Equilibrium swelling equation

Apart from the filler, the stiffening effect can also arise from the rubber network itself such as the crosslink concentration, as shown in Figure 3.9. The results shown here are based on compound formulations given in Table 3.4. The tensile modulus MlOO increased linearly with increasing crosslink concentration of the rubber network. The crosslink concentration was calculated by the Flory-Rehner equilibrium swelling equation ... 

For many years, the Flory-Rehner equilibrium swelling equation has served to relate the volume fraction of polymer in the swollen mass, Vu to the number of moles of network chains per cm, i i ... 

The final form of the Morton equation for the activated swelling process may be written by expressing the equilibrium swelling radius of the seed particles in terms of initial radius by using Eq. (10). 

B was also obtained from equilibrium-swelling data of selected block copolymers the results showed excellent agreement with the other methods (Equations 7.1 and 7.2)7... 

Our latest efforts have been to concentrate on investigating the architecture of the DIB core. As discussed before, the average number of branches per molecule (B) determined by selective link destmction and equilibrium swelling showed good agreement with the kinetic B (Equation 7.2). However, branching analysis by SEC proved to be a challenge. 

It has been shown in Chapter XI that the force of retraction in a stretched network structure depends also on the degree of cross-linking. It is possible therefore to eliminate the structure parameter ve/Vo) by combining the elasticity and the swelling equations, and thus to arrive at a relationship between the equilibrium swelling ratio and the force of retraction at an extension a (not to be confused with the swelling factor as). In this manner we obtain from Eq. (XI-44) and Eq. (39)... 

This equation calls attention to the well-established inverse relationship between the degree of equilibrium swelling of a series of rubber vulcanizates in a given solvent and the forces of retraction, or moduli, which they exhibit on stretching. The indicated approximate dependence of qm on the inverse three-fifths power of the modulus has been confirmed. 

The equation for the equilibrium swelling degree is more complicated for polyelectrolyte gels—those with ionizable groups—because of the need to include the additional ion-related terms in Eq. (5). The nion term can be substantial... 

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). 

Flory [3] formalized the equation of state for equilibrium swelling of gels. It consists of four terms the term of rubber-like elasticity, the term of mixing entropy, the term of polymer solvent interaction and the term of osmotic pressure due to free counter ions. Therefore, the gel volume is strongly influenced by temperature, the kind of solvent, free ion concentrations and the degree of dissociation of groups on polymer chains. 

Important theoretical and experimental considerations of the use of macromolecular theories for the description of coal network structures have been recently analyzed (1). Relevant equations describing the equilibrium swelling behavior of networks using theories of modified Gaussian distribution of macromolecular chains have been developed by Kovac (2 ) and by Peppas and Lucht (3) and applied to various coal systems in an effort to model the relatively compact coal network structures (1 4). As reported before (1), Gaussian-chain macromolecular models usually employed in the description of polymer networks (such as the Flory... 

The Mayo Lewis equation, using reactivity ratios computed from Eq. 18, will give very different results from the homogenous Mayo Lewis equation for mini-or macroemulsion polymerization when one of the comonomers is substantially water-soluble. Guillot [151] observed this behavior experimentally for the common comonomer pairs of styrene/acrylonitrile and butyl acrylate/vinyl acetate. Both acrylonitrile and vinyl acetate are relatively water-soluble (8.5 and 2.5%wt, respectively) whereas styrene and butyl acrylate are relatively water-insoluble (0.1 and 0.14%wt, respectively). However, in spite of the fact that styrene and butyl acrylate are relatively water-insoluble, monomer transport across the aqueous phase is normally fast enough to maintain equilibrium swelling in the growing polymer particle, and so we can use the monomer partition coefficient. 

This relation has been tested by Morton, who determined values of and y from equilibrium swelling measurements of polystyrene latices. The results obtained were in good agreement with values of and y (54) determined by other methods. Allen (J) measured the swelling of natural rubber latex with methyl methacrylate and found the dependence of swelling on particle size to agree well with the above equation. 

Equation 11 refers to equilibrium swelling conditions. Now, Flory (24) concludes from theoretical considerations that monomer is easily supplied to the polymer particles at the required rate even in the case of monomers which are little soluble in water, such as styrene. That equilibrium swelling is maintained during emulsion polymerization is supported by a comparison of values of the monomer concentrations determined in equilibrium swelling measurements with those found to prevail during polymerization and determined by analysis of reaction kinetics (see below). The results obtained by both methods are plotted in Figure 10. 

The last relation is valid for any preparation concentration, and was obtained using Eq. (7.76). Such universal relations that are independent of the details of the preparation state are useful for predicting the equilibrium swelling in a -solvent from a measurement of the modulus of the network in the dry state (and vice versa). Equation (7.78) has not been tested in a -solvent, but it does describe swelling in concentrated solution, where ideal chain statistics apply (see the small swelling part of Fig. 7.17). 

If the elastomer is swollen to below the equilibrium swelling so that no deswelling will occur upon deformation, the statistical expression for the shear modulus [equation (6-59)] can be readily modified. We define Vr as the ratio of the unswollen volume to the swollen one, which is identical to the volume fraction of polymer in the mixture. The number of network chains per unit volume then becomes NQVr and the mean square end-to-end distance of the network chain is now r02 / Vr2/3. Equation (6-59) then reads... 

Experimental Results. Equation 5 cannot be used to calculate C unless the values of the constants are known. An explicit relation for C is given by equation 6 in which the effect of the rubber pressure has been neglected. Equation 6 has been used to calculate the equilibrium amount of water absorbed by natural rubber (vulcanizate B) from a salt solution containing 10% sodium chloride. The value of A cannot be calculated since the nature and concentration of the impurity in rubber is unknown but its value was estimated from the observed equilibrium swelling of rubber in a saturated salt solution. The value of A was found to be 1.14 x 10" and the calculated value of the concentration of water in the rubber at equilibrium in the 107. salt solution was 0.0140 gm cm which was in excellent agreement with the experimental value of... 

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