Entropy-Elasticity

Moduli of elasticity also depend strongly on the environmental conditions. In many cases, water acts as a plasticizer and lowers the modulus of elasticity by increasing chain mobility. Since the diffusion of water into the material is time dependent, the modulus of elasticity also varies with time. For example, a polyamide sample in a dry state has a modulus of elasticity of = 2700 N/mm. This decreases to 1700 N/mm in humid air and to 860 N/mm after 4 months in air. 

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 behavior of weakly cross-linked rubber was described in Section 11.1 as entropy-elastic. If this material is deformed, the chains are displaced from their equilibrium positions and brought into a state which is en-tropically less favorable. Because of the weak cross-linking, the chains are unable to slip past one another. On relaxation, the chains return from the ordered position to a disordered one the entropy increases. The phenomenon can be described in various ways. Seen thermodynamically, the rubber elasticity is related to a lowering of entropy on deformation. From the molecular point of view, the molecular particles are forced to adopt an 

Entropy- and energy-elastic bodies differ very characteristically in some phenomena  

Energy-elastic bodies show small, reversible deformations of a maximum ofO.1-1 % at large moduli of elasticity (see, e,g., Table 1-5). The energy-elastic-body steel, for example, possesses an E modulus of 2.1 x 10 N/cm. The entropy-elastic-body soft rubber, on the other hand, shows a large reversible deformation of several hundred percent at low moduli of elasticity of 20-80 N/cm. The E moduli of entropy-elastic bodies thus lie at low values similar to those of gases (10 N/cm ). 

The gaslike behavior is characteristic for what are known as entropy- 

Energy-elastic bodies exhibit large moduli of elasticity for small deformations of about 0.1%. In contrast, entropy-elastic bodies possess low moduli of elasticity and high reversible deformabilities of several hundred percent. 

Energy-elastic bodies cool under strain, whereas entropy-elastic bodies become warmer. 

Energy-elastic bodies such as steel and rubber under strains of less than about 10% expand on heating, but strongly stretched rubber contracts. 

In this section we mq)lain how the conformational probabilities of the polymer chains contribute to the entropy of the rubber and, from this, how 

Consider the specimen, which is to be subject to an external tensile force F, to be a thermodynamic tem. The differential of the Helmholtz free energy A with respect to the length L of the specimen is equal to F (Problem 3.2)  

For solids other than rubbers—that is, for metals and ceramics, for example—E changes rapidly with L, and 5 not at all. We term these solids energy-elastic, and they are usually of high modulus. 

1 Boltzmann s equation relates the entropy S of a system to O, the probability of that state, so that 5 X A In n. See (3.N.3) for an application of the Boltzmann equation. 

Hence for an ideal chain (for an ideal chain the internal energy is independent of the chain end-to-end vector r), this change in entropy s with length r implies a force (see eqn 3.5) /  

On tensile deformation above the glass transition, the flexible macromolecule reacts differently from the rigid macromolecule. It changes its random coil to a more extended conformation, as shown in Fig. 5.166. This contrasts also the rotation on shear deformation, illustrated in Fig. 5.164. The main effect on extension is a decrease in conformational disorder, i.e., a decrease in entropy. This property was made use 

A typical entropy-elastic material is cross-linked natural rubber, ds-poly(l-methyl-1-butenylene) or cts-l,4-polyisoprene, as summarized in Fig. 5.166 (see also Fig. 1.15). Its extensibility is 500 to 1,000%, in contrast to the 1% of typical energy-elastic sohds. Natural rubber has a molar mass of perhaps 350,000 Da (about 5,000 isoprene monomers or 20,000 carbon backbone bonds) and is then vulcanized to have about 1% cross-links (see Fig. 3.50). A rubber with a Young s modulus of 10 Pa (depending on cross-link density) must be compared to its bulk modulus (= 1/p, 

Kinetically one can compare the rubber elastic extension to the compression of a gas, as illustrated in Fig. 5.167. The thermodynamic equations reveal that reversible rabber contraction can just as well drive a heat pump as reversible gas expansion. Raising temperature, increases the pressure of a gas, analogously it takes a greater force to keep a rabber band extended at higher temperature. The two equations for this fact are known as the ideal gas law p = PoTV(/(ToV) with PoV/r = R, the gas 

At constant temperature the change of internal energy is zero for ideal gas and rubber 

The gas (left) does no elastic work, the rubber (right) no volume work, as given in the two equations written 

The subscript signifies that the average over is taken with the chain out of the network. Since this chain is the same as all the other v chains—if they too were examined detached from the network—the value of averaged over all 

Imagine the chains now transferred back into the specimen and cross-linked, to reform the network (3.N.1). In order that the chains may be packed together to reform the specimen at volume V, the end-to-end distances have to be changed. The value of ), when averaged over the assembled chains cross-linked into the network, we signify by )j (T stands for in ). That is to say, when in our thought experiment, we move the un-cross-linked free-chains (mean value of r, ) ) into the network, some constraint has to be imposed 

The value of ) is not an intrinsic property of the molecule but depends on the specimen volume K. If V is changed (by heating, for example, or by absorption of diluent (3.N.2)) then changes accordingly is an 

In this section we explain how the conformational probabilities of the polymer chains contribute to the entropy of the rubber and, from this, how the specimen is stabilized mechanically by a network of internal forces between adjacent tie points. The conclusions of this section are summarized in Fig. 3.3 it may be opportune, at first reading, to accept Fig. 3.3 and proceed directly to 3.4. 

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Figure 3.3 Comparison of experiment (points) and theory [Eq. (3.39)] for the entropy elasticity of a sample of cross-linked natural rubber. [From L. R. G. Treloar, Trans. Faraday Soc. 40 59 (1944).]...

As noted above, this force is counterbalanced by an entropy-based, elastic force which prevents the molecule from uncoiling. Entropy elasticity was discussed in Sec. 3.4, where the elastic (subscript el) force between crosslinks is given by Eq. (3.19) to be... 

The ideal entropy-elastic system (ri = — 1, to = 0), in which the work exchanged is totally transformed into a change in entropy. A well-known example of such systems is the ideal gas. 

From the dynamic mechanical investigations we have derived a discontinuous jump of G and G" at the phase transformation isotropic to l.c. Additional information about the mechanical properties of the elastomers can be obtained by measurements of the retractive force of a strained sample. In Fig. 40 the retractive force divided by the cross-sectional area of the unstrained sample at the corresponding temperature, a° is measured at constant length of the sample as function of temperature. In the upper temperature range, T > T0 (Tc is indicated by the dashed line), the typical behavior of rubbers is observed, where the (nominal) stress depends linearly on temperature. Because of the small elongation of the sample, however, a decrease of ct° with increasing temperature is observed for X < 1.1. This indicates that the thermal expansion of the material predominates the retractive force due to entropy elasticity. Fork = 1.1 the nominal stress o° is independent on T, which is the so-called thermoelastic inversion point. In contrast to this normal behavior of the l.c. elastomer... 

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. 

Fully reversible negative expansion is shown by a strained ideal rubber, as a result of the entropy-elasticity, discussed in 5.1. 

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. 

Elastomers are cross-linked macromolecules above the glass-transition temperature. They are entropy elastic and free of viscous flow. For most applications, the rubber is blended with filler material such as silicates and carbon black before vulcanization. Carbon black is an active filler which introduces physical cross-links of macromolecular chains in addition to the chemical cross-links formed during the vulcanization process. The chemical cross-link density is temperature independent, while the strength of the physical cross-links varies with temperature. 

Entropy decrease (for entropy-elasticity) of the network chains as a result of the dilation (elastic deformation term). 

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. 

According to rheology (the science of flow), viscous flow and energy elasticity are only two extreme forms of the possible types of behavior of matter. It is appropriate to consider the entropy-elastic (or rubber elastic), viscoelastic, and plastic bodies as other special cases. 

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... 

Entropy-elastic bodies differ very characteristically from energy-elastic bodies ... 

The changes in the states of entropy-elastic bodies described in the previous section can be expressed quantitatively by phenomenological thermodynamics, starting with one of the fundamental equations in thermodynamics. The relationship of interest here relates the pressure p with the internal energy (7, the volume F, and the thermodynamic temperature T (see textbooks of chemical thermodynamics) ... 

Sec, 11,3 Entropy Elasticity respect to the length /, this gives... 

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