// Molecular Constitutive Modeling Begin with a picture (model) of the kind of material that interests you Derive how stress is produced by deformation of that picture Write the stress as a function of deformation (constitutive equation) 65 At the beginning of the course... Chapter : Newtonian Fluid Mechanics Polymer heology Molecular Forces (contact) this is the tough one stress f at P ds on ds the force on that surface choose a surface through P P We need an expression for the state of stress at an arbitrary point P in a flow. 66
// At the beginning of the course... Molecular Forces (continued) Think back to the molecular picture from chemistry: At that t time we wanted to avoid specifying much about our materials. The specifics cs of these forces, connections, and interactions must be captured by the molecular forces term that we seek. 67 At the beginning of the course... Molecular Forces (continued) We will concentrate on expressing the molecular forces mathematically; We leave to later the task of relating the resulting mathematical expression to experimental observations. First, choose a surface: arbitrary shape small stress at P ds f on ds ds nˆ What is f? f 68
// At the beginning of the course... Molecular Forces (continued) Assembling the force vector: We swept all molecular contact forces into the stress tensor. Now, we seek to calculate molecular contact forces directly from a molecular picture. f f ds nˆ ee ˆ ˆ eˆ eˆ eˆ eˆ ee ˆ ˆ eˆ eˆ eˆ eˆ ee ˆ ˆ eˆ eˆ eˆ eˆ ds nˆ ds nˆ ds nˆ pm eˆ eˆ pm p m eˆ eˆ pm p m Total stress tensor (molecular stresses) 69 Long-Chain Polymer Constitutive Modeling molecular tension force on arbitrary surface ~ f da nˆ ( ) stress tensor We now attempt to calculate molecular forces by considering molecular models. Polymer Dynamics end-to-end vector, Long-chain polymers may be modeled as random walks. 7
// Polymer coil responds to deformation A polymer chain adopts the most random configuration at equilibrium. end-to-end vector, When deformed, the chain tries to recover that most random configuration, giving grise to a spring-like restoring force. spring of equilibrium length and orientation We will model the chain dynamics with a random walk. 7 Gaussian Springs (random walk) Equilibrium configuration distribution function - probability a walk of N steps of length a has end-to-end distance From an entropy calculation of the work needed to extend a random walk, we can calculate the force needed to deform a the polymer coil ( ) kt Na f e Na If we can relate this force, the force to extend the spring, to the force on an arbitrary surface, we can predict rheological properties molecular tension force on arbitrary surface ~ f da nˆ stress tensor 7 4
// Molecular force generated by deforming chain ~ f Tension force on da Force on surface da due to chains of ETE Probability chain of ETE crosses surface da n ˆ (see next slide) Probability chain has ETE ( ) d d d Force exerted by chain w/ ETE kt Na f = number of polymer chains per unit volume 7 da I put a that in b nˆ because this c does not print to PDF right d fam Probability chain of ETE crosses surface da a intersection with da nˆ nˆ b Probability chain of ETE crosses surface da nˆ / d c = volume per polymer chain 74 5
// Molecular force generated by deforming chain ~ kt f nˆ Na ( ) d d d BUT, from before... molecular tension ~ f da nˆ force on arbitrary surface in terms of Comparing these two we conclude, kt Na Molecular force generated by deforming chain ( da 75 ) How can we convert this equation, kt Na Molecular stress in a fluid generated by a deforming chain which relates molecular ETE vector and stress, into a constitutive equation, which relates stress and deformation? We need a idea that connects ETE vector motion to macroscopic deformation of a polymer network or melt. 76 6
// Elastic (Crosslinked) Solid Between every two crosslinks there is a polymer strand that follows a random walk of N steps of length a. Distribution of ETE vectors ETE = end-to-end vector 77 How can we relate changes in end-to-end vector to macroscopic deformation? AN ANSWE: affine-motion assumption: the macroscopic dimension changes are proportional p to the microscopic dimension changes before after 78 7
// 8 Consider a general elongational deformation: F For affine motion we can relate the components of the initial and final ETE vectors as, ETE after ETE before ) ( t 79 Na kt We are attempting to calculate the stress tensor with this equation: ) ( d d d Na But where do ) ( t But, where do we get this? 8
// Probability chain has ETE between and +d: Equilibrium configuration distribution function: Configuration distribution function ( ) d d d ( ) e But, if the deformation is affine, then the number of ETE vectors between and +d at time t is equal to the number of vectors with ETE between and +d at t Na Conclusion: ( ) ( ' ) e 8 Now we are ready to calculate the stress tensor. kt Na ( t) ( ) d d d ( ) ( ) i i e i (much algebra omitted; solved in Problem 9.57) Final solution: kt eˆ eˆ i i i 8 9
// Final solution for stress: kt ˆ ˆ i eiei kt Compare this solution with the Finger strain tensor for this flow. C ( t, t) F T F Since the Finger tensor for any deformation may be written in diagonal form (symmetric tensor) our derivation is valid for all deformations. ktc Which is the same as the finite-strain Hooke s law discussed earlier, with G=kT. 8 What about polymer melts? Non permanent crosslinks Green-Tobolsky Temporary Network Model junction points per unit volume = constant ETE vectors have finite lifetimes when old junctions die, new ones are born newly born ETE vectors adopt the equilibrium distribution Probability per unit time that strand dies and is reborn at equilibrium Probability that strand retains same ETE from t to t (survival probability) P t, t 84
// What is the probability that a strand retains the same ETE vector between t and t +t? P t,tt t Probability that strand retains same ETE from t to t (survival probability) Probability that strand does not die over interval t P t, t t dp P P t, t t, t, dt ln P t, t t, t e t Pt t t C tt 85 The contribution to the stress tensor of the individual strands can be calculated from, Stress at t from strands born between t and t +dt = Probability that strand is born between t and t +dt Probability that a strand survives from t to t Stress generated by an affinely deforming strand between t and t d dt e tt t G e tt GC C ( t, t) ( t, t) dt Green-Tobolsky temporary network mode (Lodge model) 86
// Oh no, back where we started! NO! t G e tt C ( t, t) dt Green-Tobolsky temporary network mode (Lodge model) We now know that affine motion of strands with equal birth and death rates gives a model with no shear-thinning, no second-normal stress difference. To model shear-thinning, N, etc., therefore, we must add something else to our physical picture, e.g., Anisotropic drag nonaffine motion of various types 87 Anisotropic drag - Giesekus In a system undergoing deformation, the surroundings of a given molecule will be anisotropic; this will result in the drag on any given molecule being anisotropic too. Starting with the dumbbell model (gives UCM), replace 8kT with an B anisotropic mobility tensor. Assume also that the anisotropy in B is proportional to the anisotropy in. Giesekus Model B I G : see Larson, Constitutive Equations for Polymer Melts, Butterworths, 988 88
// Constitutive equations incorporating non-affine motion include: Gordon and Schowalter: strands of polymer slip with respect to the deformation of the macroscopic continuum ; see Larson, p (this model has problems in step-shear strains) strand slippage D Dt Phan-Thien/Tanner Johnson-Segalman T v v Larson: uses nonaffine motion that is a generalization of the motion in the Doi Edwards model; see Larson, Chapter 5 Wagner: uses irreversible nonaffine motion; see Larson, Chapter 5 see Larson, Constitutive Equations for Polymer Melts, Butterworths, 988 89 eptation Theory (de Gennes) etraction (Doi-Edwards) Non-affine motion 9
// Step shear strain - strain dependence, G(t), Pa, <.87.4 5. 6.68.4 8.7 5.4 Figure 6.57, p. Einaga et al.; PS soln time, s 9 Step shear strain - Damping Function damping function, h. shifted G(t), Pa Depending on the strain, a different amount of stress is relaxed during retraction After retraction the relaxation is governed by the memory function M(t-t ) etraction time.. strain The Doi-Edwards model does a good job of predicting the damping function, h() (see Larson p8) Figure 6.58, p. Einaga et al.; PS soln time, t 9 4
// Doi-Edwards Model Predicts a memory function t M ( t t) Q( t, t) dt Predicts a strain measure uˆ F uˆ F Q( t, t) 5 sin dd 4 uˆ F M ( t t) i odd Gi e i tt i G 8G i i N i i Predicts a relaxation time distribution û unit vector that gives orientation of strands at time t (Factorized K-BKZ type) M. Doi and S. Edwards J. Chem Soc. Faraday Trans II 74, 88 (978); ibid 74 56, 98 (978); ibid 75, (979); ibid 75, 8 (979) 9 Doi-Edwards Model Steady Shear SAOS M. Doi and S. Edwards J. Chem Soc. Faraday Trans II 75, 8 (979) 94 5
// Doi-Edwards Model Shear Start Up M. Doi and S. Edwards J. Chem Soc. Faraday Trans II 75, 8 (979) 95 Doi-Edwards Model Steady Elongation Elongation Startup M. Doi and S. Edwards J. Chem Soc. Faraday Trans II 75, 8 (979) 96 6
// Doi-Edwards Model Large-Amplitude Step Shear, G(t), Pa, <.87.4 5. 6.68.4 8.7 5.4 time, s damping function, h... strain Figure 6.58, p. Einaga et al.; PS soln M. Doi and S. Edwards J. Chem Soc. Faraday Trans II 74, 8 (979) 97 Doi-Edwards Model Correctly predicts:!!!! atio of / shape of start-up curves shape of h( ) (nonlinear step strain, damping function) predicts =AM shear thinning of, tension-thinning elongational viscosity Fails to predict: =AM 4.4 shape of shear thinning of, reversing flows Elongational strain hardening (branched polymers) Tentatively conclude: shear thinning is an issue of non- affine motion 98 7
// Advanced Models Long-chain branched polymers Pom-Pom Model (McLeish and Larson, JO 4 8, 998) Extended Pom-Pom (Verbeeten, Peters, and Baaijens, JO 45 8, ) Single backbone with multiple branches Backbone can readily be stretched in an extensional flow, producing strain hardening In shear startup, backbone stretches only temporarily, and eventually collapses, producing strain softening Based on reptation ideas; two decoupled equations, one for orientation, one for stretch; separate relaxation times for orientation and stretch) 99 Extended Pom-Pom (Verbeeten, Peters, and Baaijens, JO 45 8, ) Predicts elongational strain hardening 8
// Extended Pom-Pom (Verbeeten, Peters, and Baaijens, JO 45 8, ) What about polymer solutions? Dilute solutions: chains do not interact collisions with solvent molecules are modeled stochastically calculate () by a statistical-mechanics solution to the Langevin equation (ensemble averaging) Elastic Dumbbell Model W. Kuhn, 94 andom force models random collisions Drag on beads models friction 9
// Elastic Dumbbell Model Continuum modeling Momentum balance on a control volume (Navier-Stokes Equation) v v v p tt v g Inertia = surface + body Mixed Continuum/Stochastic modeling (Langevin Equation) Momentum balance on a discrete body (mass m, velocity u) In a fluid continuum (velocity field v) du m u v dt 4kT A Inertia = drag + spring + random (Brownian) Construct an ensemble of dumbbells and seek the probability of a given ETE at t Elastic Dumbbell Model Langevin Equation du m dt u v 4kT A Construct an ensemble of dumbbells and seek the probability of a given ETE at t To solve, (see Larson pp4-45). Consider an ensemble of dumbbells and seek the probability that a dumbbell has an ETE at a given time t. The equation for is the Smoluchowski equation: t 4kT kt v kt Na We can calculate stress from: ( ) d dd If we multiply the Smoluchowski equation by and integrate over space, we obtain an expression for (i.e. the constitutive equation for this model) 4
// Integration yields: see Larson, Constitutive Equations for Polymer Melts, Butterworths, 988 Upper-Convected Maxwell Model! Two different models give the same constitutive equation (because stress only depends on the second moment of, not on details of G kt 8kT Na number of dumbbells/volume bead friction factor from random walk 5 Elastic Dumbbell Model for Dilute Polymer Solutions p p Polymer contribution s s Solvent contribution p s Dumbbell Model (Oldroyd B) See problem 9.49 see Larson, Constitutive Equations for Polymer Melts, Butterworths, 988 6
// ouse Model Multimodal bead-spring model Springs represent different sub-molecules Drag localized on beads (Stokes) No hydrodynamic y interaction N+beads N springs 7 ouse Model see Larson, Constitutive Equations for Polymer Melts, Butterworths, 988 ouse wrote the Langevin equation for each spring. Each spring s equation is coupled to its neighbor springs which produces a matrix of equations to solve. Langevin Equation du m dt u v 4kT A ouse found a way to diagonalize the matrix of the averaged Langevin equations; this allowed him to find a Smoluchowski equation for each transformed mode ~ i of the ouse chain Each Smoluchowski equation gives a UCM for each of the modes ~ i GI i N i i i G kt i 6kT sin ( i ( N )) ouse Model for polymer solutions (multi-mode UCM) 8
// Zimm Model see Larson, Constitutive Equations for Polymer Melts, Butterworths, 988 Multimodal bead-spring model Springs represent different sub-molecules Drag localized on beads (Stokes) Dominant hydrodynamic y interaction ouse: solvent velocity near one bead is unaffected by motion of other beads (no hydrodynamic interaction) N+beads N springs Zimm: dominant hydrodynamic interaction) 9 What about suspensions? (Mewis and Wagner, Colloidal Suspension heology, Cambridge ) uniform flow Stokes flow Dilute solution Einstein relation. 5 m Increasing complexity; solve NS Concentrated suspensions Stokesian dynamics
// Stokesian Dynamics Brady and Bossis, Ann. ev. Fluid Mech, 988 Wagner and Brady, Phys. Today 9, p7 Langevin Equation for Dumbbells du m u v 4kT A dt Inertia = drag + spring + random (Brownian) Another Langevin Equation Stokesian Dynamics for Concentrated Suspensions du M dt F hd hydrodynamd ic F particle F Brownian Hydrodynamic = everything the suspending fluid is doing (including drag) Particle = interparticle forces, gravity (including spring forces) Brownian = random thermal events Stokesian Dynamics Brady and Bossis, Ann. ev. Fluid Mech, 988 Spanning clusters increase viscosity it Faith A. Morrison, Michig gan Tech U. 4
// Summary Molecular models may lead to familiar constitutive equations ubber-elasticity theory = Finite-strain Hooke s law model Green-Tobolsky temporary network theory = Lodge equation (UCM) eptation theory = K-BKZ type equation Elastic dumbbell model for polymer solutions = Oldroyd B equation Model parameters have greater meaning when connected to a molecular model G = kt G i, i specified by model Molecular models are essential to narrowing down the choices available in the continuum-based models (e.g. K-BKZ, ivlin-sawyers, etc.) As always, the proof is in the prediction. see Larson, esp. Ch 7 Modeling may lead directly to information sought (without ever calculating the stress tensor) Summary Molecular models may lead to familiar constitutive equations ubber-elasticity theory = Finite-strain Hooke s law model Green-Tobolsky temporary network theory = Lodge equation (UCM) eptation theory = K-BKZ type equation Elastic dumbbell model for polymer solutions = Oldroyd B equation (UCM) Caution: correct stress predictions do not imply that the molecular model is correct Stress is proportional to the second moment of (), but different functions may have the same second moments. 4 5
// Summary Materials Discussed Elastic solids Linear polymer melts with affine motion (temporary network) Linear polymer melts with anisotropic drag Linear polymer melts with various types of non-affine motion Chain slip eptation Branched melts (pom-pom) Polymer solutions Suspensions esources. G. Larson, Constitutive Equations for Polymer Melts. G. Larson, The Structure and heology of Complex Fljuids J. Mewis and N. Wagner, Colloidal Suspension heology 5 6