Simulation of Coarse-Grained Equilibrium Polymers J. P. Wittmer, Institut Charles Sadron, CNRS, Strasbourg, France Collaboration with: M.E. Cates (Edinburgh), P. van der Schoot (Eindhoven) A. Milchev (Sofia), D. Landau (Athens, Georgia) J.-P. Ryckaert (Brussels), H. Xu (Metz) Outline: F. Crevel, A. Johner,... (Strasbourg) 1. Introduction: Definitions, Coarse-graining, Focus 2. Algorithm: Effective Hamiltonian & Tricks & Benchmark 3. Static properties I: Gaussian chain of blobs 4. Static properties II: Non-Gaussian corrections 5. Dynamics: Equilibrium rate constants 6. Summary 7. Perspectives
1. Introduction Definition: Equilibrium polymers with annealed (thermalized) mass distribution Wormlike micelles Coarse-grained (bead-spring) polymer model Cylindrical micelles COARSE GRAINING k k r s Surfactant Equilibrium polymer B E General interest: - Test of theoretical models (static + dynamics) for dead polymers - Dynamics: B E 10k B T for topological constraints (reptation) - Fast equilibration for B 0, ease to work grand-canonically Focus here: - Good solvents (v > 0), d = 3, flexible chains ( isotropic solutions) - Long chains (E 1 N 1) at large densities (ρ ρ 1) - Rouse dynamics: B 1k B T
2. Algorithm: Effective Hamiltonian Bonded and non-bonded interactions: Hard Spheres-Hard Bonds Bond-Fluctuations Model (MC) - all beads r ij > σ - bonded beads: σ < r ij < 2σ MC: Local (athermal) jumps of beads CD: Elastic collisions of beads Bead-Spring Model (BSM) for MD and off-lattice MC: Repulsion: LJ/Morse, Binding: FENE/Springs Scission and recombination of bonded intra-actions: Metropolis (MC) with attempt frequency ω = exp( B): Check for energy change and constraints (ring closure,...) Local Jump Lattice model with restricted set of bonds
2. Algorithm: Tricks of the Trade Topology change is a MC step: Detailed Balance problem Weights for scission! Data structure for EP: (a) 6 6 6 1 1 (b) Recombine ibond=2 and jbond=5 (c) Break ibond= 1 3 4 1 5 5 5 2 2 5 3 5 5 2 3 2 2 4 4 2 pointer( 6) = 1 pointer(1) = 6 pointer( 5) = 1 pointer( 1) = 5 pointer( 2) = 4 pointer(4) = 2 pointer(2) = 5 pointer(5) = 2 pointer( 1) = NIL pointer( 5) = NIL - Connection of chains in bond space - Symmetric list connecting bonds - 2 operations for breaking/recombination - No sorting of monomers needed - Absolut coordinates for dynamics Polydispersity Large boxes to avoid finite-size effects: nr. of mons 10 4 N Milchev, Wittmer, Cates, J.Chem. Phys., 109, 834 (1998)
2. Algorithms: Monodisperse chain benchmark Mix of local, snake and double-pivot moves to sample the phase space: LOCAL MOVE 7 6 5 SNAKE MOVE 1 2 3 4 DOUBLE PIVOT 6 4 3 5 7 1 2 Bond-Fluctuation Model (BFM): - 8ρ 1/2, N 8192 - Athermal, flexible chains Bead-Spring Model (BSM): - ρ = 0.84, N 1024 - Similar results, poorer statistics. Beware of correlations!
3. Static properties: Linear chains Allow only linear chains Get distribution c 1 (N) for given density Flory-Huggins functional: Ω[c 1 (N); V, µ] = N=1 c 1 (N){log(c 1 (N)) + E + f end (N, ρ, E) + µn} R(ρ) Mean-Field: f end (N, ρ) = const Dilute: f end (N, ρ) = (γ 0 1) log(n) Semi-dilute: f end (N, ρ) = (γ 0 1) log(g(ρ)), g(ρ) ρξ 3 ρ 1/(3ν 0 1) Minimization: Mean-Field: c 1 (x) = exp( x) with x N/ N Dilute: c 1 (x) x γ0 1 exp( γ 0 x) Semi-Dilute: c 1 (x) exp( x) for N g(ρ), ρ ρ
3. Static properties: Linear chains Mean chain length N N (E)(ρ/ρ (E)) α ρ α exp(δe) with N exp(e/2.93), ρ exp( E/3.8) dilute: α 0 = δ 0 = 1 1+γ 0 0.46 semidilute: α 1 = 1 2 (1 + γ 0 1 3ν 0 1 ) 0.6, δ 1 = 1/2 <N> 10 4 10 3 10 2 φ = 0.001 φ = 0.005 φ = 0.01 φ = 0.05 φ = 0.1 φ = 0.4 φ = 0.5 φ = 0.6 BFM δ 1 =1/2 N * <N>/N * E=3 E=5 E=6 E=8 E=10 E=13 E=15 α 1 =0.60 δ 0 =0.46 α 0 =0.46 0 5 5 E 10-1 10-2 10-1 10 2 φ/φ
3. Static properties: Linear chains Mean chain length N N (E)(ρ/ρ (E)) α ρ α exp(δe) <N> 10 4 10 3 10 2 with N exp(e/2.93), ρ exp( E/3.8) dilute: α 0 = δ 0 = 1 1+γ 0 0.46 semidilute: α 1 = 1 2 (1 + γ 0 1 3ν 0 1 ) 0.6, δ 1 = 1/2 φ = 0.001 φ = 0.005 φ = 0.01 φ = 0.05 φ = 0.1 φ = 0.4 φ = 0.5 φ = 0.6 BFM δ 1 =1/2 N * δ 0 =0.46 <N>/N * E=3 E=5 E=6 E=8 E=10 E=13 E=15 α 0 =0.46 α 1 =0.60 0 5 5 E 10-1 10-2 10-1 10 2 φ/φ
3. Static properties: Linear chains <R g 2 > 10 5 10 4 10 3 10 2 Conformational Properties: BFM: ρ = 0.1/8 ξ 2 = 20a 2 s 2ν 0 s 1 g(ρ)=160 E = 10 E = 15 c 1 (x) 10 2 10-1 10-2 10-3 Finite-Size-Effects: BFM: E=10,ρ=0.5/8 One chain gets it all! L=10 L=20 L=25 L=30 L=40 L=100 10 2 10 3 10 4 s Same as dead polymers! 10-4 0 2 4 6 8 x=n/<n> Large boxes required!
3. Static properties: Pressure and compressibility Pressure for MF-MC data Flory-Huggins: P = Ω[c eq (N)] = ρ/ N + R(ρ) Dilute: P ρ/ N ρ 0.54 Semidilute: P R(ρ) 1/ξ 3 ρ 2.3 Compressibility: g = κ T ρ = ρ/ P Semidilute: κ T ξ 3 ρ 2.3 Experimental data from Candau group EPL 54, 58 (2001) 10 2 P κ T 10-1 10-2 10-3 10-4 0.54 E=3 E=5 E=7 E=9 E=11 E=13 MORSE beads SLOPE 1 2.3 10-5 10-3 10-2 10-1 10 ρ 0 10 5 10 4 10 3 10 2 10-1 -0.54 Additional Physics E=5 E=7 E=9 E=11 E=13 Buhler RG -2.31 10-2 10-3 10-2 10-1 ρ Conclusion: Additional physics at large densities where blobs too small
4. Static properties: Non-Gaussian Corrections P. J. Flory (Nobel Chem. 1974): Polymer chains in the melt are stongly entangled (d = 3) and are therefore Gaussian!
4. Static Properties: Non-Gaussian Corrections P. J. Flory (Nobel Chem. 1974): H. Benoit: S. F. Edwards: Polymer chains in the melt are stongly entangled (d = 3) and are therefore Gaussian! Experimental verification (Kratky plot q 2 F(q) vs. q) RPA description of screening for r ξ 1/ρ 1/2
4. Static properties: Non-Gaussian Corrections P. J. Flory (Nobel Chem. 1974): H. Benoit: S. F. Edwards: Polymer chains in the melt are stongly entangled (d = 3) and are therefore Gaussian! Experimental verification (Kratky plot q 2 F(q) vs. q) RPA description of screening for r ξ 1/ρ 1/2 Only short-range INTRAactions along chains should matter: P(s) exp( s/s p ) Notations: i=n Re i=1 n l n r nm l m s=m n P(s) = ln l m=n+s /l 2 R 2 (s) = ( r n r m=n+s ) 2 Assuming translational invariance: P(s) = 1 2 R(s) 2 2l 2 s 2 m
4. Static properties: Long-range bond-bond correlation! 10-1 10-2 E=11 E=13 N=2048 N=4096 PRL, 93, 147801 (2004) Result robust: P(s) 10-3 10-4 Slope 1/s 3/2 Noise! - different simulation models - higher Legendre-Polynomials - coarse-graining: non-locality! 10-5 10 2 s P(s; N) N = P(s) = c a s ω with c a = 6/π 3 4ρb 3 e - Scaling with ρ and v, ω = d/2 = 3/2 Return-probability! Perturbation theory for finite s/n: P(s; N) = c a s 3/2 f(s/n) with f(x 1) 1, f(x 1) 0 Small N roughly exponential power law was overlooked
4. Static properties: Segmental correlation hole! a density ρ = const b c *(s) c(r,s) Segmental correlation hole R(s) r total chain R(s) Repulsion c (s) = c(r R(s), s) s/r(s) d s 1 dν 1/ s s chains interact if d = 3, ν = 1/2! hierarchy of nested segmental correlation holes of all sizes aligned and correlated along the chain backbone
4. Static properties: U(r, s) ln p(r, s) pair-distribution between cm of segments ( ) p(r,s) p(,s). U(r, s) p(,s) p(r,s) p(,s) c(r, s)/ρ U (s) = U(r = 0, s) = s 0,5 Statistics ρr(s) d U(r,s)/U * (s) 0,4 0,3 0,2 0,1 Repulsion BFM c(r,s) R(s) 3 /s BFM s=64 BFM s=256 BFM s=1024 BFM s=n=4096 BSM s=128,n=1024 Gauss 0,0 Weak attraction -0,1 0 1 2 3 r/r(s) U (s) 1/s 1/2 for d = 3 Standard Perturbation!
4. Static properties: Size of curvilinear length Detailed perturbation calculation... 1 R2 (s) b 2 e s 24/π 3 U (s) 1/ s in d = 3 10 R 2 (s)/s 9 8 Chain end effects E=11: <N>=887 E=13: <N>=2401 E=15: <N>=6011 N=1024 N=2048 b e 2 ( 1 - κ R /s 1/2 ) with b e =3.245 10 2 10 3 10 4 s P(s) 2 s R(s) 2 1/s d/2
4. Static properties: Size distribution of EP Ω[c 1 (N); V, µ] = N=1 c 1(N) (log(c 1 (N)) + E + U (N)) Non-exponentiality: β p 1 Np p! N β p = w p κ ep / N. κ ep =1/6.44 w p 10 5 β p / (w p κ ep ) 10-1 p=2 p=3 p=4 p=5 p=6 p=7 p=8 BFM ρ=0.5/8 0 0 5 p 1/<N> 1/2 10 2 10 3 <N>
Simulation of Coarse-Grained Equilibrium Polymers J. P. Wittmer, Institut Charles Sadron, CNRS, Strasbourg, France Collaboration with: M.E. Cates (Edinburgh), P. van der Schoot (Eindhoven) A. Milchev (Sofia), D. Landau (Athens, Georgia) J.-P. Ryckaert (Brussels), H. Xu (Metz) Outline: F. Crevel, A. Johner,... (Strasbourg) 1. Introduction: Definitions, Coarse-graining, Focus 2. Algorithm: Effective Hamiltonian & Tricks & Benchmark 3. Static properties I: Gaussian chain of blobs 4. Static properties II: Non-Gaussian corrections 5. Dynamics: Equilibrium rate constants 6. Summary 7. Perspectives
4. Equilibrium dynamics: Mean-Square Displacements 10 5 10 4 ω = 1 ω = 0.1 ω = 0.01 BFM ρ=0.5/8 D s ~ r *2 /t * g(t) 10 3 10 2 r *2 ~t *1/2 E = 5 E = 10 SLOPE α=1/2 t * (E,ω) 10 2 10 3 10 4 10 5 10 6 10 7 t/mcs
4. Equilibrium dynamics: Life-Time distribution 10 3 Ψ(t) τ b /κ 10 2 BFM ρ=0.5/8,ω=1 Slope - 5/4 10-1 10-2 10-3 10-4 E=10 E=9 E=8 E=7 E=6 E=5 10-5 10-3 10-2 10-1 t/τ b Power law predicted by O Shaughnessy, Yu, PRL (1995) Recombination time = Scission time τ b = 1/k s N E-Jumps: E δe E τ b (E)
τ b <N> e -E / t h 4. Equilibrium dynamics: Scaling in Rouse limit h : mean distance between adsorbers Characteristic time t h h 4 τ b = 1 N e E ω f(ωτ h) D s = D s (ω = 0)g 10 3 10 2 10-1 10-2 10-3 BFM ρ=0.5/8 E=10 E=9 E=7 E=5 E=3 E=1 E=0 t h =h 4 1/x 1/2 10-4 10-1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 x=ω t h D s /D s (ω=0) E=10 E=9 E=7 E=5 E=3 E=1 E=0 y=0.75x 1/3 ( ) τ (ω=0) τ b 10-3 10-2 10-1 10 2 10 3 10 4 10 5 t * (ω=0)/τ b (ω) Complete scaling with ω difficult due to non-universal physics at short times Relaxation time goes stronger down as expected
6. Summary Statics of flexible EP (linear, linear + rings) essentially understood in terms of Flory-Huggins + blob scaling picture from dead polymers Rings allowed c 0 (N) 1/N 5/2 Non-universal aspects for flexible chains Non-Gaussian corrections lead to swelling of chain segments! Similar results for lattice and off-lattice. Peculiar large density effects. Rouse dynamics for large E, large ω = exp( B) with t (ω) τ b (ω) 7. Perspectives MD code (DPD, Lattice Boltzmann) with stiffness and (weak) branching Larger barrier needed to obtain entangled dynamics: - Levy flight dynamics (Langevin, Bouchaud)?! - Viscosity around ρ, role of rings?