Gaussian Elasticity Research Articles

Swelling and Elastic Properties of Polyelectrolyte Gels

Ionized poly(acrylic acid) gels were studied both at concentrations close to the concentration of preparation and at swelling equilibrium. In the first experimental condition, the introduction of electrostatic interactions decreases the shear modulus. The addition of salt screens these interactions and allows one to recover the shear modulus of unneutralized gels. The correlation of these effects with light scattering results suggests that they are related to a change of the gel microstructure with electrostatic interactions. The swelling equilibrium of these gels if found to scale like the ratio of the ionization degree to the Debye-Huckel screening parameter with an exponent 6/5. The shear modulus at swelling equilibrium is given by the simple aline deformation law for not too high swelling degrees (< 200). For larger swelling ratios, the shear modulus increases with swelling ratio due to deviations from Gaussian elasticity. These results can be partly explained by a recently proposed model. Finally, the cooperative diffusion coefficient can be measured by kinetics of swelling experiments and its behavior does not follow the predictions of the same model, possibly due to the coupling of cooperative diffusion with the establishment of a Donnan equilibrium

Diffusion of linear deuterated polystyrene chains in crosslinked polystyrene networks

The diffusion of linear deuterated polystyrene (DPS) chains into a cross-linked polystyrene matrix has been studied by secondary ion mass spectrometry (SIMS). The cross-linked matrices were prepared by radiation cross-linking with {gamma}-rays from a {sup 60}Co source. The doses used were 40, 75, and 156 Mrad, which resulted in an approximate network density, N{sub c} (number of monomer units between cross-linking points), of 9,300, 4,900, and 2,400. The homopolymer DPS chains were of molecular weight M{sub w} = 85,000, 104,000, 303,000, and 550,000 (corresponding to degrees of polymerization of N{sub b} = 759, 929, 2,705, and 4,910). Diffusion of the linear chains into the cross-linked matrix was observed at all molecular weights and cross-linking densities studied. Diffusion was significantly slower into the cross-linked systems than into the matrix chains before cross-linking (M{sub w} = 1,030,000). A free energy expression for the system calculated by assuming additivity of Gaussian rubber elasticity and the Flory-Huggins energy of mixing was used in combination with the Hartley-Crank equation to calculate concentration profiles for linear chains diffusing into the matrix. The predictions were then compared with the data obtained by SIMS. The only free parameter used in fitting the data was the relaxed or referencemore » state network volume fraction. Good fits to the data were obtained when the relaxed state of the networks accounted for the free sol chains still present after cross-linking. The diffusion of the linear chains became very slow when N{sub b} {approximately} N{sub c}, but no distinct halt to the diffusion was observed at this linear chain length. The tracer diffusion coefficients for the linear chains, which were measured separately, were found to be independent of the cross-linking density and to scale with M{sub w}{sup {minus}2.0}, in agreement with the repetition model prediction.« less

The elasticity of nematic networks

AbstractNematic rubbers are composed of crosslinked polymer chains with stiff rods either incorporated into their backbones or pendant as side chains. When nematic effects axe strong, such rubbers exhibit discontinuous stress‐strain relationships and spontaneous shape changes. We model such a rubber using Gaussian elasticity theory, including the nematic interaction via a mean field. Results are presented for the cases of uniaxial extension and compression. Under uniaxial extension the rubber can undergo a first order phase transition to a uniaxial nematic phase. Under uniaxial compression first or second order transitions are possible to genuinely biaxial nematic states with biaxial strains. When nematic effects are very small (i.e. T &gt;&gt; Tc where Tc is the nematic‐isotropic phase transition temperature of the rubber) we postulate that the model is a good approximation to a conventional, non‐nematic elastomer, and fit our model to data from an isoprene rubber.

Elasticity of nematic networks and nematic effects in conventional rubbers

Nematic rubbers are composed of cross-linked polymer chains with stiff rods either incorporated into their backbones or pendant as side chains. When nematic effects are strong, such rubbers exhbit discontinuous stress-strain relationships and spontaneous shape changes. Such a rubber is modelled using Gaussian elasticity theory, including the nematic interaction via a mean field. Results are presented for the cases of uniaxial compression and biaxial extension, and unusual phase transitions are obtained. When nematic effects are very small (i.e., T>>T c , where T c is the nematic-isotropic phase transition temperature of the rubber) it is postulated that the model is a good approximation to a conventional, nonnematic elastomer and the model is fitted to data from an isoprene rubber

General conditions governing the formation of cylindrical [formula omitted] microemulsion aggregates

On the basis of classical surface thermodynamics in a general version derived by Melrose, and the Helfrich expression for the bending free energy of an interface, we consider the mechanical and physicochemical conditions which control the formation of cylindrical microemulsion aggregates of water in oil in the presence of an excess water phase (Winsor II case). We find that ultralow interfacial tensions (about 10 −7 N/m or less) have to be attained for cylindrical geometry in the range about the spontaneous curvature in order to obtain spherocylindrical water aggregates in appreciable amounts. According to the generalized Young-Laplace equation for interfaces with bending free energy, this corresponds to an excess pressure of a few N/m 2. Furthermore, mechanical stability of the cylinder-shaped aggregates requires the interfacial tension at very large radii, γ ∝, to fulfill the condition 2k c H 0 2 ⩽ γ ∞ ⩽ ( 8 3 )k c H 0 2 , where k c is the ordinary bending elasticity constant and H 0 the spontaneous curvature. Approximately, this condition means that 1 2 ⩽ k c kT ⩽ 2 3 . In addition, the Gaussian curvature elasticity constant, k c , should be positive and fairly large as compared with k c ; otherwise spheroidal droplets will predominate. It is inferred that elongated spherocylindrical aggregates might account for the bicontinuity of Winsor II and Winsor III microemulsions observed at comparatively low water contents when H 0 differs appreciably from zero.

Assessment of the intrinsic birefringence of partially oriented poly(ethylene terephthalate) yarns

AbstractPartially oriented poly(ethylene terephthalate) yarns (PET POY) have been studied with respect to their birefringence, sonic modulus, and stress‐optical properties in an effort to extract the values for the intrinsic birefringences of the crystalline (Δnc°) and amorphous (Δna°) regions. The data have been analyzed within the framework of both the sonic modulus‐birefringence treatment of Samuels and Dumbleton, and the Gaussian rubber elasticity theory using the usual shrinkage stress‐birefringence relations. The following values have been determined: Δnc° = 0.22 and Δna° = 0.19. The work was undertaken to resolve the discrepancy in the published literature values of these two parameters.

Elasticity and structure of chemically crosslinked polyurethanes

AbstractA combination of electron‐microscopy, light‐scattering, and stress‐birefringence studies on chemically crosslinked polyurethanes point toward the existence of rodlike regions (“bundles”), approximately 3000–8000 Å in length and involving about 5% of the volume, in which molecular orientations are correlated. The elastic behavior of these networks—as indeed that of most rubberlike networks—deviates substantially from the Gaussian behavior. The empirical representation of the data in Mooney‐Rivlin plots yields C1 and C2 constants which depend on the type of imposed strain. It is thus impossible to identify C1 with the Gaussian behavior and C2 with the deviation there‐from. Instead, it is found that the elastic behavior can be adequately described if it is assumed that, as a result of the bundle structure, about 5% of the segments of each chain are not free to assume the normal random‐walk configurations. The determination of the number of chains in the network from the elastic behavior remains ambiguous, however, and the behavior upon swelling is not (yet) adequately reproduced by the theory. It is conceivable that in many cases deviations from Gaussian elasticity behavior may be caused by an intermolecular structuring effect, rather than by various minor deficiencies in the Gaussian model for the single chain statistics or by anisotropic excluded volume effects, as has been proposed in the past. In the present case, the amount of bundle structure, as well as the C2/C1 values, increase with the number of urethane couplings per chain, and this suggests that the interaction of the highly polar urethane couplings is responsible for the structuring. In other networks one often finds a dependence of C2/C1 on the previous history of the sample, which suggests that an accidentally trapped order may be responsible for the elastic behavior.