Polymer entropy elasticity

Solutions of rigid polymer molecules (e.g., poly-/)-phenylene terephthalate) may also exhibit extrudate swelling because they too are entropy elastic, molecules exit the capillary in a fairly oriented state and become randomly oriented downstream. 

So far the micro-mechanical origin of the Mullins effect is not totally understood [26, 36, 61]. Beside the action of the entropy elastic polymer network that is quite well understood on a molecular-statistical basis [24, 62], the impact of filler particles on stress-strain properties is of high importance. On the one hand the addition of hard filler particles leads to a stiffening of the rubber matrix that can be described by a hydrodynamic strain amplification factor [22, 63-65]. On the other, the constraints introduced into the system by filler-polymer bonds result in a decreased network entropy. Accordingly, the free energy that equals the negative entropy times the temperature increases linear with the effective number of network junctions [64-67]. A further effect is obtained from the formation of filler clusters or a... 

Eq. (13.37) shows that the modulus of a rubber increases with temperature this is in contrast with the behaviour of polymers that are not cross-linked. The reason of this behaviour is that rubber elasticity is an entropy elasticity in contrast with the energy elasticity in "normal" solids the modulus increases with temperature because of the increased thermal or Brownian motion, which causes the stretched molecular segments to tug at their "anchor points" and try to assume a more probable coiled-up shape. 

Ortiz, C., and Hadziioannou, G. (1999). Entropie elasticity of single polymer chains of poly(methacrylic acid) measured by atomic force microscopy. Macromolecules, 32, 780-787. 

Gels usually consist of small amount of polymer as a network and a lai amount of solvent. Therefore when we discuss the dynamics erf polymer gels, we are tempted to deal with these Is from the stand point of the dynamics of polymer solutions. However, since the polymer chains in a gel are connected to each other via chemical bonds and/or some kinds of sj cific interaction, sudi as, hydrogen bonding or hydrophobic interaction, the gel has to be treated as a continuum. In addition, gels behave as an assembly of springs due to the entropy elasticity of polymer chains between the crosslink points. Therdbre, the dynamics of polymer gels is well described in terms of the theory of elasticity... 

The aim of any grafting is to increase the rubber efficiency, i.e. the ratio between the gel content and the rubber content, and to enable the rubber particles to bond to the polystyrene phase in order to ensure the transmission of external forces from the energy-elastic phase to the entropy-elastic phase. The graft polymer acts as emulsifier and stabilizes the dispersed rubber particles in the two-phase system. 

The kinetics of swelling is successfully described as a collective diffusion process. Tanaka et al. (Tanaka et al. 1973) developed a theory for the dynamics of polymeric gels. They realized that the polymer chains are cormected by chemical bonds and a gel has to be treated as a continuum. In addition, the network behaves as an assembly of springs due to their entropy elasticity. 

Values for tte internal variabtes in thetmodynamic, internal equilibriwn are generally uniquely defined by the values for the external variables. For instance, in a simple, thermomechanical system (i.e. one that reacts mechanically solely volume-elastically) the equilibrium concentrations of the conformational isomers are uniquely described by temperature and pressure. In this case the conformational isomerism is not explicitly percqitible, but causes only overall effects, for example in the system s enthalpy or entropy. Elastic macroscopic effects may, however, occur when the relationship between internal and external variables is not single-valued. Then the response-functions of the system diverge or show discontinuities. The Systran undergoes a thermodynamic transformation. The best-known example of sudi a transformation based on conformational isomerism is the helix-coil transition displayed by sonte polymers in solution. An example in the scdid state is the crystal-to-condis crystal transition discussed in this paper. The conditions under which such transformations occur are dealt with in more detail in Sect 2.2. 

The moduli of elasticity determined by stress / strain measurements are generally much lower than the lattice moduli of the same polymers (Table 11-3). The difference is to be found in the effects of entropy elasticity and viscoelasticity. Since the majority of the polymer chains in such polymer samples do not lie in the stress direction, deformation can also occur by conformational changes. In addition, polymer chains may irreversibly slide past each other. Consequently, E moduli obtained from stress/strain measurements do not provide a measure of the energy elasticity. Such E moduli are no more than proportionality constants in the Hooke s law equation. The proportionality limit for polymers is about 0.l%-0.2% of the... 

The deformation characteristic of more ductile polymer materials at ambient temperatures like most thermoplastics or all elastomers is highly non-linear, e.g., either mostly viscoelastic or entropy-elastic or a combination of both. Compared to concepts of LEFM relatively rarely used for polymer materials different concepts of non-linear elastic firacture mechanics such as elastic-plastic fracture mechanics (EPFM) or post-yield fracture mechanics (PYFM) are somewhat widely applied, therefore. One of the most important concepts of EPFM is the J integral concept. Notwithstanding the J integral is primarily defined to be valid... 

The polymers used in thermoforming must be of fairly high relative molecular mass since the heated sheet must be form stable. The significant property is melt elasticity. The elastic effects are produced by entropy-elastic forces between physical cross-links (molecular entanglements). The polymers used include polystyrene, ABS, acrylics, polycarbonate, PVC, polypropylene, and linear potyethylene. 

Entropy Elasticity. Small stresses effect large (reversible) deformations. This is displayed by gases and polymers, values of E are low. 

The number of possible shapes a polyethylene molecule can now be calculated easily by assuming that each bond must be in one of these three rotational positions. Then the statistics of the molecule is based on the conformational entropy per rotatable bond of this type, k In 3, where k is Boltzmann s constant (see Sect. 2.2.4 and Fig. A.5.4). Per mole of bonds k is replaced by R, the gas constant. For a molecule with n rotatable bonds, Scontamation = n x 9.2 J K mol. The calculation at the bottom of Fig. 1.37 reveals that the different conformations possible for a macromolecule with 20,000 flexible backbone bonds and three rotational isomers per bond are more than astronomical. This large amount of disorder (entropy) is at the root of the special behavior of polymers. It permits melting of the macromolecule without breaking into small parts, and is the reason for entropy elasticity (viscoelasticity and... 

The analyses of several polymers by dynamic mechanical analysis, DMA, are described in Sect. 4.5 in connection with the brief description of the DMA equipment. It was observed in such experiments that neither the viscosity nor the modulus are constant, as is assumed for the discussion of energy and entropy elasticity, outlined in Sects. 5.6.4 and 5.6.5, respectively. One finds a stress anomaly when the elastic limit of a material has been exceeded and plastic deformation occurs. Other deviations have the stress depend both on strain and rate of strain. Finally, a time anomaly exists whenever the stress/strain ratio depends only on time and not on the stress magnitude. 

Vulcanization effectively fixes the ends of flexible polymer chains to form three-dimensional networks. By this way, the stress relaxation of stretched polymer chains can be avoided, and the high entropy elasticity of the rubber can be produced. Moreover, the fixed chain ends also increase both melting points and glass transition temperatures of short flexible chains. 

The entropy elasticity of real polymer coils in a good solvent is... 

Thus, in contrast to lattice moduli, moduli of elasticity obtained from stress/strain measurements are not measures of the energy elasticity, because of the effects of entropy-elasticity and viscoelasticity. Moduli of elasticity have more the character of being solely proportionality constants in a Hooke-type law. The proportionality limits are 0.05 % extension for steel and 0.1-0.2% for polymers. Above these so-called proportionality... 

A significant decrease in the modulus of elasticity is observed at the glass-transition temperature and at the flow temperature because of the increasing influence of entropy-elasticity with rising temperature. Within each physical state, however, only a relatively small variation of the modulus of elasticity with polymer structure is observed (Table 11-3). 

The augend in (8.1) represents the pairwise repulsion energy of chain links in the self-consistent field approximation and the addend represents the polymer chain elastic energy determined via entropy (F =-TS) of its conformation in RW statistics. 

The Vc and Me values for crosslinked polymer networks can also be evaluated from stress-strain diagrams on the basis of theories for the rubber elasticity of polymeric networks. In the relaxed state the polymer chains of an elastomer form random coils. On extension, the chains are stretched out, and their conformational entropy is reduced. When the stress is released, this reduced entropy makes the long polymer chains snap back into their original positions entropy elasticity). Classical statistical models of entropy elasticity affine or phantom network model [39]) derive the following simple relation for the experimentally measured stress cr ... 

The first four chapters, making up the fundamental part, contain reviews of the latest knowledge on polymer chain statistics, their reactions, their solution properties, and the elasticity of cross-linked networks. Each chapter starts from the elementary concepts and properties with a description of the theoretical methods required to study them. Then, they move to an organized description of the more advanced studies, such as coil-helix transition, hydration, the lattice theory of semifiexible polymers, entropy catastrophe, gelation with multiple reaction, cascade theory, the volume phase transition of gels, etc. Most of them are difficult to find in the presently available textbooks on polymer physics. 

In other words, the free-energy cost incurred due to interaction between the chains exceeds the contribution due to the entropy elasticity of the chain (Figure 3). In the case of the widely spaced, (or mushroom ) conformation, the height of the polymer layer above the substrate is independent of the grafting density of the chains (number of attached chains per unit area). In the brush conformation, however, the height scales as the cube root of the grafting density (Figure 4), a, which itself is equivalent to 1/d. ... 

Because of the entropic origin, the above property is called the entropy elasticity. It is not limited to Ganssian chains. Any chain that has a finite size, inclnding ideal chains and real chains, has this elasticity. By the same reason, a rubber is elastic. A rubber is a cross-linked polymer. A partial chain between two cross-links behaves elastically, giving rise to the elasticity of the material as a whole. 

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