Lecture: Dynamics of Polymer Solutions and Melts
- 1. Textbook: Plastics: Materials and Processing (Third Edition), by A. Brent Young (Pearson, NJ, 2006). Structure and Properties of Engineering Polymers Lecture: Dynamics of Polymer Solutions and Melts Nikolai V. Priezjev
- 2. Dynamics of Polymer Solutions and Melts 1. Length scales and fractal structure of polymer solutions 2. Brownian diffusion of a spherical particle in a viscous fluid 3. Review of dynamics of unentangled polymer solutions (Rouse and Zimm models) 4. Structure and dynamics of entangled polymer melts (primitive path analysis) M. Rubinstein & R. Colby, Polymer Physics (2003)
- 3. The Koch Curve construction Fractal Dimension r2 r1 D rm ~ D = Fractal Dimension 2 ~ rg holds for any subsection of the ideal chain (no interaction between monomers far apart along the chain; random walk; D=2) with g monomers and size r. Real chains (self- avoiding random walk) are swollen D=5/3. Rrl RN D ~/1
- 4. Fractal Structure of Mucins in Semidilute Solution x Correlation length x=50nm at mass concentration c=1% (monomers interact with solvent and with monomers from the same section of its own chain) b Molecular brush on scale b=5-10nm On scale b<r<x excluded volume interactions swell mucins. Rg On larger length scales excluded volume interactions are screened. Radius of gyration Rg ~ 200-300nm (depends on concentration c) Contour length L=20mm (1mm=1000nm) c < c* c = c*~ 0.1% c*<c<<1
- 5. – velocity of particle due to applied force Brownian (Diffusive) Motion Dtrtr 60 2 D – diffusion coefficient f v v f vf Einstein relation: kT D Stokes Law: R 6 R – radius of the particle – viscosity of the fluid R kT D 6 – Stokes-Einstein relation mean-square displacement: viscous drag -f – friction coefficient kT R D R 22 Time required for a particle to move a distance of order of its size: M. Rubinstein & R. Colby, Polymer Physics (2003)
- 6. N beads connected by springs with root-mean square size b. – friction coefficient of a bead R = N – total friction coefficient of the Rouse chain N kTkT DR R diffusion coefficient of the Rouse chain 2 2 NR kTD R R R Rouse time for t < R – viscoelastic modes for t > R – diffusive motion Time required for a chain to move a distance of order of its size R Rouse Model (1953) Rg M. Rubinstein & R. Colby, Polymer Physics (2003) Ideal chain: no interaction between mono- mers that are far apart along the chain
- 7. Rouse Model (continued) N beads connected by springs with root-mean square size b. – friction coefficient of a bead 21 0 21 2 NN kT b R kT b 2 0 – Kuhn monomer relaxation time (time scale for motion of individual beads) For ideal chain: = 1/2 From the Stokes Law bs kT b s 3 0 2 3 2 0 N kT b N s R Rouse model: draining limit 2 2 NR kTD R R R Fractal structure where Rouse 1953 s – solvent viscosity M. Rubinstein & R. Colby, Polymer Physics (2003) bNR Ideal chain: no interaction between mono- mers that are far apart along the chain
- 8. Zimm Model (1956) Hydrodynamic interactions couple the motion of monomers with the motion of solvent. Chain drags with it the solvent in its pervaded volume. Friction coefficient of chain of size R in a solvent with viscosity s is RsZ Zimm diffusion coefficient: R kTkT D sZ Z Zimm time: 3 0 3 3 3 2 NN kT b R kTD R ss Z Z 3 12 for <1 Zimm time is shorter than Rouse time in dilute solutions. Hydrodynamic interactions are important in dilute solutions. 2/3 ~Nz 5/9 ~Nz in q-solvent in good solvent bNR Rg s M. Rubinstein & R. Colby, Polymer Physics (2003) Fractal structure
- 9. Self-Similar Dynamics of Polymer Chains Rouse Model Chains are fractal – they look the same on different length scales and move in the same way on different time scales. Zimm Model Longest relaxation time: 2 0NR Rouse time Zimm time 3 0NZ Smaller sections of the polymer chain with g=N/p monomers relax just like a g-mer. p-th mode involves relaxation of N/p monomers. 2 0 p N p 3 0 p N p At time p modes with index higher than p have relaxed, while modes with index lower than p have not relaxed. At time p there are p un-relaxed modes per chain each contributing energy of order kT to the stress relaxation modulus: p Nb kT G p 3 = Vpol/Vsol – volume fraction of the polymer in solution )//( 3 pNb number density of sections with N/p monomers
- 10. Stress Relaxation Modulus After a Step Strain G(t) is the “stress relaxation modulus” G(t) can be determined by applying a constant strain, s, and observing stress relaxation over time: / )( )( )( t s s eG t tG )()( 0 GdttG viscosity: M. Rubinstein & R. Colby, Polymer Physics (2003)
- 11. p Nb kT G p 3 Index p of the mode that relaxes at N t p 2/1 0 Stress relaxation modulus interpolation at intermediate times 0 < t < relax 2/1 0 3 t b kT tG N t p 3/1 0 p N t p 0 3/1 0 3 t b kT tG R tt b kT tG exp 2/1 0 3 Z tt b kT tG exp 3/1 0 3 Stress relaxation modulus approximation for all t > 0 Rouse Model Zimm Model Self-Similar Dynamics of Polymer Chains
- 12. t/0 100 101 102 103 104 105 106 107 b 3 G(t)/kT 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 -1/2 Stress Relaxation Modulus: Rouse Model R tt b kT tG exp 2/1 0 3 Scaling approximation relaxation time of p-th mode N p p t Nb kTtG 1 3 exp)( 2 2 2 6 p N kT b p Exact Rouse solution: (points) (solid line) N = 103 number of monomers per polymer chain Rouse model applies to melts (=1) with short unentangled chains N b 0 )( dttG M. Rubinstein & R. Colby, Polymer Physics (2003)
- 13. Normal Mode Analysis: MD simulations of Bead-Spring Model Dynamics of entangled linear polymer melts: A molecular-dynamics simulation, Kremer and Grest, J. Chem. Phys. 92, 5057 (1990) Rouse modes; p = mode # 2 0 p N p p=1 p=1 p=1 p=2 p=2 p=2
- 14. Mean Square Displacement of Monomers Rouse Model Section of N/p monomers moves by its size b(N/p)1/2 during its relaxation time p 2 22 0 p N brr jpj 2/1 0 22 /0 tbrtr jj Mean square monomer displacement for 0 < t < relax Zimm Model for ideal chain = 1/2 3/2 0 22 /0 tbrtr jj p/N = (t/t0)-1/2 p/N = (t/t0)-1/3 Sub-diffusive motion 2 0jj rtr t 0 b2 1/2 1 R R2 1 2/3 Sections of N/p monomers move coherently on time scale p Z Monomer motion in Zimm model is faster than in Rouse model ok for melts ok for dilute solutions Dtrtr 60 2
- 15. Summary of Unentangled Dynamics Rouse model – local monomer friction and no hydrodynamic interactions. It is applicable to unentangled polymer melts. Rouse friction coefficient of an N-mer is N and diffusion coefficient )/( NkTDR Zimm model – motion of monomers is hydrodynamically coupled. Polymer chain drags solvent in its pervaded volume. It is applicable to dilute solutions. Diffusion coefficient )/( RkTD sZ Polymer diffuses distance of order of its size during its relaxation time 2 NR kT R 3 R kT s Z Hydrodynamic interactions in semidilute solutions are important up to the scales of hydrodynamic screening length. On larger length scales both excluded volume and hydrodynamic interactions are screened by surrounding chains.
- 16. Viscoelastisity of Entangled Polymer Melts T Elastic T = terminal relaxation time Viscous flow Relaxation Modulus for Polymer Melts The stress relaxation G(t) for two polymers with different molecular weights. At short time the curves are identical. At intermediate times we have a plateau with a constant modulus – the plateau modulus. The plateau ends at a terminal time T which depends strongly on molecular weights (N) according to a power law T ~ Nm where the exponent m ≈ 3.4 Two Q’s arises: 1) Why is m ≈ 3.4 almost universal? 2) why do we have an almost purely elastic behaviour at the plateau? Polymerviscosity Molecular weight (chain length) 2 1 GP 4.3 0 ~)( NGdttG TP )0(0
- 17. Experimental Shear Relaxation Moduli Poly(styrene) GP Low N High N ~ 1/tG. Strobl, The Physics of Polymers
- 18. Viscosity of Polymer Melts Poly(butylene terephthalate) at 285 ºCFor comparison: for water is 10-3 Pa s at room temperature. Shear thinning behaviour Extrapolation to low shear rates gives us a value of the “zero- shear-rate viscosity”, o. o From Gedde, Polymer Physics
- 19. ~ TGP o ~ N3.4 N0 ~ N3.4 Universal behavior for linear polymer melts Applies for higher N: N>NC Why? G. Strobl, The Physics of Polymers, p. 221 Data shifted for clarity! Viscosity is shear-strain rate dependent. Usually measure in the limit of a low shear rate: o 3.4 Scaling of Viscosity: o ~ N3.4
- 20. FENE bead- spring model: 2 2 FENE o 2 o 1 r V (r) kr ln 1 2 r k = 30 2 and ro = 1.5 Lennard-Jones potential: 12 6 LJ r r V (r) 4 Simulation of the tube: polymer melt with linear chains of N beads MD simulations of a chain with N=400 monomers in an entangled polymer melt. Forty configurations of the chain are shown at equally spaced time intervals 600.
- 21. Entangled Polymer System: Reptation and the Tube Model M. Rubinstein & R. Colby, Polymer Physics (2003)
- 22. Entangled Polymer System: Reptation and the Tube Model M. Rubinstein & R. Colby, Polymer Physics (2003)
- 23. Entangled Polymer System: Reptation and the Tube Model M. Doi and S. F. Edwards, The Theory of Polymer Dynamics, pps. 189-194, Oxford Science: New York (1986). Constraint Release & Re-entangled Mechanisms A 2D projected viewpoint A chain in a fixed network of obstacles
- 24. Q1: The tube model and the idea of reptation (P. de Gennes) Every segment in the tube have a mobility, μseg restricted by the surrounding “resistance”. The tube with N segments have a mobilty, μtube= μseg/N Brownian motion within the tubes confinement – use Einstein relation to calculate a diffusion coefficient => => Dtube= kBT μtube= kBT μseg/N A polymer chain escapes from its own tube of length L after a time T const N D L LD 3 tube 2 T 2 tubeT d This relationship is close to the best experimental fit 4.3 ~ N
- 25. Q2: The rubber plateau and entanglements T Elastic T = terminal relaxation time Viscous flow Relaxation Modulus for Polymer Melts 2 1 GP Rubber plateau arises from chain entanglements. It can be shown that in a rubber, a cross-linked polymer, the elastic modulus depends on the average molecular mass between cross-links Mx, R, T and the density ρ: x P M RT G Adopting an identical relation and treating the entanglements as temporary cross-links with a lifetime of the order of τT we can calculate an average mass of the molecular mass between the entanglements (Mx). strand The value of GP is independent of N for a given polymer GP = nkT, where n is # of strands per unit volume.
- 26. Rheology and Topology of Entangled Polymers Rheology and Microscopic Topology of Entangled Polymeric Liquids, Everaers et al. 303, 823 Science (2004) Primitive Path Analysis (PPA):
- 27. Summary of Entangled Dynamics Viscous flow T Elastic T = terminal relaxation time 2 1 GP GP is independent of N for a given polymer: GP ~ N0. T~/GP ~ N3.4
- 28. Polymer Self-Diffusion X Time = 0 Time = t Reptation theory can also describe the self-diffusion of polymers, which is the movement of the centre-of-mass of a molecule by a distance x in a matrix of the same type of molecules. In a time tube, the molecule will diffuse the distance of its entire length. But, its centre-of-mass will move a distance on the order of its r.m.s. end-to-end distance, R. In a polymer melt: <R2>1/2 ~ aN1/2 R
- 29. Polymer Self-Diffusion Coefficient X tubetube self NaaN t x D 22212 = )( ~~ / A self-diffusion coefficient, Dself, can then be defined as: Larger molecules are predicted to diffuse much more slowly than smaller molecules. But we have derived this scaling relationship: 3 Ntube ~ Substituting, we find: 2 3 2 ~~ N N Na Dself https://www.slideshare.net/NikolaiPriezjev
- 30. Testing of Scaling Relation: D ~N -2 M=Nmo -2 Experimentally, D ~ N-2.3 Data for poly(butadiene) Jones, Soft Condensed Matter, p. 92 “constraint release”
- 31. Application of Theory: Electrophoresis
- 32. Relevance of Polymer Self-Diffusion When welding two polymer surfaces together, such as in a manufacturing process, it is important to know the time and temperature dependence of the diffusion coefficient D. Good adhesion is obtained when the molecules travel a distance comparable to R, such that they entangle with other molecules. R https://www.slideshare.net/NikolaiPriezjev
- 33. Stages of Interdiffusion at Polymer/Polymer Interfaces Interfacial wetting: weak adhesion from van der Waals attraction Chain extension across the interface: likely failure by chain “pull- out” Chain entanglement across the interface: possible failure by chain scission (i.e. breaking) https://www.slideshare.net/NikolaiPriezjev
- 34. Strength Development with Increasing Diffusion Distance K.D. Kim et al, Macromolecules (1994) 27, 6841 Full strength is achieved when d is approximately the radius of gyration of the polymer, Rg. Rg d https://www.slideshare.net/NikolaiPriezjev