The Flory-Rehner Equation

The entropy change caused by mixing polymer and swelling agent. The entropy change from this source is positive and favors swelling. 

The heat of mixing of polymer and swelling agent, which may be positive, negative or zero. The Flory-Rehner equation may be written 

The important point is that the original Flory-Rehner equation contains no thermoelastic front factor term, yet simple analogy with the mechanical equations shows that one must exist, even for unfilled elastomers. The need becomes more obvious on considering reinforced elastomers, where equilibrium swelling is greatly reduced. In an empirical manner, equation (10.6) may be modified to 

The equilibrium swelling theory of Flory and Rehner (76) treats simple polymer networks in the presence of small molecules. The theory considers forces arising from three sources  

The value of equation (9.92) here lies in its complementary determination of the quantity n [see equation (9.4) for simplicity]. Both equations (9.4) and (9.92) determine the number of elastically active chains per unit volume (containing, implicitly, corrections for front factor changes). By measuring the equilibrium swelling behavior of an elastomer (x values are known for many polymer-solvent pairs), its modulus may be predicted. Vice versa, by measuring its modulus, the swelling behavior in any solvent may be predicted. 

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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... 

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). 

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. 

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. 

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... 

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,... 

It is remarkable, that the prediction of a volume phase transition of nonionic polymer gels was done by Dusek and Patterson by discussing the Flory-Rehner equation (Dusek and Patterson 1968)). They predicted that a strong concentration dependence of x facilitates the occurrence of phase transition in polymer gels. The phase transition should be analogous to the coil-globule transition of a single polymer chain. 

The equilibrium degree of swelling or the gel volume of shows a large change in response to external conditions. Only when the volume change is discontinuously and the coexistence of two phases is experimentally proofed, we should use the term phase transition . By applying the stability criteria on the Flory-Rehner equation one can calculate the conditions of phase transition (binodal and spinodal curve, critical point). For details see e.g. (Onuki 1993). 

Evaluating the other two derivatives occurring in Eq. (3.125) after substitution for AGmix and AGei from Eqs. (3.120) and (3.121), respectively, we obtain (cf. Eq. (3.48)) the Flory-Rehner equation (Flory and Rehner, 1943,1950) ... 

Another use of swelling tests on rubbers quite unrelated to chemical resistance is to estimate the degree of cure, higher swelling indicating a lower cross-link density. The cross-link density is estimated from the Flory-Rehner equation ... 

If the Xi23 value is considered as a criterion of existence of a maximum of equilibrium swelling of polymer in the mixed solvent, a maximum of Q should correspond to the minimum of %i23. For practical use of Eq. [6.2.2] it is necessary to know parameters 5 13,5 23, and Xn- The values X13 and X23 can be determined by the Flory-Rehner equation, with data on swelling of a crosslinked elastomer in individual solvents 1 and 2. The evaluation of the Xn value can be carried out with use of results of the experimental evaluation of vapor pressure, viscosity and other characteristics of a binary mixture. ... 

The values of parameters Xn and X23 were calculated from the Flory-Rehner equation, using data on swelling of crosslinked polymer in individual solvents, the values Xn X23 > and %i23 were estimated from Eqs. [6.2.3-6.2.5). Then equilibrium swelling of elastomer in SL was calculated from the equation similar to Flory-Rehner equation, considering the mix-... 

In the second variant of evaluation of M data on the temperature dependence of volume fraction of polymer, in the swollen gel were used. Results of calculations of from the Flory-Rehner equation in some cases also gave negative values. The evaluation of... 

The calculation of equilibrium swelling of SPU-1 (after its deformation up to 700 % and subsequent unloading) in a given solvent was carried out using the Flory - Rehner equation. In our calculations we used values of the parameter of interaction between solvents... 

The crosslink density of the hydrogels was calculated using Flory-Rehner equation. The experimental procedure involved weighing of PAAM before immersing into water. The swollen samples were taken out and dried at room temperature after equilibrium was reached. Final weight of the swollen sample was recorded and compared to that of the original weight. The Flory-Rehner equation [19] is explained as ... 

A word must be said about the use of equations (8) and (9). In each case, the modulus of the single networks determines Nj and N2 for equation (8) and the Flory-Rehner equation determines N and N2 for equation (9) from the equivalent single network. In this manner, several effects existing in the single networks are minimized, such as physical crosslinks, incomplete crosslinking, and branching. Thus, differences from theory will emphasize new effects due to sequential IPN formation. 

Figure 1 shows values of N(swelling),(y-axis) determined via the Flory-Rehner equation vs N(stoichiometry), (x-axis) determined by the average quantity of DVB in the IPN. While the IPN s are characterized by two pairs of numbers in Fig. 1, the single networks shown are identified by a single number representing the quantity of DVB used. The fit for the IPN s is extremely poor. 

While the Flory-Rehner equation is not expected to fit homo-IPN data, it is of interest to note where the data lies with respect to the theoretical diagonal. In Figure 1, the data of Millar, and Shibayama and Suzuki tend to lie above the line, while Thiele and Cohen and Siegfried et al s. data tend to lie below it. Yet the Thiele-Cohen equation, applied to the same data in Fig. 2, showed a marked improvement, pulling the data towards the theoretical line whether it originally lay above or below the Flory-Rehner line in Fig, 1, The modified Thiele-Cohen equation, employed in Fig. 3,... 

The average molecular mass of the network chains between two neighboring network nodes Me and the crosslink density Vc of crosslinked polymer networks could be calculated by means of the Flory-Rehner equation [38] on the basis of swelling measurements. If 02 is the volume fraction of the polymer in the swollen system and Vi the molar volume of the solvent, the following equation holds for the crosslink density Vc in a tetra-functional network ... 

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