and enthalpy changes that occur with increasing chain length. 

Dudowicz J, Freed KF (2012) Lattice cluster theory of associating polymers. I. Solutions of linear telechelic polymer chains. J Chem Phys 136(6) 064902... 

Lattice-Cluster Theory Considering Molecular Geometry... 

Dudowicz J, Freed KF (1991) Effect of monomer structure and compressibility on the properties of multicomponent polymer blends and solutions 1. Lattice cluster theory of compressible systems. Macromolecules 24 5076-5095... 

In an effort to address some of the FH and gas-lattice models simplifications, the lattice cluster theory (LCT) was developed (Dudowicz et al. 1991 Dudowicz... 

Buta et al. (2001) tested the Monte Carlo approach for the lattice cluster theory to derive the thermodynamic properties of binary polymer blends. They considered the two polymers to have the same polymerization indices, i.e., M = 40, 50, or 100. The results confirm that this lattice cluster theory had a higher accuracy compared to the Flory-Huggins theory and the Guggenheim s random mixing approximation. However, some predictions for the specific heat were found to be inaccurate because of the low order cutoff of the high temperature perturbative expansion. 

The H-F Eq. 18.6 has two parts the (j)-dependent configurational entropy derived from the lattice model without free volume and the enthalpic part taken from the Hildebrand s theory of regular solutions (Shinoda 1978 Reichart et al. 1997 Maranas et al. 1998). More recent version of Eq. 18.6 was used for the interpretation of SANS data, and it will be discussed in reference to the lattice cluster theory (LCT) (Freed and Dudowicz 2005). 

Freed et al. developed the lattice cluster theory (LCT) specifically to account for diversity of segmental structures, affecting blend miscibility, viz., the critical point [ c, cj)c, chain swelling [T ], as well as the scale and intensity of composition fluctuations (Freed and Bawendi 1989 Foreman and Freed 1997 Freed and Dudowicz 1998, 2005 Dudowicz et al. 2002). As an example, several monomeric and polymeric stmctures are shown in Fig. 18.15, and LCT predictions of the phase behavior the authors are discussed in details. 

The phase diagrams of PO-type random copolymers may be more complex than those for homopolymers with different, but uniform, molecular structure. Thus, similar to SLCT, but simplified further, is the basic lattice cluster theory (BLCT). The version was developed for the random copolymers or random and block copolymer systems. 

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