Part III. Polymer Dynamics molecular models I. Unentangled polymer dynamics I.1 Diffusion of a small colloidal particle I.2 Diffusion of an unentangled polymer chain II. Entangled polymer dynamics II.1. Introduction to Tube models II.2. «Equilibrium state» in a polymer melt or in a concentrated solution: II.3. Relaxation processes in linear chains II.4. Star Polymers II.5. Constraint Release processes (CR) II.6. Models branched polymers Source: Polymer physics Rubinstein,Colby 1
Doi and Edwards model Relaxation modulus: Related viscosity: for M < M c for M > M c in the real case 2
Zero-shear viscosity vs. log (M) Slope of 3.4 Discrepancy between theory and experiments: inclusion of the contour length fluctuations process 3
Initial D&E model (with CR) 4
II.3.2. Contour length fluctuations Equilibrium length of an arm: Entropical force: Topological force: L eq = the most probable length Equilibrium length L eq real length This part of the initial tube is lost 5
Linear chains: Doi and Edwards model including the contour length fluctuations (CLF): - the environment of the molecule is considered as fixed (no CR). Rouse-like motion of the chain ends: Displacement along the curvilinear tube: For t < τ R Rem/ Before t = τ Rouse, the end-chain does not know that it belongs to the whole molecule 6
L eq,0 L(t) Partial relaxation of the stress: At time t = τ R : For τ e < t < τ R With µ close to 1 7
In the same way: Surviving section Only the unrelaxed part of the chain is considered Rescale of the reptation time in order to account for the fluctuations process Slope: 3.4 Log(M) 8
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D&E model with contour length fluctuations (with CR) 10
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II.3.3.: Linear chains: model of des Cloizeaux: Based on a time-dependent coefficient diffusion: D e 12
Des Cloizeaux model (with CR) 13
II.4. Star polymers Symmetric star: Fixed point x=1 x=0 No Reptation Only fluctuations 14
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II.4.1. Fluctuations times of a branch: Activated fluctuations times: - First passage problem - Entropically unfavorable - Exponential increase of the relaxation time constant 17
L eq U(L chain ) New coordinate system: 0 L eq L chain L chain L eq s 0 At equilibrium (U min.): Nb 2 a.l eq (Rem/ mistake in the previous course) 18
1 x 0 or Early fluctuations process: For x such U(x) < kt: Free motion of the chain ends Rouse process (see linear chains) 19
Total fluctuations times: Transition between early and activated fluctuations times: Milner McLeish 1997: τ activated τ fluc τ activated x τ fluc (results with DTD (see Next section) τ early τ 1 τ 1 must be defined: x 20
UCL approach: U(x trans ) = kt τ activated τ fluc U=kT τ early Comparison between the 2 methods: Comparison between both approaches: Milner - McLeish UCL star 28kg/mol star 144kg/mol 21
II.4.2. Zero-shear viscosity of a star polymer: 22
II.4.3. Linear polymers described as star molecules: = a 2-arms star polymer, able to relax by reptation Using this approach: Monodisperse linear PBD, from 8.8kg/mol to 326 kg/mol 23
II.4.4. Asymmetric star polymers: 1) 2) 3) 4) - Reptation can occur - the smallest branch brings important additional friction: 24
II.4.5. Multi-arm star polymers: Colloidal effect of the core 25 Extra relaxation process not described by the tube theory
II.5. Constraint Release processes (CR) (relâchement des contraintes) The relaxation of a test chain will depend on the relaxation of its neighboring chains. entanglements 26
Constraint release 27
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Example: blends of 2 monodisperse linear polymers: M w : 308 kg/mol and 21 kg/mol Watanabe et al., Macromol. 2004 υ 2 : long chains proportion The relaxation time of the long chains depends on their proportion 30
II.5.1. Double reptation process (Tsenoglou des Cloizeaux): The simplest way to account for CR effect µ A (t): the theoretical remaining (oriented) part of the initial tube segments of a molecule A Φ(t): the total fraction of survival initial tube segments entanglements Probability to have an entanglement between a chain A and a chain B The probability that such an entanglement still exists at time t 31
Monodisperse sample: F A (t): the measured unrelaxed part of the molecule A Polydisperse sample: Also affected by the CR process Continuous form: M w distribution Dilution exponent. From 1 to 1.3 (generalized double reptation) 32
Predicted behaviour of monodisperse linear samples: with (---) or without the CR process: 33
II.5.2. Dynamic Tube Dilution process (DTD) (Marrucci, Ball - McLeish): The polymer fraction already relaxed = solvent For t 0 After the relaxation of the blue chains stroboscopic effect: at time long enough (or low frequencies), some details of the polymer dynamics can be neglected. Rescale of the equilibrium parameters: 34
Rem/ : Effective molecular weight between two entanglements. In the real case, the relaxed polymer fraction does not act immediately as a solvent. Its relaxation time must be short enough compared to the observed time. 35
Effect of the DTD on the reptation time: Marrucci 85 - Monodisperse linear chain: no effect. - Bidisperse linear chains: effect only if their M w is well separated. 36
Watanabe 2004 Wang, 2003 Proportion of long chains 37
Effect of the DTD on the fluctuations times: x=1/2 The fluctuations times along the curvilinear axis of the tube are well separated in time. x=1/2 Large effect of the DTD process 50 45 Without the solvent effect Log(τ fluc ) 40 35 30 25 20 15 10 5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x 38
Monodisperse star polymers: with Bidisperse star polymers: with x 1 and x 2 are related by the fact that they correspond to the same potential: U(x 1 ) U(x 2 ) x 2 1 x 1 0 x 1 x 2 39
Blend of polyisoprenes star polymers: M arm = 28000g/mol 144000 g/mol 76% 28000 80% 28000 50% 28000 80% 28000 (without considering the Rouse process) Data: Blottière, B; McLeish, T. C. B; Hakiki, R. N; Milner, S. T; Macromolecules 1998, 31, 9295 40
Relaxation modulus of a monodisperse star polymer: Fluctuations times: expressed as a function of x. The relaxation modulus must be determined by account for each x contribution: Relaxation modulus of a bidisperse polymer: All the x_transition must be defined a priori Milner-McLeish Becomes very quickly fastidious! 41
Example: Relaxation modulus of a bidisperse linear polymer: Rouse contribution Before the fluct. of chain 1 After the rept. of chain 1 but before the rept. of chain 2 Before the fluct. of chain 1 Just after the rept. of chain 1 rept. of chain 1 rept. of chain 2 (Park and Larson 2004) 42
Relaxation modulus: Time-marching algorithm No analytical function can be found for complex polymers Molecules Time t Reptation Fluctuations Polymer «solvent» Explicit time-marching algorithm: G(t) G (ω), G (ω) Φ(t i-1 ) τ reptation (x,t i ) τ fluctuation (x,t i ) p survival (x, between t i-1 and t i ) Φ(t i ) G(t i ) t i-1 t i t p survival (x, t i ) = p survival (x, t i-1 ). p survival (x, between t i-1 and t i ) E. van Ruymbeke, R. Keunings, C. Bailly, J.N. N. F. M., 128 (June 2005) 43
Relaxation modulus: Time-marching algorithm calculated by summing up all contributions over types of arms and positions along the arms 1 x 0 x 0 1 0 x All types of arms x i not relaxed by fluctuations (fractions ϕ i ) x i not relaxed by reptation x i not relaxed by the environment if not relaxed otherwise 44
Extra condition on the DTD process: The necessary time for a chain to move from its skinny tube to its dilated tube, is, at least, equal to the corresponding Rouse time: For t 0 After the relaxation of the blue chains Motions are not instantaneous Example: 97.5 % linear PBD with M w = 24kg/mol + 2.5% PBD star with M arm = 342kg/mol :experimental data (1) ---: theoretical data G (ω), G (ω) The stars relax too quickly (1) : Roovers, J.;Macromolecules, 1987, 20, 148 With the extra condition 45 ω
II.6. Models branched polymers: Comb polymer symmetric star H polymer Asymmetric star Linear chains Tree-like polymer Pompom polymer Ring polymer + bidisperse systems and polydisperse linear polymers The quality of the samples are very important! Future challenge: symmetric comb polymer asymmetric comb polymer Final objective: industrial samples 46
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Example of question related to these structures: How does this molecular section relax? Reptation Contour length Fluctuations 2 Fluctuations modes: x branch =1 x branch =0 Relaxed branches Additional friction Hierarchy of motions 48
Larson (Macromol. 2001): M a M b 49
Structure - LVE relationship for complex polymers H PS polymers (Roovers 1984): M = 19k, 45k, 103k, 132k or 205k M M M M M M b = M a G (ω) G (ω) tanδ ω G,G (ω) ω ω Linear PS M=132k and 800k ω 50
Comb structures: 1) Relaxation of the branches (contour length fluctuations) 2) Relaxation of the backbone (CLF or reptation) 51
Structure - LVE relationship for complex polymers Ma Pompom PBD polymers (Archer 1998): G,G (ω) G,G (ω) 47k 14.8k Mb G (ω) M = 176k, 129k, 86k 47k 19.5k ω G,G (ω) G,G (ω) 90k 12k ω ω G (ω) 90k 19.5k 52 EVR et al. Macromol. 2006 ω
Polydispersity effect: Polydispersity fixed to 1.05 G, G (ω) [Pa] G, G (ω) [Pa] Log(M) Backbone: H=1 to 1.2 a Arms: b H=1 to 1.2 Explicit timemarching algorithm: G (ω) [Pa] ω (1/sec) G (ω) [Pa] ω (1/sec) Φ DTD (t) calculated at each time step c d ω (1/sec) Backbone and arms: H=1 to 1.2 ω (1/sec) 53
Cayley-tree polymers 3 fluctuations modes: Normal friction With additional friction from the G3 Coordinate system: Continuity between the fluctuation processes: τ fluc (x,g2 =x br ) = τ fluc (x G3=1) τ fluc (x G1 = x br ) = τ branch (x G2 =1) 1 2 3 M 1 M 2 M 3 x=1 x=x br2 x=x br3 M G2 Reference coordinate system G 2 coordinate system x G2 =1 x G2 =0 M G1 x G1 =1 x G1 =0 G 1 coordinate system 54
Tree-like polymers (Hadjichristidis) G3 5-6-5 G3 32-6-5 G2 19-23, 30% diluted G2 19-23 G1 G2 E. van Ruymbeke et al., Macromol. 2007 55
Ring structures: A A B B A B Relaxation time Self-similar stress relaxation Obukhov et al, PRL 1994 Rubinstein, Colby, 2003 56
Data: D. Vlassopoulos T ref = 170 o C PS, 160 kg/mol We do not observe a plateau in the G data. 57
Towards supramolecular assemblies (Hadjichristidis) Star PBD 58 EVR et al., Macromol. 07
Telechelic systems: sticky end x=1 Network Linear (M=20k), 3arm stars (Ma=10k), 12arm stars (Ma=10k), 3arms star 12arms star x=0 Linear chains D. Vlassopoulos 59
Towards supramolecular assemblies With J.F. Gohy, C.A. Fustin, F. Stadler, C. Bailly 60
Practical work: - Test of the model of Doi and Edwards with fluctuations and with CR or - Test of the model of des Cloizeaux with CR You will receive 3 curves of G,G for monodisperse or bidisperse linear PBD chains. You will have to determine the Mw of the molecules Rem/ From G(t) to G,G : Use of the Schwarzl functions 61