Lattice theories

The lattice theory explains the behaviour of polymer solutions sufficiently well for the characterization techniques described here. It is also surprisingly useful (since the theory is for dilute solutions) in explaining the behaviour of polymer blends. 

If the polymer and solvent mix then the free energy of mixing AG ,j, is negative. The magnitude of at constant pressure can be obtained from the second law of thermodynamics in terms of the enthalpy of mixing (Ai/ jJ, the entropy of mixing (AS j,) and temperature (T) by  

The combinatorial entropy of mixing of a polymer molecule and a solvent can also be obtained from a consideration of the statistical distribution of both molecules over a two-dimensional lattice. For an ideal solution this is given by  

It follows that, for ideal solutions, the free energy of mixing is given by  

It is evident there is a similarity in the expressions for the combinatorial entropy and the free energy of mixing for small and large molecules with a solvent, with mole fractions being replaced by volume fractions for polymer solutions. The solution behaviour is dependent on the volume fraction of polymer in solution so it is possible to relate the thermodynamic changes to the molecular weight of the polymer in solution as follows  

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Of particular interest has been the study of the polymer configurations at the solid-liquid interface. Beginning with lattice theories, early models of polymer adsorption captured most of the features of adsorption such as the loop, train, and tail structures and the influence of the surface interaction parameter (see Refs. 57, 58, 62 for reviews of older theories). These lattice models have been expanded on in recent years using modem computational methods [63,64] and have allowed the calculation of equilibrium partitioning between a poly-... 

J. A. Barker, Lattice Theories of the Liquid State Pergamon, New York (1963). 

More fundamental treatments of polymer solubihty go back to the lattice theory developed independentiy and almost simultaneously by Flory (13) and Huggins (14) in 1942. By imagining the solvent molecules and polymer chain segments to be distributed on a lattice, they statistically evaluated the entropy of solution. The enthalpy of solution was characterized by the Flory-Huggins interaction parameter, which is related to solubihty parameters by equation 5. For high molecular weight polymers in monomeric solvents, the Flory-Huggins solubihty criterion is X A 0.5. 

Clearly Fig. 7 must actually have a maximum at high asymmetry since this corresponds to negligible anchor block size and therefore to no adsorption (ct = 0). The lattice theory of Evers et al. predicts this quantitatively [78] and is, on preliminary examination, also able to explain some aspects of these data. From these data, the deviation from power law behavior occurs at a number density of chains where the number of segments in the PVP blocks are insufficient to cover the surface completely, making the idea of a continuous wetting anchor layer untenable. Discontinuous adsorbed layers and surface micelles have been studied theoretically but to date have not been directly observed experimentally [79]. 

Bittner ER (2006) Lattice theory of ultrafast excitonic and charge-transfer dynamics in DNA. 

Fig. 17 B/E-p dependence of the critical temperatures of liquid-liquid demixing (dashed line) and the equilibrium melting temperatures of polymer crystals (solid line) for 512-mers at the critical concentrations, predicted by the mean-field lattice theory of polymer solutions. The triangles denote Tcol and the circles denote T cry both are obtained from the onset of phase transitions in the simulations of the dynamic cooling processes of a single 512-mer. The segments are drawn as a guide for the eye (Hu and Frenkel, unpublished results)...

Mean-field lattice theory proved to be capable of predicting the phase behaviour of the ternary block copolymer polyethylene-b-poly(propylene oxide)-b-poly(ethylene oxide), PE-b-PPO-b-PEO in the selective solvent water [164], The ethylene block is known to be highly hydrophobic, and its hydrophobicity does not depend strongly on temperature. The difference in hydrophobicity between PPO and PEO, on the other hand, is moderate... 

The first term in Eq. (100) may be calculated from the free energies of formation of pure salts and the four linearly additive binary terms may be evaluated from information on the four binary systems. Only the last term cannot be directly evaluated from information on lower-order systems. A useful approximation to P in this term may be made by comparing it with analogous terms in the quasi-lattice theory.13,18 This approximation is... 

The lattice theory deals with rod-like particles which do not have interactions with their neighbors except, of course, repulsions occur when the particles overlap. Above a certain criticsd concentration (V2 ) that depends on the axi d ratio x the theory predicts the system will adopt a state of partial order (biphasic region). Below V2 the system... 

Aspler and Gray (65.69) used gas chromatography and static methods at 25 C to measure the activity of water vapor over concentrated solutions of HPC. Their results indicated that the entropy of mixing in dilute solutions is mven by the Flory-Huggins theory and by Flory s lattice theory for roddike molecules at very nigh concentrations. 

The truncation of the high temperature series in Eq. (4) at order is a vahd concern when applying the theory at low temperatures. This concern also extends to GD theory, which implicitly involves a truncation at order p. At some point, these perturbative treatments must simply fail, but we expect the lattice theories to identify faithfully the location of the entropy crisis at low temperatures, based on numerous previous comparisons between measurements and GD theory. Experience [96] with the LCT in describing equation of state [97] and miscibility [98] data indicates that this approach gives sensible and often accurate estimates of thermodynamic properties over wide ranges of temperatures and pressures. In light of these limitations, we focus on the temperature range above Tg, where the theory is more reliable. 

A ternary system consisting of two polymer species of the same kind having different molecular weights and a solvent is the simplest case of polydisperse polymer solutions. Therefore, it is a prototype for investigating polydispersity effects on polymer solution properties. In 1978, Abe and Flory [74] studied theoretically the phase behavior in ternary solutions of rodlike polymers using the Flory lattice theory [3], Subsequently, ternary phase diagrams have been measured for several stiff-chain polymer solution systems, and work [6,17] has been done to improve the Abe-Flory theory. 

Recently, Chagnes et al. [22] treated the molar conductivity of LiCl04 in y-buty-rolactone (y-BL) on the basis of the quasi-lattice theory. They showed that the molar conductivity can be expressed in the form A = (A°°) — fe c1/3 and confirmed it experimentally for 0.2 to 2 M LiCl04 in y-BL. They also showed, using 0.2 to 2 M LiCl04 in y-BL, that the relation k = Ac = (A°°) c — k cA was valid and that /cmax appeared at cmax = [3(xf00)74fe ]3 where d/c/dc=0. 

Fint is the free energy of non-Coulomb interactions of monomer units. Finl can be expressed, for example, in terms of the Flory-Huggins lattice theory [21]. In the general case, when network is immersed in solvent which includes 1 different components some of which can be polymeric with the degree of polymerization Pi(Pi 1, i = L 2,... k), Fim in the Flory-Huggins theory has the following form [21-22] ... 

Flory-Huggins /u. The Flory-Huggins n value measures the interaction between polymer and solvent (plasticizer). It derives from the so-called lattice theory, which represents a statistical approach to the behavior of polymer molecules in solution (10, 14, 75, 16, 22). The n value may be experimentally determined for any polvmer-plasticizer system (where the plasticizer can dissolve the polymer) by osmotic pressure measurements according to the relation ... 

Thermochemical cycles extend to much less routine applications than those associated with (3.106). As an illustrative example, Sidebar 3.10 summarizes the Bom-Haber cycle, by which a key quantity of ionic lattice theory is obtained from fiendishly indirect thermochemical measurements. 

The model of Marchetti et al. is based on the compressible lattice theory which Sanchez and Lacombe developed to apply to polymer-solvent systems which have variable levels of free volume [138-141], This theory is a ternary version of classic Flory-Huggins theory, with the third component in the polymer-solvent system being vacant lattice sites or holes . The key parameters in this theory which affect the polymer-solvent phase diagram are ... 

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