Flory-Rehner network theory

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

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

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 ... 

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 ... 

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. 

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) ... 

Let us consider a polymeric network that contains solvent, usually called a polymeric gel. There are several types of gels. A previously cross-linked polymer subsequently swollen in a solvent follows the Flory-Rehner equation (Section 9.12). If the network was formed in the solvent so that the chains are relaxed, the Flory-Rehner equation will not be followed, but rubber elasticity theory can still be used to count the active network segments. 

Most of the above cross-linked polymers were considered in the dry state, although the Flory-Rehner theory (Section 9.12) made use of equilibrium swollen gels in the evaluation of the cross-link density. Generally, a polymeric gel is defined as a system consisting of a polymer network swollen with solvent. It must be understood that the solvent is dissolved in the polymer, not the other way around. 

For a polymer network at equilibrium swelling one can use the Flory-Rehner theory for the free energy of the network to obtain the following expression for R ... 

By microscopic measurements of the degree of swelling (Flory-Rehner theory) or rather by compression measurements (rubber elasticity theory (RET)) the network parameters, such as cross-linking density or the molecular weight of network chains Me can also be determined vc = p2lMc) (Flory et al. 1976 James and Guth 1943 Wall 1942, 1943, 1951 Hermans 1947). 

Early in 1944 a symposium on the Theory of Long-range Elasticity was held at the Cleveland Meeting of the American Chemical Society. Flory s plenary lecture, Network Structure and the Elastic Properties of Vulcanized Rubber , was published in Chemical Reviews [21]. It summarized all the work done through 1944 and presented new comparisons between theory and experiment. The classic case of the elongational modulus and its dependence on swelling is shown in Figure 8 from that paper (Fig. 2.3). It is based on the Flory-Rehner equation ... 

As discussed previously, equilibrium swelling is reached when the net gel swelling pressure is zero. For a non-ionic gel prepared in the bulk state, the Flory-Rehner theory represents total gel swelling pressure as the sum of the mixing and elastic pressures [46]. Total gel swelling pressure may be written as follows for networks crosslinked in solution, by combination of equations (2) and (4) [13] ... 

Theoretically, the cross-linking degree is taken into consideration in the following way, based on the Flory-Rehner theory [92]. The coexistence curve may be calculated from the chemical potentials of the linear chain inside and outside of the network. The chemical potential of the hnear chain is given by ... 

The concept of affine deformation is central to the theory of rubber elasticity. The foundations of the statistical theory of rubber elasticity were laid down by Kuhn (JJ, by Guth and James (2) and by Flory and Rehner (3), who introduced the notion of affine deformation namely, that the values of the cartesian components of the end-to-end chain vectors in a network vary according to the same strain tensor which characterizes the macroscopic bulk deformation. To account for apparent deviations from affine deformation, refinements have been proposed by Flory (4) and by Ronca and Allegra (5) which take into account effects such as chain-junction entanglements. 

Several theories have been proposed to calculate the molecular weight between crosslinks in a hydrogel membrane. Probably the most widely used of these theories is that of Flory and Rehner [5]. This theory deals with neutral polymer networks and assumes a Gaussian distribution of polymer chains and tetrafunctional crosslinking within the polymer network. 

There is only an alternative way of determining crosslink density by a non empirical method, using the theory of equilibrium swelling of a network in a solvent (Flory and Rehner, 1943). This method, usual in the domain of rubbers, needs the knowledge of a polymer solvent interaction coefficient proportional to (8p — 8S)2, which is not very easy to determine accurately. Furthermore, damaging by swelling stresses and the need to work at elevated temperatures complicate the analysis seriously for the usual thermosets... 

An early model based on crosslinked rubbers put forward by Flory and Rehner (1943) assumed that chain segments deform independently and in the same manner as the whole sample (affine deformation) where crosslinks were fixed in space. James and Guth (1943) then described a phantom-network model that allowed free motion of crosslinks about the average affine deformation. The stress (cr) described from these theories can be described in the following equations ... 

Statistical network models were first developed by Flory (Flory and Rehner, 1943, Flory, 1953) and Stockmayer (1943, 1944), who developed a gelation theory (sometimes referred to as mean-field theory of network formation) that is used to determine the gel-point conversions in systems with relatively low crosslink densities, by the use of probability to determine network parameters. They developed their classical theory of network development by considering the build-up of thermoset networks following this random, percolation theory. 

Flory PJ (1977b) The molecular theory of mbber elasticity. Contemp Top Polym Sci 2 1-18 Flory PJ, Rehner J Jr (1943a) Statistical mechanics of cross-linked polymer networks. I. Rubberlike elasticity. J Chem Phys 11 512-520... 

In the 1940s, Flory and Rehner (Flory and Rehner 1943a, b Flory 1953) were the first to formulate the theory of swelling of network structures . Here the change of ambient conditions can be represented by a change of the Gibb s free energy AF (see also Chap. 3). 

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 ... 

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