Entanglements. M < M e. M > M e. Rouse. Zero-shear viscosity vs. M (note change of slope) Edwards degennes Doi. Berry + Fox, slope 3.4.

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1 Entanglements Zero-shear viscosity vs. M (note change of slope) M < M e Rouse slope 3.4 M > M e Edwards degennes Doi slope 1 Berry + Fox, 1968

2 Question: Which factors affect the Me: T, P, M, flexibility, concentration (solution), branching, microstructure?

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4 Dilution: network swells in good solvent G(Φ)= G 0 Φ β M e (φ)=m e F -α b=2 or 7/3 a=b-1 Adam, 1975 Colby, Rubinstein, 1990

5 J. D. Ferry Viscoelastic properties of polymers

6 Elastic element: G Viscous element: Viscoelastic element: G G Characteristic relaxation time Key relation in linear viscoelasticity

7 Obtain time: D kt Einstein 1905 ζ = friction R f u Stokes 1905 D 2 R Fick 1905 R kt 3

8 Summary: Synthesis: N, b Size: R=N 1/2 b Distribution: Modulus: G=νkT Diffusion: D kt Hooke: F=Kx Newton: Viscoelasticity:

9 Importance of the time scale granite Halite (mineral form of salt) Even some apparent solids flow if they are observed long enough =4.5x10 19 Pa.s

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11 Question: How many characteristic relaxation times exist in a polymer? What are the implications for viscosity?

12 short time (local motion) long time (global motion)

13 A gross classification of viscoelastic materials: Glass / crystal Rubber Melt / solution Important point: For a solid, macroscopic motion if frozen (long time is infinite) whereas short-time (local) motion takes place

14 Glass: - Amorphous ( frozen ) solid T 2 >T 1 T 1 Liquid Glass (T 1 ) Crystal - Non-equilibrium (aging) - Low temperatures (or short times) - Glass transition: out of equilibrium (metastable) - Cooling rate dependence - Local motions: free volume, cooperativity -1 T g T

15 specific volume V The Glass Transition At the glass transition the polymer chain segments undergo increased local motions, which consume heat and leads to an increase in the volume requirement for the chain. heat capacity change c p specific volume change V thermal expansion heat capacity c heat capacity T g

16 Local motion in a dense polymer (melt)

17 Vitrification: Cotton candy -1 sugar T g T Amorphous polymers: Semi-crystalline polymer: Crystal Journal of Crystal Growth 48, 1980, p. 303/20 Glass: G~10 9 Pa Crystal: G<10 7 Pa

18 Crystallization and Melting > T m < T c

19 Question: Which are the implications of the existence of local short-time motion in solids? Think of self-healing (recent group seminar)

20 Rubber (elastomer) Thermosetting polymers (chemical network) - solid-like (permanent or temporary) - network Vulcanisation: - T > T g Hevea tree Latex - liquid Natural rubber Solid-like (gel) G kt G~ Pa

21 Indian boot Amazon (~500 bc)

22 Question: How does vulcanization relate to the Indian boot example? Think of Goodyear (~1839)

23 Compare metal vs elastomer

24 Thermoplastic polymers (physical network): - Liquid - temporary network - Large scale motions entanglements G 0 N RT M e High temperatures (T>Tg), Low times Tube model: de Gennes, Doi and Edwards (1986)

25 Global motion (center of mass) in a flowing polymer melt

26 Compare permanent and temporary networks 0 P(t) or (t) Temporary network Pt ( ) exp t 0 time 0 rel = time The molecules move in order to recover their stable coil-shape structure (t) Rubber (permanent network) time

27 Question: Can the same polymer behave as a glass and as a melt? Think of Tg

28 Important point: We can tune the state of a polymer (and hence modulus and time) with T T<Tg solid, glass T>Tg solid, rubber T >> Tg liquid Alternatively, at a given temperature: very short time: glass ; very long time: liquid This means that there is an interplay of the characteristic time of the material and the temperature: principle of time-temperature Superposition (tts) - to be discussed below

29 Importance of the time scale of the material in reference to observation time elastic Characteristic relaxation time of the material viscous time The same material behaves like a liquid or a solid, depending on the observation time (experimental interrogation) Use of dimensionless number: Observation time versus Relaxation time

30 Deborah number: (M. Reiner) De t exp The mountains flowed before the Lord (song by prophetess Deborah, Bible, Judges 5:5) De>>1 Elastic behaviour De<<1 viscous behaviour De ~ O(1) viscoelastic behaviour Silly Putty

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32 Amplitude : angular frequency ( t) : phase angle 0 ( t) 0 t impose (t) sin t probe ( t) sin( t ) 2 Elastic response Viscous response 0 G' ( ) sin( t) 0 G"( ) cos( t)

33 Deborah number in an oscillatory experiment: De De > 1 t exp < λ Elastic element: 0 G ~ Viscous element: ~ G and η are material constants (do not depend on external mechanical stimulus)

34 strain [1] stress [Pa] Ideal elastic solid t γ sinωt γ 0 0 σ(t) G γ(t) G γ0 sin ωt time t [s]

35 strain [1] stress [Pa] Ideal viscous liquid 0 t γ sinωt γ σ(t) η γ(t) η γ 0ωcos 0 ωt η γ ωsin ωt time t [s]

36 G', G'' [a.u.] Maxwell-Model Simplest viscoelastic liquid: Maxwell element = F = D = F + D d dt 1 G d dt 0 cos t 1 G d dt G sin G 2 1 t cost G(G,, ) sin t G(G,, ) cost = 10 2 s angular frequency [s -1 ] G' G'' 90 Importance of time (frequency): Span time Span responding length scale Spectroscopy

37 strain [1] stress [Pa] Kelvin-Voigt-Model Simplest viscoelastic solid: Voigt-Kelvin element = F = D σ σ F σ D G γ η γ σ(t) G γ 0 sin 0 ωt η γ ωsin ωt 2 arctan G 2 F D time t [s]

38 Mesoscopic modeling: bead-spring models Spring (constant k G) : elastic element Bead (friction ζ η): viscous element

39 I. Unentangled polymer dynamics (M<Me) I.1 Diffusion of a small colloidal particle: random walk Simple diffusive motion: Brownian motion 2 r ( t) r (0) 6Dt Friction coefficient: Diffusion coefficient A constant force applied to the particle leads to a constant velocity: F v Einstein relationship: D kt Friction coefficient Stokes law: 6 R Stokes-Einstein-Sutherland (Sphere of radius R in a Newtonian liquid with a viscosity ) D kt 6R Determination of R H kt 6D

40 I.2 Diffusion of an unentangled polymer chain: I.2.1. Rouse model: - Each bead has its own friction - Total friction: R N - Rouse time: R D 2 2 R R D kt / N kt t < Rouse : viscoelastic modes - Stokes law: b s - Kuhn monomer relaxation time: 0 R R R N beads of length b NR kt N 2 t > Rouse : diffusive motion b kt 2 kt b 3 s R N 0 2

41 Rouse relaxation modes: The longest relaxation time: R N 0 2 For relaxing a chain section of N/p monomers: 0 N p 0 p 2 (p = 1,2,3,,N) N relaxation modes 1 R i i 1 4 R /16 2 R /4 1 R

42 - At time t = p, number of unrelaxed modes = p. - Each unrelaxed mode contributes energy of order kt to the stress relaxation modulus: G kt b p 3 N - Mode index at time t = p : G t Approximation: Gt p p kt t 3 b 0 (density of sections with N/p monomers: ) 3 ( N / p) b p 0 1/ 2 1/ 2 1/ 2 kt t t exp 3 b 0 R N, for 0 < t < R, for t > 0 Volume fraction

43 G t kt t 3 b 0 1/ 2, for 0 < t < R

44 Viscosity of the Rouse model: 1/ 2 kt t t G tdt exp 0 3 dt b 0 0 R kt 1/ 2 kt kt 3 0 R x expx dx R 3 0N N b b b b N b Or: R N R 0 G( ) 2 kt Nb 3 G( ) R R N b

45 Exact solution of the Rouse model: G t N kt t exp Nb p1 p 3 with p 2 2 bn 6 ktp 2 2 Each mode relaxes as a Maxwell element

46 Example: Frequency sweep test: G'( ) "( ) 1/ 2 G For 1/ R < < 1/ 0 G '( ) G"( ) 2 For < 1/ R

47 Rouse model: unentangled chains in polymer melts: Relaxation times rel M c M For M < M c : M 2 For M > M c : M 3.4 Main issue with Rouse model: phantom chain (no interactions with environment (hence, it does not work for dilute solutions)

48 I.2.2. Account for hydrodynamic interactions: Zimm model Viscous resistance from the solvent: the particle must drag some of the surrounding solvent with it. Hydrodynamic interaction: long-range force acting on solvent (and other particles) that arises from motion of one particle. When one bead moves: interaction force on the other beads (Rouse: only through the springs) The chain moves as a solid object: Z (no friction distribution in chain) kt kt DZ Z sr Zimm time: 2 3 R s 3 sb 3/ 2 3/ 2 Z R N 0N (In a theta solvent) D kt kt Z s R

49 Rouse time versus Zimm time: R R R R sb D kt / N kt / N b kt R s 3 N 2 Z R R R sb N N D kt / kt / R kt Z Z s 3/ 2 3/ Zimm has a weaker M-dependence than Rouse. - Zimm < Rouse in dilute solution - Zimm works better in dilute solution

50 Slope: 5/9 to 2/3 (theta condition)

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52 Rheological Regimes: (Graessley, 1980) Dilute solutions: Zimm model H.I. no entanglement Unentangled chains (Non diluted): Rouse model no H.I. no entanglement Entangled chains: Reptation model no H.I. entanglement

53 III. Polymer: From diluted to concentrated state c<c* c=c* c>c* Dilute solutions: Concentrated solutions or melts: eff c c 4 c* 3 Na M Mass 3 c R c Mass c Mass : mass per volume c: number of molecules per volume Semi-dilute solutions: c Mass c* 1 c* N a M 4 R 3 M: [g/mol] c*: mass per volume c Mass > c*: Interpenetration of the chains c*: overlap concentration 3