Elastomers PDF

27. Elastomers 27.1. Introduction Elastomers (crosslinked rubbers), possess the remarkable ability of being able to stretch to 5–10 times their original length and then retract rapidly to near their original dimensions when the stress is removed. Three requirements for elastomers: The polymer m...

27. Elastomers 27.1. Introduction Elastomers (crosslinked rubbers), possess the remarkable ability of being able to stretch to 5–10 times their original length and then retract rapidly to near their original dimensions when the stress is removed. Three requirements for elastomers: The polymer must be above its glass transition temperature Tg. The polymer must have a very low degree of crystallinity (xc→0). The polymer should be lightly crosslinked Examples: ethylene/propylene rubbers. (copolymerization decreases crystallinity) Natural rubber: polymerization of cis-1, 4-polyisoprene. (cis configuration reduces crystallinity; melting point is low ~35 oC) Vulcanization: crosslinking rubbers with Sulfur (temperature 120-180 oC) m can be 1 or 2 with an accelerator. 1 The properties of a particular elastomer are controlled by the nature of the crosslinked network. Ideally, this can be envisaged as an amorphous network of polymer chains with junction points from which at least three chains emanate. Every length of polymer chain can be thought of as being anchored at two separate sites. In reality, this is a great oversimplification and the likely situation is sketched in Figure 21.1. In a polymer of finite molar mass, there will be some loose chain ends. Also it is possible that intramolecular crosslinking can take place leading to the presence of loops. Neither of these two structural irregularities will contribute to the stiffness of the network. On the other hand, entanglements can act as effective crosslinks, even in an unvulcanized rubber, at least during short-term loading. 2 27.2. Mechanical behavior The unique deformation behavior of elastomeric materials has fascinated scientists for many years, and there are even reports of investigations into the deformation of natural rubber from the beginning of the nineteenth century. Elastomer deformation is particularly amenable to analysis using thermodynamics, as an elastomer behaves essentially as an ‘entropy spring’. It is even possible to derive the form of the basic stress–strain relationship from the first principles by considering the statistical thermodynamic behavior of the molecular network. 27.3. Statistical theory of elastomer deformation. Assumptions: 1. Freely jointed chain 2. The change in the components of the displacement vector will be proportional to the overall change in specimen dimensions means, i.e. 𝑥 ′ = 𝜆1 𝑥, 𝑦 ′ = 𝜆2 𝑦, 𝑧 ′ = 𝜆3 𝑧 3. Constant volume: 𝜆1 𝜆2 𝜆3 = 1 The entropy of an individual chain: 𝑆 = 𝑐 − 𝑘𝛽 2 𝑟 2, where c is a constant. Before deformation: 𝑆 = 𝑐 − 𝑘𝛽 2 (𝑥 2 + 𝑦 2 + 𝑧 2 ) 3 After deformation: 𝑆 ′ = 𝑐 − 𝑘𝛽 2 (𝜆12 𝑥 2 + 𝜆22 𝑦 2 + 𝜆23 𝑧 2 ) Change in entropy: ∆𝑆𝑖 = 𝑆 ′ − 𝑆 = −𝑘𝛽 2 [(𝜆12 − 1)𝑥 2 + (𝜆22 −1)𝑦 2 + (𝜆23 − 1)𝑧 2 ] the probability per unit volume W(x,y,z) of finding one end of a freely jointed chain at a point (x,y,z) a distance r from the other end is If the number of these chains per unit volume is defined as N, the number dN, which have ends initially in the small volume element dxdydz at the point (x,y,z), can be determined from the Gaussian distribution function as the total entropy change ΔS per unit volume of the sample during deformation is given by ∆𝑆 = −1/2𝑁𝑘 (𝜆12 + 𝜆22 + 𝜆23 − 3) 4 This equation relates the change in entropy to the extension ratios and the number of chains between crosslinks per unit volume. It was shown earlier (Section 21.2) that ideally there is no change in internal energy U when an elastomer is deformed and, since the deformation takes place at constant volume, the change in the Helmholtz free energy per unit volume ΔA is ∆𝐴 = −𝑇∆𝑆 = 1/2𝑁𝑘𝑇 (𝜆12 + 𝜆22 + 𝜆23 − 3) for isothermal deformation. This will be identical with the isothermal reversible work of deformation w per unit volume and hence 𝑤 = 1/2𝑁𝑘𝑇 (𝜆12 + 𝜆22 + 𝜆23 − 3) Limitations and use of the theory Limitations: freely jointed chains (Gaussian distribution). The distribution cannot be applied when the chains become extended. Another problem concerns the value of N. This will be governed by the number of junction points in the polymer network that can be either chemical (crosslinks) or physical (entanglements) in nature. There will be chain ends and loops that do not contribute to the strength of the network, but, if their presence is ignored, it follows that if all network chains are anchored at two crosslinks then the density ρ of the polymer can be expressed as 𝑁𝑀𝑐 𝜌= 𝑁𝐴 where Mc is the number-average molar mass of the chain lengths between crosslinks NA is the Avogadro constant. 𝜌𝑁𝐴 𝜌𝑅 𝑁= = 𝑀𝑐 𝑀𝑐 𝑘 𝜌𝑅𝑇 𝑤= [(𝜆12 + 𝜆22 + 𝜆23 − 3) 2𝑀𝑐 5  𝜌𝑅𝑇 𝐺= 2𝑀𝑐 The parameter G relates w, the work of deformation per unit volume (which has dimensions of stress) to the extension ratios and so G also has dimensions of stress. It is, therefore, often referred to as the shear modulus of the elastomer. G has some interesting properties. As might be expected, G increases as Mc is reduced. This means that the material becomes stiffer as the crosslink density increases and the network becomes tighter. A rather more surprising prediction of the equation, that has been substantiated experimentally, is that, unlike almost every other material, the modulus of an elastomer increases as the temperature is increased. This is yet another consequence of the deformation of elastomers being dominated by changes in entropy rather than of internal energy. Network defects, such as the entanglements, loops and chain ends shown in Figure 21.1, will have an important effect upon the mechanical behaviour of elastomers. Entanglements will act as physical crosslinks and tend to increase the modulus. Loops will not contribute to the network elasticity and so do not affect the modulus. Chain ends also will have no significant effect upon the network elasticity. 6