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Unformatted text preview: Lecture 13 Networks and Gels Rubber Elasticity and Polymer Melt Rheology references for this lecture: Rubenstein and Colby, Chapter 7 Painter and Coleman, Essentials Chapter 13 (Painter and Coleman Fundamentals Chapter 11) Strobl CHapter 9 1 Monday, November 15, 2010 Polymers and Entropy Driven Elasticity • • • • The chain folded structure and/or coiled up chains explain the ability of polymers to be extensively deformed, compared to metals and ceramics Once they are deformed, what is the driving force to recover their original shape? Polymer (rubber) elasticity is entropy driven For a polymer exhibit elasticity, there needs to be some sort of network on a time scale relevant to the deformation 2 Monday, November 15, 2010 Elastomers are entropy “springs” uncoiling coiling relaxed structure(s) many energy states large Ω high entropy Δ S < 0 stretched structure few energy states small Ω low entropy Δ S = k ln [Ωstretched / Ωrelaxed] < 0 3 Monday, November 15, 2010 Entropy driven elasticity • • • • Polymer (rubber) networks exhibit predominately entropy-driven elasticity Uniaxially stretched rubber, under constant load, contracts reversibly upon heating (WHY?) Rubber gives out heat reversibly when stretched The entropy decreases, due to the orientation of the chain in the direction of deformation 4 Monday, November 15, 2010 But wait… • • How do the chains get tied together? What do we really mean by network formation? 5 Monday, November 15, 2010 Networks •Polymer chain networks interpenetrate • If you stretch a chain or a network from its equilibrium end to end distance, there is a retractive force to return the molecule to its equilibrium coil dimension, this is the origin of rubber elasticity 6 Monday, November 15, 2010 Polymer networks • • Polymer networks are very important soft solids, used in a range of biomedical and tissue engineering applications In contrast to plastic materials, which can be molded and shaped (sometimes repeatedly) but lack reversible elasticity, in cross-linked polymers deformation is reversible 7 Monday, November 15, 2010 Elastomers • • • • • An elastomer is a polymer which exhibits rubber elastic properties, for example, a material that can be stretched to several times its original length without breaking, and, upon release of the stress, immediately returns to its original length (reversible deformation, not limited to stretching) Crosslinking of unsaturated bonds can occur using sulfur (vulcanization), peroxides or other radicals (other mechanism possible too) Natural rubber is a crosslinked, elastomeric network Different modes of crosslinking? How will different modes affect elastic properties? 8 Monday, November 15, 2010 Crosslinked Networks • • crosslinks – chemical or physical – covalent; ionic; H-bonding; microphase segregation, entanglements? crosslinking – increased molecular weight – swell in solvents • organogel • hydrogel 9 Monday, November 15, 2010 Chemical Crosslinking • Forming chemical bonds between polymer chains • Adding crosslinks increases elasticity, toughness • If enough crosslinking, the material can lose crystallinity and never have a Tm • Can result in a network structure • As extent of crosslinking increases (or crosslinking density), the material becomes less deformable • Can crosslink across unsaturated bonds (in polybutadiene, for example), or by adding a crosslinking agent (during or post-polymerization) (vulcanization) 10 Monday, November 15, 2010 Physical Crosslinking • Physical crosslinks: crosslinking as a result of non-covalent interactions, especially microphase segregation • • Segregation as a result of secondary interactions (hydrogen bonding in polyurethanes) Example: a thermoplastic elastomer has two blocks - a short, rigid block of polyurethane, and a longer, flexible chain. The polyurethane blocks segregate into small domains. This prevents the chains from slipping during deformation, and also acts as small particulate filler to toughen the sample. • Segregation as a result of Tg differences • Example: polystyrene and 1,3-diene copolymers at RT, polystyrene is below its Tg, so the polystyrene blocks are glassy, while the polydiene is amorphous. The polystyrene blocks phase segregate into small domains that act to crosslink the material. 11 Monday, November 15, 2010 What is happening? 12 Monday, November 15, 2010 Entropy Driven Elasticity • • • • • • In a network at its equilibrium position, the chains are not fully stretched When they become stretched, their entropy decreases, so they release the heat that was being “used” by the chain to rotate about different conformations When we allow the network to relax to its equilibrium position (for a bulk network, this will be the balance between the chains wanting to be in the theta state, and the shrinkage stress due to the volume change that occurs upon crosslinking); the chains will coil up again, and this requires energy (heat) to allow the many different conformations Because of high segmental mobility (we are above the Tg) of rubbers, instantaneous deformations are possible Reversible deformation is a consequence of the fact that rubbers are lightly crosslinked materials, typically The segments between the crosslinks can move freely and easily rotate around their sigma bonds 13 Monday, November 15, 2010 Some thermo… • Consider the free energy of a system at constant temperature and volume (Helmholtz free energy) F = E − TS • If we stretch a sample to length L, the free energy changes; this change can be related to the applied force, f: ⎛ ∂F ⎞ ⎛ ∂E ⎞ ⎛ ∂S ⎞ f = ⎜ ⎟ = ⎜ ⎟ − T⎜ ⎟ ⎝ ∂L ⎠V ,T ⎝ ∂L ⎠V ,T ⎝ ∂L ⎠V ,T • In an ideal crystalline material, the change in internal energy is enthalpic in origin (stretching bonds): ⎛ ∂E ⎞ f ≈⎜ ⎟ ⎝ ∂L ⎠V ,T • In an ideal rubber, the change in internal energy is entropic in origin (stretching chains): ⎛ ∂S ⎞ f ≈ −T⎜ ⎟ ⎝ ∂L ⎠V ,T 14 Monday, November 15, 2010 Some more thermo… • • • The change in entropy, resulting from the change in the distribution of conformations, is responsible for generating the restoring force This is true of ideal elastomers, but real systems are more complex For now, we’ll approximate the system as ideal ⎛ ∂F ⎞ ⎛ ∂E ⎞ ⎛ ∂S ⎞ f = ⎜ ⎟ = ⎜ ⎟ − T⎜ ⎟ ⎝ ∂L ⎠V ,T ⎝ ∂L ⎠V ,T ⎝ ∂L ⎠V ,T This term is zero for ideal elastomer 15 Monday, November 15, 2010 Entropy driven elasticity • • • Flory constructed a simple method to separate energetic from entropic contribution to the elastic force Consider a typical temperature dependence of a retractive force, f, for a network of constant volume V at constant elongation L. The slope of the curve at T is: ⎛ ∂f ⎞ slope = ⎜ ⎟ ⎝ ∂T ⎠V ,L • The change in the ordinate from the point on the curve to the intercept of the tangent with the f-axis is the entropic contribution to the force ⎛ ∂E ⎞ fE = ⎜ ⎟ ⎝ ∂L ⎠T ,V (in ideal networks, this is approximately zero) ⎛ ∂f ⎞ ⎛ ∂S ⎞ f S = T⎜ ⎟ = −T⎜ ⎟ ⎝ ∂T ⎠V ,L ⎝ ∂L ⎠T ,V Monday, November 15, 2010 (in crystalline materials, this is low or approximately zero) 16 Molecular Models of Rubber Elasticity • • • • • • In the affine model, chain segments deform independently; the relative deformation of each network strand is the same as the relative macroscopic deformation imposed on the whole network In the affine model, the crosslinks are assume fixed in space, and the chains between the crosslinks have conformational freedom In the phantom network model, a certain free motion of the crosslinks about their average affine deformation positions is also allowed These two models can be used to predict the modulus of the network These two are limiting cases - that is the affine network theory gives the upper modulus boundary and the phantom network gives the lower boundary Assumes the dimensions of these chains are unperturbed by excluded volume effects 17 Monday, November 15, 2010 Rubber Elasticity What this picture doesn’t capture well (due to being 2D) is that we are talking about unentangled networks 18 Monday, November 15, 2010 Affine Model ASSUMPTIONS • The chain segments between crosslinks can be represented by Gaussian statistics of unperturbed (phantom) chains • Networks consist of N-chains per unit volume • The entropy of the network is the sum of the entropies of the individual chains • All different conformation states have the same energy • The deformation on the molecular level is the same as that on a macroscopic level (affine) • The unstressed network is isotropic • The volume remains constant during deformation 19 Monday, November 15, 2010 Affine Model 2 2 2 3 R2 3 Rx + Ry + Rz S N , R = − k 2 + S ( N , 0) = − k + S ( N , 0) 2 Nb 2 Nb 2 () 2 2 2 2 2 2 3 Rx + Ry + Rz 3 Rx 0 + Ry 0 + Rz0 ΔS = S N , R − S N , R0 = − k +k 2 Nb 2 2 Nb 2 ()( ) (sum over every strand in the network) Rx = λx Rx 0 Ry = λy Ry 0 ΔSnet = − l λ= l0 Rz = λz Rz0 nk 2 (λx + λ2y + λ2z − 3) 2 If there’s only an entropic contribution to the free energy: ΔF = −TΔSnet = nkT 2 (λx + λ2y + λ2z − 3) 2 Typically, dry networks are incompressible, so volume does not change upon deformation: V = Lx 0 Ly 0 Lz0 = Lx Ly Lz = λx Lx 0 λy Ly 0 λz Lz0 = λx λy λzV For uniaxial deformation: Monday, November 15, 2010 λx λ y λ z = 1 ΔFnet = −TΔSnet = λx = λ λy = λ z = 1 λ ⎞ nkT ⎛ 2 2 ⎜ λ + − 3⎟ 2⎝ λ⎠ 20 Affine Model ΔFnet = −TΔSnet = f= ∂ΔFnet ∂ΔFnet 1 ∂ΔFnet nkT ⎛ 1⎞ = = = ⎜λ − 2 ⎟ ∂Lx Lx 0 ⎝ λ⎠ ∂ ( λLx 0 ) Lx 0 ∂λ σ= f f nkT ⎛ 1 ⎞ nkT ⎛ 2 1 ⎞ = x= ⎜λ − 2 ⎟ = ⎜λ − ⎟ A Ly Lz Lx 0 Ly Lz ⎝ λ⎠ V ⎝ λ⎠ G= • ⎞ nkT ⎛ 2 2 ⎜ λ + − 3⎟ 2⎝ λ⎠ nkT ρRT = vkT = V Ms • This equation relates the true stress and the extension ratio The modulus ρRT/Ms is proportional to T • The modulus is also inversely proportional to Ms (Mc) • • Rubbers with high crosslink density (low Mc) are stiff With constant stress, the modulus increases with increasing temperature Monday, November 15, 2010 21 Phantom Model • In the phantom model, crosslinks are not fixed in space, but can fluctuate • Effective chains are used to represent the way elasticity is transmitted from macroscopic scales down to the individual chains • The phantom network model (with fluctuating junctions but fixed endpoints) is equivalent to an affine network model with a combined effective chain instead of just the chain between two non-fluctuating crosslinks The expression for free energy extension includes a term, f, which expresses functionality at the crosslink site: • ⎛ 2 ⎞ nkT ⎛ 2 2 ⎞ ΔFnet = ⎜1 − ⎟ λ + − 3⎟ ⎜ f⎠ 2 ⎝ λ⎠ ⎝ • The stress becomes: ⎛ 2 ⎞ nkT ⎛ 2 1 ⎞ ⎛ 2 1⎞ σ = ⎜1 − ⎟ ⎜ λ − ⎟ = G⎜ λ − ⎟ ⎝ f⎠ V ⎝ λ⎠ λ⎠ ⎝ 22 Monday, November 15, 2010 Phantom Chain Model • The stress becomes: ⎛ 2 ⎞ nkT ⎛ 2 1 ⎞ ⎛ 2 1⎞ σ = ⎜1 − ⎟ ⎜ λ − ⎟ = G⎜ λ − ⎟ ⎝ f⎠ V ⎝ λ⎠ λ⎠ ⎝ • If f=4, the phantom model predicts a modulus half that given by the affine model G= G= nkT ρRT = vkT = V Ms affine ⎛ 2 ⎞ ρRT ⎛ 2 ⎞ nkT ⎛ 2 ⎞ ⎜1 − ⎟ = vkT⎜1 − ⎟ = ⎜1 − ⎟ phantom V⎝ f⎠ f ⎠ Ms ⎝ f⎠ ⎝ G= ρRT Mx Here, Mx is greater than Ms, and is the combined strand length in the phantom network 23 Monday, November 15, 2010 Comparison: Rubber Elasticity Models • Affine gives upper limit modulus: σ true = • nkT ⎛ 2 1 ⎞ ⎜λ − ⎟ V⎝ λ⎠ Phantom gives lower limit modulus: ⎛ 2 ⎞ nkT ⎛ 2 1 ⎞ ⎛ 2 1⎞ σ = ⎜1 − ⎟ ⎜ λ − ⎟ = G⎜ λ − ⎟ ⎝ f⎠ V ⎝ λ⎠ λ⎠ ⎝ • Affine and Phantom fail to predict the stress- strain behavior at large strain affine nkT ρRT G= = V Ms G= ρRT ⎛ 2 ⎞ ⎜1 − ⎟ phantom Ms ⎝ f⎠ 24 Monday, November 15, 2010 Deviations from Theory • In classical theories – Chains are assumed to be endless, though loose chain ends transfer stress less efficiently than other chain parts – Other types of defects exist in networks 25 Monday, November 15, 2010 Deviations from Theory • Deficiencies in the statistical mechanics: – The segment vectors do not follow Gaussian statistics when the rubber is highly stretched - experimental results do not follow theoretical values (the number of states in high deformations is smaller than predicted by theories) – Gaussian theory does not describe highly crosslinked materials very well 26 Monday, November 15, 2010 Mooney-Rivlin Model • • Alternatively, we can use a phenomenological model (doesn’t include molecular interpretations) The Mooney-Rivlin model uses strain invariants to derive an expression for stress; in classical models, 2C1=G and C2=0 ⎛ ⎛ 1⎞ 1⎞ σ = 2C1⎜ λ − 2 ⎟ + 2C2 ⎜1 − 3 ⎟ ⎝ ⎝ λ⎠ λ⎠ A Mooney-Rivlin plot, the stress divided by the classical model prediction is on the y axis, and the inverse of the deformation is on the x-axis; C2 shows deviation from classical theories 27 Monday, November 15, 2010 Comparison of the Models 28 Monday, November 15, 2010 Entangled Networks • • • • • The phantom model should be the lower modulus limit, and the affine model the upper modulus limit, but the modulus of many real systems is considerably larger than either model predicts Remember, both the phantom and affine models assume that the chains do not entangle, that chain motion is not restricted by other chains In reality, network chains impose topological constraints on one another (entanglements) Imagine a single chain in a polymer network, where many chains form loops around it (entanglements) These loops or entanglements limit the movement of that single polymer chain to an effective tube 29 Monday, November 15, 2010 Plateau Modulus for Polymer Melts • Recall that the elastic shear modulus of a network depends on molecular weight between crosslinks, Mc. In a polymer melt, G therefore depends on Me . • Just as for a polymer network: ρRT Ge = Me 30 Monday, November 15, 2010 Entanglement Parameters ρRT Ge = Me 31 Monday, November 15, 2010 Entangled Networks Entanglements confine the movement of a polymer chain The molar mass of the entanglement strand is Me=NeM0, and this strand replaces the network strand in our determination of the modulus for networks made from long strands Although the entanglements are not permanent, the time scale over which they dis-entangle is not instantaneous; think of it like “temporary” crosslinks Ge = ρRT Me ⎛1 1⎞ G ≅ Gx + Ge = ρRT⎜ + ⎟ Mx Me ⎠ ⎝ 32 Monday, November 15, 2010 Entangled Networks This graph shows computer simulations of network modulus for networks with three different strand lengths that do not entangle (open) and that do entangle (filled) This discrepancy is due to entanglement 33 Monday, November 15, 2010 Now that we’ve covered polymer elasticity… • It’s time to talk about the viscous nature of polymer melts (polymer melt rheology) 34 Monday, November 15, 2010 Boston Molasses Disaster 35 Monday, November 15, 2010 Shear Shear: stress (τ) : applied force per unit area strain (γ) : deformation (as a result of stress) θ τ= Fs A0 δ γ = = tan θ z A perfectly elastic undergoing nondestructive shear will deform almost instantly proportionally to G ! E δ z $ -K G " # 36 Monday, November 15, 2010 Viscosity Unlike an elastic solid, a fluid continues to deform under the action of shear stress, dissipating the energy as flow This time, the top surface moves a distance δ in t sec τ= . Fs A0 . θ . δv γ= = 0 zz δ z dv d ⎛ dx ⎞ γ xy = =⎜⎟ dz dt ⎝ dz ⎠ 37 Monday, November 15, 2010 Viscosity Viscosity: resistance of a fluid to deformation under shear stress stress (τ) : applied force per unit area . Strain rate (γ) : change in strain over time Newtonian fluid: linear response to stress non-Newtonian fluid: non-linear response to stress . τ = ηγ n n < 1 pseudoplastic (shear thinning) n = 1 Newtonian n > 1 dilatent (shear thickening) pseudoplastic newtonian fluid ! η G dilatent ." thixotropic - fluid decreases viscosity with time under shear rheopectic - fluid increases viscosity with time under shear 38 Monday, November 15, 2010 Viscosity 39 Monday, November 15, 2010 Viscosity ηm = K L ( DP )1.0 w ηm = K H ( DP ) 3.4 w Monday, November 15, 2010 40 Entanglements and Polymer Chain Dynamics 41 Monday, November 15, 2010 Development of Reptation Scaling Theory Pierre de Gennes (Paris) developed the concept of polymer reptation and derived scaling relationships. Sir Sam Edwards (Cambridge) devised tube models and predictions of the shear relaxation modulus. In 1991, de Gennes was awarded the Nobel Prize for Physics. 42 Monday, November 15, 2010 Concept of Chain Entanglements If the molecules are sufficiently long (N >100 - corresponding to the entanglement mol. wt., Me), they will entangle with each other. Each molecule is confined within a dynamic tube Tube Monday, November 15, 2010 G.Strobl, The Physics of Polymers, p. 283 Network of Entanglements There is a direct analogy between chemical crosslinks in rubbers and “physical” crosslinks that are created by the entanglements. The physical entanglements can support stress (for short periods up to a time τT), creating a “transient” network. 44 Monday, November 15, 2010 Snakes! There are obvious similarities between a collection of snakes and the entangled polymer chains in a melt. The source of continual motion on the molecular level is thermal energy, of course. 45 Monday, November 15, 2010 Reptation Theory • Polymer molecules “dis-entangle” after a time τT. • Chain entanglements create restraints to other chains, defining a “tube” through which they must travel. The process by which a polymer chain moves through its tube formed by entanglements is called reptation Reptation (from the Latin reptare: to crawl) is a snake-like diffusive motion that is driven by thermal motion. • • • • • Models of reptation consider each repeat unit of the chain as diffusing through a tube with a drag coefficient, ξ The tube is considered to be a viscous medium surrounding each segment. For a polymer consisting of N units: ξpol = Nξ 46 Monday, November 15, 2010 Polymer Reptation • The distance traveled, l, by a particle diffusing in a medium in a time t goes by as t1/2 (related to random walk) l 2 Dt • We can use these equations to determine the time it takes a polymer chain to diffuse out of its tube kT Dtube = Nξ lc2 τd = Dtube • Since l ~N, we get: • The relaxation time is related to ratio of η to G at the transition between elastic and viscous behavior τ~η/Ge η ~ τd ~ N3 47 Monday, November 15, 2010 Relaxation Modulus for Polymer Melts Elastic Viscous flow t Gedde, Polymer Physics, p. 103 Monday, November 15, 2010 Experimental Evidence for Reptation Fluorescently-stained DNA molecule manipulated with optical tweezers After a brisk tug, the DNA chain relaxes back along the path of its reptation tube Initial state Stretched Chain follows the path of the front 49 Chu et al., Science (1994) 264, p. 819. Monday, November 15, 2010 Next time… • Combining the viscous and elastic behavior of polymer networks to get viscoelasticity 50 Monday, November 15, 2010 ...
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This note was uploaded on 11/29/2010 for the course BME 104 taught by Professor Kasko during the Fall '10 term at UCLA.

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