Chapter 3 Entropy elasticity (rubbery materials) Review basic thermal physics Chapter 5.1 to 5.5 (Nelson)
Outline: 3.1 Strain, stress and Young modulus 3. Energy density 3.3 Typical stress-strain curve 3.4 Affine deformation: 1D and 3D 3.5 Rubber elasticity 3.6 Uniaxial deformation 3.7 Stress vs elongation 3.8 Theory vs experiment 3.9 Small strain limit
Goals: understand properties of rubbery material link the microscopic and macroscopic properties use statistical physics to understand entropy elasticity
Rubber Elasticity of Polymer Network Polymer network: long polymer chains that are crosslinked with each other form a continuous molecular framework. All polymer networks (which are not in the glassy or partially crystalline states) exhibit the property of high elasticity (= large reversible deformations at relatively small applied stress). Rubbers are known to be highly extendable before breaking. The nature of rubber elasticity is different from the elasticity of other material such as metals.
3.1 Strain, stress and Young modulus beam (length L and cross-sectional area A) stretches by L under tension force F E Strain ε = L Stress σ = F L A Young s modulus E Pa Pa 3. Energy density F kx F AE L L U 1 kx Energy density u U V vol 1 E
3.3 Typical stress-strain curves steel 9 10 A B 0.01 Pa C E 10 11 Pa L L rubber 7 310 Pa A 5 C B E 10 6 Pa L L A - upper limit for stress-strain linearity B - upper limit for reversibility of deformations C - fracture point Rubber: strain ( ΔL L) much larger than for steel. Steel: stress σ much larger than for rubber. Young moduli much larger for steel than for rubber. Steel: linearity and reversibility are lost practically simultaneously Rubber: a very wide region of nonlinear reversible deformations. Steel has a wide region of plastic deformations (between points B and C) Rubber: practically no region of plastic deformations.
3.4 Affine Transformation: 1D The macroscopic change in dimension is mirrored at the molecular level. We define an extension ratio, λ, as the dimension after a deformation divided by the initial dimension: Bulk: L 0 L Chain: L L 0 0 0
Affine Transformation: 3D Bulk: z L z0 L z L z z0 L x0 y L y L y y0 L x L x x0 x Single Chain: L y0 z z R z0 R 0 R R x0 R y0 y y x x R0 Rx0 Ry0 Rz0 R xrx0 yry0 zrz0
.5 Rubber Elasticity: Model Assumptions Consider macroscopic block of rubber with N net the number of chains that form the block. Chains are crosslinked at their ends. Each chain in the network is a FJC. 1) The segment (subchain) between two adjacent cross-links are treated by Gaussian approximation and have N links. The enormous deformability of networks like rubber rises from the entropy elasticity of the polymer chains connecting the cross-links of the network ) Each cross-link remains fixed (= No slippage between chains & Microscopic stress equals macroscopic stress). 3) Network undergoes affine deformation when stretched. macroscopic strain = microscopic strain. extension ratio of individual chains = extension ratio of bulk rubber. 4) Rubber network is isotropic in the un-deformed state (properties are the same in all directions). 5) Rubber network is incompressible.
Entropy Change in Deforming the Network For single chain: R x = λ x R x0, R y = λ y R y0, R z = λ z R z0 Each chain has N effective segments of length b. Entropy of a chain Entropy change ΔS of a chain: R R R S N R k S k S Nb Nb 3 R 3 x y z, B N,0 B N,0,, 3 S N R S N R k 1 1 1 x x0 y y0 z z0 0 B (affine deformation) R R R Nb For the whole network, we sum over all chains (N net ): S 3 k Nnet Nnet Nnet B net x 1 Rx0 1 y0 1 z0 i y R i z R Nb i i1 i1 i1 Each of the N net chains is an Nn 1 N R R Nb 3 3 N N N ideal FJC (isotropic network): x R 0 y0 z0 net net net et i i i i1 i1 i1 i1 Nb net Entropy change for the whole network (total of N net chains): Nne tkb Snet x y z 3
Deforming the Network: Entropy and Free Energy Entropy change for the whole net B S net x y z 3 network (total of N net chains): N k net B Free energy change: F 3 net T Snet x y z N k T assuming that the interaction energy is negligible Dry networks are incompressible, volume does not change when deformed V L L L L L L V 1 x0 y0 z0 x y z x y z x y z rubber networks are typically nearly incompressible
3.6 Uniaxial Deformation along x (constant volume) Constant volume (λ x λ y λ z = 1) and stretched in the x direction 1 x y z Free energy change: NnetkBT Fnet 3 Elongation force: f x F F 1 F N k T 1 net net net net B Lx ( Lx0 ) Lx0 Lx0 Nominal stress: force divided by cross section area yz f 1 N k T 1 N k T 1 n L L L L L V x net B net B y0 z0 y0 z0 x0 Equation of state relating stress, length, and temperature
A note on stress nominal stress: true stress: n t force cross-sectional area of releaxed network force cross-sectional area of stretched network Uniaxial Deformation of Incompressible Network fx NnetkBT 1 n L L V y0 z0 t f N k T L L V x net B 1 y z Polymer scientists usually use nominal stress
3. 7 Stress versus Elongation 1 n nkbt n nk T B chain density: n N net V Nonlinear relationship between stress and elongation Elasticity of rubber network results from sum of chain entropies Stiffness of rubber network increases with temperature (characteristic for entropic elasticity) Fundamental insight: The elasticity of rubber does not depend on the chemical make-up of the polymer nor on how it is crosslinked. Stiffness of network depends on the crosslink density (= proportional to the density of chains n). The retractile force increases linearly with crosslink density of network. To make a stiffer network, more crosslinks should be added so that the lengths of the segments become shorter.
3.8 Theory vs experiment For moderate elongations (1. < λ < 5) the agreement is acceptable. Typical discrepancy between theory and experiment are ~0%. However, deviations tend to be systematic. And are due to topological constraints to the subchains conformation. For large elongations (λ > 5) theory and experimental values diverge. Gaussian approximation of end-to-end distance distribution breaks down. A real chain cannot be stretched beyond its contour length L C. Treloar, Physics of Rubber Elasticity (1975) 1 n nkbt Good agreement for small elongations (λ < 1.). The network behaves like a Hookean spring with stress proportional to strain, σ = Eε. prediction is good in the small extension region. Overall, quiet successful model, especially because it is universal
3.9 Small strain limit Lx Lx0 Lx Parameter λ 1 equals the strain ε ( 1) L L L x0 x0 x0 Examine the stress in the limit of small strain (ε 0 or λ 1) Taylor expansion for λ = 1: 1 3( 1) 3 Stress in the limit of small strain: 3 B n k T Young s modulus E Young s modulus of rubber network: 3 B E n k T Young s modulus increases linearly with temperature and linearly with the density of cross-links
Rubber band stiffness when heated
Rubber bands explained by Richard Feynman
Rubber Band Thermodynamics with IR Camera