Lattice chain theory

Miscibility of the polymer with a given solvent is well explained in the mean-field theory. The theory is an extension of the lattice fluid theory originally developed to explain the miscibility of two low-molecular-weight liquids. Flory pioneered the application of the mean-field lattice fluid theory to polymer solutions. The simplest version of this lattice chain theory is generally referred to as... 

The theory of relaxation spectra in polarized luminescence for various dynamic models of a flexible polymer chain has been developed by several groups of workers. Wahl has proposed a theory for the model of Gaussian subchains. The authors and coworkers used dynamic chain models consisting of rigid or deformable elements with continuous visco-elastic mechanism of mobility and rotational-isomeric lattice chain... 

Dubois-Violette, Monnery, Geny et al. and Taran have developed a theory of PL for the rotational-isomeric tetrahedral lattice chain model A rela-... 

Figure 2.17. Surface tension of linear alkanes as a function of (a) temperature and (b) redueed temperature. Cp Cg, etc. refer to alkanes with 1, 2,. .. CH2 groups per chain. Lattice hole theory, redrawn after Schlangen et al. (J. Phys. Chem. 100 (1996) 3607.)...

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... 

A relaxation spectrum similar to that of Fig. 4.2 is obtained for the diffusional motion of a local-jump stochastic model of IV+ 1 beads joined by N links each of length b, if a weak correlation in the direction of nearest neighbor links is taken into account for the probability of jumps (US). On the other hand, relaxation spectra similar to that of the Rouse theory (27) are obtained for the above mentioned model or for stochastic models of lattice chain type (i 14-116) without the correlation. Iwata examined the Brownian motion of more realistic models for vinyl polymers and obtained detailed spectra of relaxation times of the diffusional motion 117-119). However, this type of theory has not gone so far as to predict stationary values of the dynamic viscosity at high frequencies. 

We also note that the line joining rt and ry contains an arbitrary number of links and that a weight exp( — AJ) is associated with each link. Formula (11.3.6) is a direct consequence of these observations, and, for a given lattice, it establishes an exact correspondence between an interacting-chain theory and a zero-component field theory. 

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... 

The previous discussion, whose microscopic origins are in a lattice gas theory of mixtures, is applicable to mixtures of compact, molecules with comparable dimensions. In order to treat mixtures of long, chain, flexible polymers in small molecule solvents, the mean-field theory described above must be modified to take into account that the polymers and the solvent molecules are not of the same size and that the polymers are flexible macromolecules. The resulting Flory-Huggins free energy per monomer, ffn, is written... 

Since Chapter 6 presents detailed discussion of Simha-Somcynsky lattice-hole theory, only an outline is provided here. The theory was derived for spherical and chain molecule fluids [Simha and Somcynsky, 1969 Somcynsky and Simha, 1971]. The model lattice contains a volume fraction y of occupied sites and h= —y of nonoc-cupied sites, or holes. From the Helmholtz free energy, F, the S-S equation of state was obtained in the form of coupled equations ... 

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... 

Sanchez and Lacombe supposed that in a binary polymer blend, free volume occupied Nq lattice sites, and the bulk polymer density p = NKN + Nq), where N = ILNiri and r, was the chain length of /th fraction, then they developed the lattice fluid theory to calculate Helmholtz free energy (Sanchez and Lacombe 1974 Sanchez 1978), as given by... 

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 Flory-Huggins theory uses the lattice model to arrange the polymer chains and solvents. We have looked at the lattice chain model in Section 1.4 for an excluded-volume chain. Figure 2.1 shows a two-dimensional version of the lattice model. The system consists of si,e sites. Each site can be occupied by either a monomer of the polymer or a solvent molecule (the monomer and the solvent molecule occupies the same volume). Double occupancy and vacancy are not allowed. A hnear polymer chain occupies N sites on a string of N-l bonds. There is no preference in the direction the next bond takes when a polymer chain is laid onto the lattice sites (flexible). Polymer chains consisting of N monomers are laid onto empty sites one by one until there are a total tip chains. Then, the unoccupied sites are filled with solvent molecules. The volume fraction of the polymer is related to rip by... 

The decisive advantage of the original Elory-Huggins theory [1] lies in its simplicity and in its ability to reproduce some central features of polymer-containing mixtures qualitatively, in spite of several unrealistic assumptions. The main drawbacks are in the incapacity of this approach to model reality in a quantitative manner and in the lack of theoretical explanations for some well-established experimental observations. Numerous attempts have therefore been made to extend and to modify the Elory-Huggins theory. Some of the more widely used approaches are the different varieties of the lattice fluid and hole theories [2], the mean field lattice gas model [3], the Sanchez-Lacombe theory [4], the cell theory [5], different perturbation theories [6], the statistical-associating-fluid-theory [7] (SAET), the perturbed-hard-sphere chain theory [8], the UNIEAC model [9], and the UNIQUAC [10] model. More comprehensive reviews of the past achievements in this area and of the applicability of the different approaches are presented in the literature [11, 12]. 

When one fits the Hory-Huggins theory to experiment [7], nontrivial dependence of Hory-Huggins interaction parameters on temperature and volume fractions also result, but might have other reasons than those noted above in particular, it is important to take into account the disparity between size and shape of effective monomers in a blend, and also the effects of variable chain stiffness and persistence length [25, 26]). To some extent, such effects can be accounted for by the lattice cluster theories [27-30], but the latter still invokes the mean-field approximations, with the shortcomings noted above. In the present article, we shall focus on another aspect that becomes important for the equation of state for polymer materials containing solvent pressure is an important control parameter, and for a sufficiently accurate description of the equation of state it clearly does not suffice to treat the solvent molecules as vacant sites of a lattice model. In most cases it would be better to use completely different starting points in terms of off-lattice models. 

To describe the microscopic structm-e and thermodynamic properties of a pol5Tner near the interface with a solid lattice, a theory was developed based on the theory of solutions. However, as distinct from a solution, the system analyzed consists of a great number of chains. It is assumed that in the surface layer, various orientations of chain links are realized, which deviates from the isotropic one only in a very narrow region near the interface (approximately 6... 

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