Flory—Rehner theory

The swelling of a network is described by the Flory-Rehner theory (38) which gives... 

The structure of hydrogels that do not contain ionic moieties can be analyzed by the Flory Rehner theory (Flory and Rehner 1943a). This combination of thermodynamic and elasticity theories states that a cross-linked polymer gel which is immersed in a fluid and allowed to reach equilibrium with its surroundings is subject only to two opposing forces, the thermodynamic force of mixing and the retractive force of the polymer chains. At equilibrium, these two forces are equal. Equation (1) describes the physical situation in terms of the Gibbs free energy. 

Peppas and Merrill (1977) modified the original Flory-Rehner theory for hydrogels prepared in the presence of water. The presence of water effectively modifies the change of chemical potential due to the elastic forces. This term must now account for the volume fraction density of the chains during crosslinking. Equation (4) predicts the molecular weight between crosslinks in a neutral hydrogel prepared in the presence of water. 

The present article attempts to clarify the nature of the discontinuous transition of gels. First, in Sect. 2 we give an outline of the fundamental aspect of the volume phase transition on the basis of the Flory-Rehner theory of gels, with special attention to how the discontinuous transition comes about within the phenomenological treatment Then, in Sect 3 previous experimental results... 

Fig. 5. Linear swelling ratio calculated for neutral NIPA gels on the basis of the Flory-Rehner theory. The structural parameters used in the calculation were determined from the preparation condition of (a) gel A, and ( >) gel B. The curve (c) represents the critical transition. (Reproduced with permission from Ref 20)...

The phase coexistence observed around the first-order transition in NIPA gels cannot be interpreted by the Flory-Rehner theory because this theory tacitly assumes that the equilibrium state of a gel is always a homogeneous one. Heterogeneous structures such as two-phase coexistence are ruled out from the outset in this theory. Of course, if the observed phase coexistence is a transient phenomenon, it is beyond the thermodynamical theory. However, as will be described below, the result of the detailed experiment strongly indicates that the coexistence of phases is not a transient but rather a stable or metastable equilibrium phenomenon. At any rate, we will focus our attention in this article only on static equilibrium phenomena. 

Peppas and Merrill [65] modified the Flory-Rehner theory to describe the preparation of hydrogels in the presence of water to obtain the following relationship ... 

If the objective of the experiment is to estimate the network chain density by the statistical theory, and if swelling is resorted to as a means for alleviating the above experimental difficulties, then the measurement of stresses becomes redundant the theory of equilibrium swelling of Flory and Rehner (127,128) makes an estimate of the desired network chain density possible with only the knowledge of the equilibrium swelling volume. The usual procedure is to calculate the volume fraction of polymer in the swollen rubber (ar) under the assumption that the filler does not swell. The Flory-Rehner theory then yields, for a network containing no sol fraction ... 

For determining the cross-linking degree by Eq. 12 or 13, the x must be known for the polymer. Even if x is determined for a polymer system, Flory-Rehner theory has been questioned due to the assumption of a Gaussian chain distribution (see Sec. II.C). Thus Peppas et al. [66] have made a modification by introducing the terms 1 — (molecular weight of poly-... 

Neuburger NA, Eichinger BE. Critical experimental test of the Flory-Rehner theory of swelling. Macromolecules 1988 21 3060-3070. 

AGei (Flory 1944 Flory and Rehner 1943a, 1943b). A basic premise of the Flory-Rehner theory is the separability and additivity of both contributions ... 

Flory-Rehner theory assumes that the mixing part of the chemical potential of the network is the same as the mixing part of the uncross-Unked polymer with an infinite molecular weight ... 

D. Patterson in 1968 based on an analysis of Flory-Rehner theory. It took ten years for the phenomenon to be experimentally observed after prediction. It was found by T. Tanaka that, when a critical amount of an organic solvent was added to a water-swollen poly(acrylamide) gel, the gel collapses. Many gels of synthetic and natural polymers have been studied. Subsequent experiments showed that a volume phase transition (swelling/collapse) could also be brought about by changes in other environmental parameters such as pH, ionic strength, and temperature. 

Nonionic hydrogels can be analyzed with Flory-Rehner theory to determine the quantitative parameters described previously. In Flory-Rehner analysis, hydrogel swelling is dictated by the thermodynamic free energies of mixing and... 

As expected,preferential sorption was observed, with the essential distinction of molar volumes Vi and Vj of components of the mixed solvent (hexane-DBP, DOS-DEP). For swelling of the crosslinked elastomer SCN-26 in the mixture of components, having similar molar volumes Vj and Vj (e.g., amyl acetate-dimethyl phthalate) the preferential sorption of components of SL is practically absent. Influence of Vj and V2 and the influence of double interaction parameters on the sorption of binary liquids by crosslinked elastomers was examined by the method of mathematical experiment. Therewith the set of equations describing swelling of crosslinked elastomers in binary mixture, similar to the equations obtained by Bristow from the Flory-Rehner theory and from the work of Schulz and Flory, were used ... 

Calculations were made according to the Flory-Rehner theory, Eq. (2) Reprinted from Haraguchi et al. [62], Copyright 2007, with permission of ACS... 

Swelling properties of water-soluble microgels can be modified by copolymerization with hydrophobic comonomers. In such cases, the desaiption of the sweUing ratio as a function of temperature by the Flory-Rehner theory was only partially satisfying and fails for higher comonomer contents. ... 

A hydrogel network is obtained through the physical or chemical interactions that occur between polymer chains. According to the Flory—Rehner theory, fundamental structural parameters of hydrogels are (Fig. 10.2) (Hu et ah, 2009 Flory, 1953) ... 

The solubility parameter of a new polymer may be determined by any of several means. If the polymer is cross-Unked, the solubility parameter may be determined by swelling experiments (7). The best solvent is defined for the purposes of the experiment as the one with the closest solubility parameter. This solvent also swells the polymer the most. Several solvents of varying solubility parameters are selected, and the cross-linked polymer is swelled to equilibrium in each of them. The swelling coefficient, Q, is plotted against the various solvent s solubility parameter, the maximum defining the solubility parameter of the polymer. The theoretical extent of swelling is predicted by the Flory-Rehner theory on the basis of the cross-link density and the attractive forces between the solvent and the polymer (see Section 9.12). 

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