Equilibrium Elastic Properties

The reservations about determining equilibrium moduli and compliances for unloaded polymers, occasioned by slow approaches to elastic equilibrium, apply also to filled materials, but by making measurements at rather long times (or even dynamic measurements at low frequency, at elevated temperatures) the modulus or compliance can be obtained with sufficient accuracy to study its dependence on filler content and size. For noninteracting rigid fillers, the shear modulus appears to be independent of the particle size and increases with volume fraction of filler in accordance with an empirical equation of Eilers and van Dijk -  

Here Geo is the equilibrium modulus of the unfilled polymer, j is the volume fraction of filler, and / , is a maximum volume fraction corresponding to close packing, which may be between 0.74 and 0.80. For t 0.70, this equation is equivalent to the result of a theoretical formulation by van dcr Pocl (which can be evaluated only numerically) relating the shear and bulk moduli of a composite with spherical particles to the shear and bulk moduli and Poisson s ratios of the two component materials. The derivation of van dcr Poel has been corrected and simplified by Smith.For a hard solid in a rubbery polymer, the ratio of the shear moduli is so large that the result is insensitive to its magnitude. An example is shown in Fig. 14-13 for data of Schwarzl, Brcc, and Nederveen for nearly monodisperse sodium chloride particles of several different sizes embedded in a cross-linked polypropylene ether. Extensive comparisons of data with equation 18 have been made by Landcl, - - - who has also employed an alternative relation  

In Fig. 14-13, shear moduli measured at low temperature where the polymer is essentially in the glassy state arc also plotted, and are observed to increase linearly with (/ . In this case, also, agreement is obtained with the theory of van der Poel with a ratio of shear moduli of 8.4 (filler/polymer) and Poisson s ratio of 0.25 and 0.5 respectively for the two phases. Measurements of the bulk modulus Ke for a poly- 

Shear storage modulus G at 1 Hz for a cross-linked polypropylene ether containing sodium chloride particles with sharp size distributions in the ranges shown, plotted against volume fraction of particles at two temperatures where G should be close to Ge and Gg respectively. Curves calculated from theory of van der Poel.  

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Since the excellent work of Moore and Watson (6, who cross-linked natural rubber with t-butylperoxide, most workers have assumed that physical cross-links contribute to the equilibrium elastic properties of cross-linked elastomers. This idea seems to be fully confirmed in work by Graessley and co-workers who used the Langley method on radiation cross-linked polybutadiene (.7) and ethylene-propylene copolymer (8) to study trapped entanglements. Two-network results on 1,2-polybutadiene (9.10) also indicate that the equilibrium elastic contribution from chain entangling at high degrees of cross-linking is quantitatively equal to the pseudoequilibrium rubber plateau modulus (1 1.) of the uncross-linked polymer. 

A considerable number of experimental studies, as well as theoretical developments, have been done on the equilibrium elastic properties of regular model silicone networks in absence of pendant chains. The goal of most of these studies has been to test quantitatively the molecular basis of the theory of rubber elasticity. One of the major concerns has been the influence of topological interactions between chains on elastic properties of the networks. However, despite the considerable amount of experimental work, there is still considerable debate concerning the validity and applicability of different models. 

Abstract This chapter deals with the mechanical properties of single polymer chains, aggregates, and supramolecular complexes. The topics discussed cover a broad range from fundamental statistical mechanics of the equilibrium elastic properties of single polymer chains to details of the behavior of binding pockets in... 

Treloar, L. R. G., Physics of Rubber Elasticity , 3rd edn.. Clarendon Press, Oxford, 1975. Presents the main developments in the field of the equilibrium elastic properties of rubber, and the associated theoretical background. 

A solid, by definition, is a portion of matter that is rigid and resists stress. Although the surface of a solid must, in principle, be characterized by surface free energy, it is evident that the usual methods of capillarity are not very useful since they depend on measurements of equilibrium surface properties given by Laplace s equation (Eq. II-7). Since a solid deforms in an elastic manner, its shape will be determined more by its past history than by surface tension forces. 

More recently, Smith et al. have developed another model based on spontaneous curvature.163 Their analysis is motivated by a remarkable experimental study of the elastic properties of individual helical ribbons formed in model biles. As mentioned in Section 5.2, they measure the change in pitch angle and radius for helical ribbons stretched between a rigid rod and a movable cantilever. They find that the results are inconsistent with the following set of three assumptions (a) The helix is in equilibrium, so that the number of helical turns between the contacts is free to relax, (b) The tilt direction is uniform, as will be discussed below in Section 6.3. (c) The free energy is given by the chiral model of Eq. (5). For that reason, they eliminate assumption (c) and consider an alternative model in which the curvature is favored not by a chiral asymmetry but by an asymmetry between the two sides of the bilayer membrane, that is, by a spontaneous curvature of the bilayer. With this assumption, they are able to explain the measurements of elastic properties. 

What we would like to do is use these thermodynamic properties to calculate an equilibrium elastic moduli. The bulk modulus is by definition the constant of proportionality that links the infinitesimal pressure change resulting from a fractional change in volume (Section 2.2.1). In colloidal terms this becomes... 

For coarse-grained models of linear biopolymers—such as DNA or chromatin— two types of interactions play a role. The connectivity of the chain implies stretching, bending, and torsional potentials, which exist only between directly adjacent subunits and are harmonic for small deviations from equilibrium. As mentioned above, these potentials can be directly derived from the experimentally known persistence length or by directly measuring bulk elastic properties of the chain. 

SORPTION OF SOLVENT MIXTURES IN ION EXCHANGE RESINS INFLUENCE OF ELASTIC PROPERTIES ON SWELLING EQUILIBRIUM AND KINETICS... 

A typical evolution of equilibrium mechanical properties during reaction is shown in Fig. 6.1. The initial reactive system has a steady shear viscosity that grows with reaction time as the mass-average molar mass, Mw, increases and it reaches to infinity at the gel point. Elastic properties, characterized by nonzero values of the equilibrium modulus, appear beyond the gel point. These quantities describe only either the liquid (pregel) or the solid (postgel) state of the material. Determination of the gel point requires extrapolation of viscosity to infinity or of the equilibrium modulus to zero. 

The equilibrium (relaxed) elastic properties of polymers in the rubbery state display two very important features ... 

The relationship between the structure of the disordered heterogeneous material (e.g., composite and porous media) and the effective physical properties (e.g., elastic moduli, thermal expansion coefficient, and failure characteristics) can also be addressed by the concept of the reconstructed porous/multiphase media (Torquato, 2000). For example, it is of great practical interest to understand how spatial variability in the microstructure of composites affects the failure characteristics of heterogeneous materials. The determination of the deformation under the stress of the porous material is important in porous packing of beds, mechanical properties of membranes (where the pressure applied in membrane separations is often large), mechanical properties of foams and gels, etc. Let us restrict our discussion to equilibrium mechanical properties in static deformations, e.g., effective Young s modulus and Poisson s ratio. The calculation of the impact resistance and other dynamic mechanical properties can be addressed by discrete element models (Thornton et al., 1999, 2004). 

A detailed analysis of the theoretical concepts of equilibrium elasticity and its role in the stability of various objects is presented by Kitchener [25], Given bellow are the simplest equations of the modulus of the equilibrium elasticity permitting to elucidate the main dependences of the elasticity properties on surface activity and surfactant concentration as well as on film thickness. 

Under dynamic conditions, where equilibrium between the surface and the film bulk cannot be realised, some specific elasticity properties are expressed. This is Marangoni s effect. Assuming that under such conditions there is an equilibrium only in some parts between the film bulk and its surface, it is possible to employ Eq. (7.6) for the material balance to calculate the modulus of elasticity. Hence, instead of the whole film volume, only the zone where equilibrium with the film surface is established, should be considered. The faster the process of film thinning, the smaller this volume is and the larger the modulus of film elasticity. In the limiting case, when it is completely impossible to achieve equilibrium between the film bulk and its surface, the elasticity of the adsorption surfactant layers takes place. 

The mechanism of the equilibrium elasticity acts until it is possible to provide a surfactant re-partition between the exterior and interior of the film. In a NBF such a repartition is not possible and this mechanism of elasticity ceases to act. The elasticity properties of bilayer films, in which the hydrodynamic and adsorption processes are characterised with normal time of relaxation, are due to Marangoni effect in the insoluble adsorption layers. That is why stable foams with black films are very sensitive to different local disturbances (heating, vibration, etc.). 

Our first non-Newtonian liquids are solutions or melts of a polymer (Figure C4-4). In equilibrium the polymer strands tend to form more or less spherical coils. However, when the liquid is sheared, the coils are stretched and tend to become aligned. This stretching increases their energy, and when the shear is removed, they rebound . So the fluid has elastic properties in addition to viscous ones. An important property of the polymer in solution is its relaxation time. This is a measure for the time that it takes to rebound. It is of the order of nanoseconds for small molecules, of seconds for polymers in processing equipment, and of centuries in construction materials. Yes, these last ones also flow or creep . 

The strain energy generated by placing a sphere in a hole of different size in a continuous medium depends on the elastic properties of the medium, specifically the Young s modulus ( ) and Poisson s ratio (a). Since, for all liquids. Young s modulus is zero and Poisson s ratio 0.5, the strain energy of substitution into a melt should be zero. This yields the following relationship between the equilibrium constants for reactions (1), Kq, and... 

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