Flory-Rehner

Table 21.1 shows the recipe for obtaining NR specimen without filler. The specimen was cured at 145°C for 30 min. The cross-link density was measured by allowing it to swell in toluene and calculated to be 1.47 X 10 mol cm using the Flory-Rehner equation. The average number of... 

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. 

Here V is the total volume and D is the specific volume of the polymer. Substitution and rearrangement yields the final form of the Flory-Rehner model [6],... 

As mentioned, the Flory-Rehner model describes the situation in which crosslinks are introduced in the dry state. Peppas and Merrill derived a model reported by Peppas and Barn-Howell [3] which accounts for the introduction of crosslinks in the swollen state as in the case of solution polymerization. The final... 

As pointed out earlier, an elastomer cross-linked above its gel point will not dissolve in a solvent, but will absorb it and swell. The swelling will continue until the forces of swelling balance the retroactive forces of the extended chains of the network. The cross-link density can then be calculated from the degree of swelling using the Flory-Rehner equation ... 

Crosslink density of PU films. The PU films were subjected to swelling tests in dimethylformamide. The crosslink densities of the films were thereafter estimated from equilibrium swelling data using a modified version of the Flory-Rehner equation. The swelling tests as well as the calculation of crosslink density are described in detail in a previous paper (6). 

The degree of crosslink density in vulcanized elastomeric gum compounds can be estimated by applying the Flory-Rehner network theory [226]. But, their... 

Discontinuous Transition of Gels Within the Flory-Rehner... 

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... 

Discontinuous Transition Within the Flory-Rehner Phenomenological Theory... 

The Flory-Rehner phenomenological theory [9,10] has beei most widely used to analyze the volume phase transition of gels. This theory, extended [11] to take into account the concentration dependence of the polymer-solvent interaction parameter x, can predict basic features of the phase transition. 

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. 

Other relationships between % and an observable physical property such as osmotic pressure [20, 43], freezing point depression of polymer [20, 52] or solvent [20, 53], and gas liquid chromatography [46-54], were established in like fashion. The relationship determined for swelling of cross-linked polymer to thermodynamic equilibrium in excess liquid has particular significance for the subject of this review. It is given here in the form of the Flory-Rehner equation. 

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 ... 

An important methodology established early on is an equilibrium volume swelling approach and application of the Flory-Rehner equation [134, 135]. While this is satisfactory for simple one component elastomer networks it is not particularly convenient experimentally and is of dubious value for multi-component elastomers. 

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 ... 

Since vr0jvT varies only slowly with vr0, while v is a rather sensitive function of the inverse swelling ratio, the result of substituting vr into the Flory-Rehner equation is a vapp which is too large, but which still reflects rather faithfully any changes in vr0 brought about by varying the network chain density. Thus we may write as an approximation... 

While one of the < 2 values is equal to the peak position of envelope I, the other value cannot be determined accurately from Figure 1 due to the uncertainty of the peak position and low intensity of the broad peak (envelope II) Yeo has thus calculated (3) the interaction (X) parameter based on the modified Flory-Rehner equation (16) ... 

Elastic term (Oeias) the earliest elasticity mode based on Gaussian chain distributions (Flory—Rehner and Flory theories) improved elasticity model based on constrained junction theory an elastic expression that accounts for the limits of elongation Prausnitz et al. [36] Saito et al. [42]... 

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. 

The Flory-Rehner equation (2,13) was used to interpret the swelling of the polyurethane networks in pure dioxane in order to calculate, the molecular weight between crosslinks. Here, we set qQ= 1 and x the network-solvent interaction parameter, constant and assumed equal to 0.35 (3). Networks with degrees of swelling (q ) of 6.41, 11 and 21.1 yielded Mc values of 7,700,... 

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