Polymers Dynamics by Dielectric Spectroscopy

Transcription

1 Polymers Dynamics by Dielectric Spectroscopy

2 What s a polymer bulk? A condensed matter system where the structural units are macromolecules

3 Polymers Shape of a Macromolecule in the Bulk Flory's prediction in the 50's: 'RANDOM COIL' P. J. Flory Nobel Prize 1974 <Re 2 > N monomers <R 2 > n steps PATHWAY OF A RANDOM WALK

4 POLYMERIC MATERIAL MACROMOLECULAR STRUCTURE STRUCTURAL UNITS: MACROMOLECULES ~ 300Å CHAIN: RANDOM COIL IN THE BULK PACKING OF CHAINS: LIQUID-LIKE

5 POLYMERIC MATERIAL MACROMOLECULAR ATOMIC STRUCTURE STRUCTURAL UNITS: MACROMOLECULES ~ 300Å 30Å CHAIN: RANDOM COIL ALSO IN THE BULK PACKING NO LONG-RANGE OF CHAINS: LIQUID-LIKE ORDER

6 ATOMIC STRUCTURE: LIQUID-LIKE Polymer GLASSES Metallic Glass Fe 40 Ni 40 P 14 B 6 Molecular Glass S(Q) S(Q) S(Q) Q (Å -1 ) Q (Å -1 ) Q (Å -1 )

7 SMALL LENGTH SCALES LARGE LENGTH SCALES ATOMIC STRUCTURE RANDOM COIL STRUCTURE 30Å 300Å GLASSY PROPERTIES CHAIN PROPERTIES

8 Orthoterphenyl [C 18 H 14 ] STRUCTURE FACTOR RELAXATION MAP α-relaxation T g orthoterphenyl β-relaxation orthoterphenyl

9 Polyisoprene [-CH 2 -CH=C(CH 3 )-CH 2 -] n STRUCTURE FACTOR RELAXATION MAP 0 T g log [τ (s)] -2 α-relaxation -4-6 β-relaxation Boson Peak / T(K) CH 3 -rotation

10 300Å POLYMER MELTS CHAIN - RANDOM COIL - STRUCTURE

11 Coil diameter 100Å Polyisoprene [-CH 2 -CH=C(CH 3 )-CH 2 -] n RELAXATION MAP R N End-to end vector log [τ (s)] end-to-end relaxation α-relaxation T g β-relaxation CH 3 -rotation Boson Peak / T(K)

12 Large Scale Dynamics

13 Let Gaussian chains move The Rouse Model P. Rouse, J. Chem. Phys. 21 (1953) 1272

14 The Rouse Model

15 j r j r j+1 j +1 The Rouse Model Macromolecular chain with short-range interactions in a melt Excluded volume interactions screened out b We define points ( j) along the chain of coordinates r j r j+1 r j 2 = b 2 Sub-chain between j and j+1 large enough End-to-End distribution Gaussian

16 j r j r j+1 j +1 The Rouse Model Macromolecular chain with short-range interactions in a melt Excluded volume interactions screened out b We define points ( j) along the chain of coordinates r j r j+1 r j 2 = b 2 Sub-chain between j and j+1 large enough End-to-End distribution Gaussian Entropic contribution to the free energy: ΔF = 3 Spring-like force: F = 3k B T r b 2 j+1 r j ( ) MAPPING ONTO A BEAD-SPRING -CHAIN 2 k B T r j +1 ( r ) 2 j N ' a 2

17 The Rouse Model Rouse chain: target chain in a melt of similar chains Intermolecular interactions: Mean-field approach Friction force: constant friction coefficient F friction = ξ d r dt Stochastic random force: Friction related to the random force (fluctuation-dissipation theorem) f j (t) f i (t' ) = 6ξk B Tδ ji δ(t t' ) f ξ No correlation between random forces

18 The Rouse Equation of Motion ξ ( ) 3k B T r b 2 j 1 r j r j 1 v j r j f j v j r j+1 3k B T r b 2 j+1 r j ( ) Inertial term ξ d r j dt = 3k B T r b 2 j+1 + r j 1 2 r j ( ) + Brownian motion of coupled oscillators The Rouse Model m d 2 r j dt = F spring 2 j + F friction j + f j F spring j = 3k B T ( r b 2 j+1 + r j 1 2 r j ), j 1,N F friction j = ξ d r j dt m d 2 r j for dt 0 2 t >> m ξ ~ s. f j

19 The Rouse Model Transformation to normal coordinates (Rouse modes) X p X o (t) X p (t) Independent equations of motion X p (t) = 1 N r j (t) = N r (t)cos pπ j N j 1 2, p = 0,1,...N - 1 j=1 N 1 X o (t) + 2 X p (t)cos πp N j 1 2 p=1 X o (t) = 1 N r j (t) N j=1 center of mass of the chain describes the motion of a subchain of wavelength 1/ 2 and length-scale ~ Nb2 p ~ N p

20 The Rouse Equation in normal coordinates The Rouse Model ξ p d X p (t) dt = K p X p (t) + g p Independent equations K p = 12k B Tξ p b 2 ξ ξ p = (2 δ op )Nξ g p = ξ N p Nξ j=1 sin 2 pπ 2N f j (t) cos jπp N

21 Correlation function of the Rouse modes The Rouse Model X p (t) X q (0) = 1 ξ p ξ q No correlation between random forces t 0 (t t') 0 t'' τ dt' e p τ dt'' e q g p (t' ) g q (t'' ) The most important magnitude g p (t' ) g q (t'' ) δ pq δ(t' t'' ) No spatial correlation orthogonality of Rouse modes Key results No time correlation X p (t) X p (0) exp t τ p

22 The Rouse Model X p (t) X p (0) = X 2 p (0) exp t τ p X p 2 (0) = b 2 8N sin 2 ( pπ 2N) Relaxation times τ 1 p = 4W sin 2 pπ 2N W = 3k B T ξb 2 Rouse rate for p << N τ p 1 π 2 W N p 2 N M w τ p M w 2 Large wave-length N p slower relaxation times τ 1 (the slowest time) = the Rouse time τ R

23 The Rouse Model

24 The Rouse Model: Experimental evidences Dynamics of End-to-End vector R N (t) = r N (t) r 1 (t) R N (t) = 4 Correlation of τ p = τ 1 p 2 ; τ 1 τ R N 1 p:odd R N (t) X p (t) cos pπ 2N R N (t) R N (0) 8b2 N π 1 2 p exp t p:odd 2 (Rouse time) τ p for p << N R N (t) R N (0) 8b2 N 1 tp2 π 2 2 exp p p:odd τ R Dominated by τ R

25 Following The Rouse Model: Experimental evidences Molecular (electrical) dipoles of macromolecules Most polymers: Type A polymers: R N (t) R N (0) µ = 0 P (t) P (0) R N (t) R N (0) by dielectric spectroscopy randomly oriented µ µ P always same direction R N can be followed by dielectric experiments experimentally determined can be checked τ R τ R M w 2 R N µ P µ µ

26 The Rouse Model: Experimental evidences Following Type A polymers: R N (t) R N (0) µ by dielectric spectroscopy always same direction P Polyisoprene PI Poly(propilene glycol) PPG Poly(propilene oxide) PPO Poly(alkylene oxide) PAO s R N R N µ P µ µ

27 Following ϕ(t) = R N (t) R N (0) R 2 N (0) τ p = τ 1 p 2 ; τ 1 τ R The Rouse Model: Experimental evidences R N (t) R N (0) Frequency domain: ε" Φ"(ω) = = 8 ε o ε π 1 2 p:odd by DS = 8 π 1 2 p exp t p:odd 2 p 2 ωτ p 1+ω 2 τ p 2 τ p

28 The Rouse Model: Experimental evidences Following τ p = τ 1 p 2 ; τ 1 τ R R N (t) R N (0) Frequency domain: ε" Φ"(ω) = = 8 ε o ε π 1 2 p:odd p 2 by DS ωτ p 1+ω 2 τ p 2 Φ"(ω) ωτ p 1+ω 2 τ p 2 ω max τ p =1 p=1 p=3 p=5 p= log ω

29 The Rouse Model: Experimental evidences Following R N (t) R N (0) by DS τ p = τ 1 p 2 ; τ 1 τ R Frequency domain: ε" Φ"(ω) = = 8 ε o ε π 1 2 p:odd p 2 ωτ p 1+ω 2 τ p 2 Φ"(ω) contribution p=1 contribution p=3 contribution p=5 contribution p=7 8 1 π 2 p 2 ωτ p 1+ω 2 τ p log ω

30 The Rouse Model: Experimental evidences Following R N (t) R N (0) by DS τ p = τ 1 p 2 ; τ 1 τ R Frequency domain: ε" Φ"(ω) = = 8 ε o ε π 1 2 p:odd p 2 ωτ p 1+ω 2 τ p 2 Φ"(ω) ω max τ R =1 contribution p= π 1 ωτ p 2 p 2 1+ω 2 2 p:odd τ p log ω

31 The Rouse Model: Experimental evidences Following R N (t) R N (0) by DS Normal mode polyisoprene CH 3 α-relaxation -[CH 2 -CH=C-CH 2 ]- n Frequency domain: ε" Φ"(ω) = = 8 ε o ε π 1 2 p:odd p 2 ωτ p 1+ω 2 τ p 2

32 The Rouse Model: Experimental evidences Following R N (t) R N (0) by DS polyisoprene CH 3 -[CH 2 -CH=C-CH 2 ]- n Normal mode Normal mode Frequency domain: Including polydispersity ε N " (ω) = Δε M n M 2b2 pπ ωτ cot 2 p N 2M 1+ω 2 τ p (M) g(m)dm 2 0 P:odd

33 The Rouse Model

34 The Rouse Model: Experimental evidences Following R N (t) R N (0) polyisoprene CH 3 -[CH 2 -CH=C-CH 2 ]- n by DS 10-2 (τ N /τ α )/M Rouse M w [g/mol] The Rouse model fails for very high molecular weights

35 Dynamic structure factor Entanglements: Experimental evidences M w =2000g/mol (n~70) M w =14200g/mol (n~500) polyethylne -[CH 2 -CH 2 ]- n

36 The Rouse Model

37 SUMMARY The Rouse Model: Experimental evidences Rouse fails at M>M c Rouse approximations do not properly describe the many-body interactions of a melt of long macromolecules Random forces acting on beads are likely spatial- and time-correlated

38 Generalized Langevin Equation (GLE) Models based on GLE: Renormalized Rouse Models (RRM) m d 2 r j dt 2 = r W { r i } j ( ) dτ t Γ αβ jn (τ;t τ )v β n (t τ ) n 0 e α + F j (t) Problems: How to construct a Memory function from first principles Until now only phenomenological approaches

39 Entanglements: tube/reptation model The tube concept Let s assume: tagged chain mobile rest of the chains (matrix) fixed Chain motion restricted to a tube-like region d Tube diameter d~2d max d max = 2π Q max Inter-macromolecular distances

40 Entanglements: tube/reptation model The tube concept d Let s assume: 8 10 tagged chain mobile K rest of the chains (matrix) 6 10 fixed K 4 10 Chain motion restricted to 4 a tube-like Inter region Tube diameter S(Q) (counts) d~2d max 4 K Q (Å -1 ) Q max 1Å 1 H C H Intra H C d max = 2π Q max Inter-macromolecular distances d 6Å

41 Edwards & degennes Entanglements: tube/reptation model Real situation: all chains move Intermolecular interactions, uncrossability and chain connectivity d~50-70å FICTITIOUS TUBE along the contour length of a tagged chain

42 Entanglements: tube/reptation model Edwards & degennes Real situation: all chains move FICTITIOUS TUBE along the contour length of a tagged chain The tube becomes effective at t τ e d r 2 ( τ ) e d 2 r 2 ( t) t 1 2 (Rouse) τ e d 4 Independent of N

43 Following R N (t) R N (0) by DS Survival probability of unrelaxed chain segments within the tube Reptation Model µ D (t) = R N (t) R N (0) R 2 N (0) = 8 1 π 2 p exp t 2 p:odd τ d Formally equal to the Rouse expression but with τ d N 3

44 Crossover from Rouse to Reptation Following R N (t) R N (0) polyisoprene CH 3 -[CH 2 -CH=C-CH 2 ]- n by DS 10-2 (τ N /τ α )/M Reptation M 3 Experimnts M 3.4 Rouse M w [g/mol] The Rouse model fails for very high molecular weights

45 High molecular weight crossover: Pure Reptation? Following R N (t) R N (0) polyisoprene CH 3 -[CH 2 -CH=C-CH 2 ]- n by DS Reptation M 3

46 Summary Entanglements: tube/reptation model Tube/reptation models capture the main entanglement effects Corrections are usually needed to describe experimental data: Contour length fluctuations Constraint release... However, whether the tube picture is indeed correct still remains a matter for debate

47 Temperature dependence Following R N (t) R N (0) by DS and TSDC polyisoprene CH 3 -[CH 2 -CH=C-CH 2 ]- n

48 Temperature dependence Following R N (t) R N (0) polyisoprene and α-relaxation by DS and TSDC

49 Chain dynamics: Determination of monomeric friction Molecular Dynamics Simulations Rouse Model fails at high-q (short distances) Local potentials and chain stiffness enter the game U ΔE~4k B T Δε -120º 0º Δε~k B T 120º gauche - trans gauche +

50 Chain dynamics: Determination of monomeric friction Molecular Dynamics Simulations Rouse Model fails at high-q (short distances) Friction depends on the wavelength of chain modes log(ξ) PEP T= 350K 413K 492K Neutron scattering values

51 Advanced Topics Following R N (t) R N (0) and α-relaxation by DS and TSDC Normal Mode under confinement Normal Mode in nanocomposites Type A polymers with non-linear architectures

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