Marcus M üller and Kostas Ch. Daoulas

1. Soft, coarse-grained models for multi-component polymer melts: free energy and single chain dynamics • outline: • soft, coarse-grained models • free energy of self-assembled structures • kinetics (single-chain and collective) Marcus Müller and Kostas Ch. Daoulas Development and Analysis of Multiscale Methods Minneapolis, Nov 3, 2008

2. diffusion ~ 10-9 -10-6 s bond vibrations ~ 10-15 s conformational rearrangements ~ 10-12 - 10-10 s coarse-grained models: time and length scales ordering kinetics ~ hours/days Edwards, Stokovich, Müller, Solak, de Pablo, Nealey, J. Polym. Sci B 43, 3444 (2005)

3. diffusion ~ 10-9 -10-6 s bond vibrations ~ 10-15 s conformational rearrangements ~ 10-12 - 10-10 s coarse-grained models: time and length scales a small number of atoms is lumped into an effective segment (interaction center) MC,MD, DPD, LB,SCFT minimal coarse-grained model that captures only relevant interactions: connectivity, excluded volume, repulsion of unlike segments • incorporate essential interactions through a small number of effective parameters, chain extension, Re, compressibility kN and Flory-Huggins parameter cN • soft potentials, elimination of degrees of freedom efficient techniques for large systems (106 segments)

4. soft, coarse-grained models starting point for field-theoretic and particle-based description relevant interactions: connectivity, excluded volume, repulsion of unlike segments systematic coarse-graining procedures (RG) demonstrates that effective interactions become weaker as one increases the degree of coarse-graining no (strict) excluded volume, effective segments can overlap, rather: enforce low compressibility on length scale of interest, ## terms of order generate pair-wise interactions, more general density functionals can be employed for polymer solutions or solvent-free models of bilayer membranes molecular architecture: Gaussian chain with

5. soft, coarse-grained models “ ´’’ express density through particle coordinates particle-based description amenable to MC, BD, DPD or SCMF-simulations • regularize d-function by lattice with discretizationDL MC or SCMF-simulation • regularize d-function by a weighting function (WDA) DPD-like models Daoulas, Müller, JCP 125, 184904 (2006), PM-methods (electrostatics), PIC (plasma physics) computationally fast but not translationally invariant Laradji, Guo, Zuckermann PRE 49, 3199 (1994) Groot,Warren, JCP 107, 4423 (1997) slower in dense systems (factor 10-100) but translationally invariant

6. Single-Chain-in-Mean-Field (SCMF) simulations algorithm: 1. simulate an ensemble of many independent molecules in real, fluctuating, external fields, wA and wB, for a predefined number of MC steps 2. calculate (coarse-grained) densities, fA and fB, on a grid, and calculate new, external fields 3. goto 1. if it converges and the ensemble is very large the average densities will relax towards a solution of the (equilibrium) SCF equations Do SCMF simulations describe fluctuations for a finite ensemble of molecules? SCMF simulations provide a controlled approximation quasi-instantaneous field approximation depends on discretization Daoulas, Müller, JCP 125, 184904 (2006)

7. 1/r-behavior of total correlation at short scales and gtot(r )=1 for r>x ginter exhibits correlation hole on length scale Re and depth long-ranged correlations between bonds correlation hole and long-ranged correlations r Wittmer, Meyer, Baschnagel, Johner, Obukhov, Mattoni, Müller, Semenov, PRL 93, 147801 (2004)

8. invariant degree of polymerization corresponds to SCFT definition of length without referring to a definition of a segment: interdigition • Ginzburg-parameter that controls regime of critical, Ising-like fluctuations in a binary blend, or shift of ODT from first-order transition in block copolymer • broadening of interfaces by capillary waves • bending rigidity of interfaces, formation of micro-emulsions near Lifshitz-points • depth of correlation hole and amplitude of long-range bond-bond correlations • tube diameter, packing length for Gaussian coils (Fredrickson-Helfand) or (tricritical)

9. large requires soft potentials typical experimental values: • invariant degree of polymerization • typical length scale typical value for a coarse-grained model with excluded volume (BFM, bead-spring): necessary condition: or less (otherwise crystallization, glass) typical values for a coarse-grained model with soft cores eff. interaction centers (segments)/chain chain discretization dense melt of long chains reptation re-entrant melting of underlying soft-spheres fluid + lattice effects choose chains are crossable Rouse-dynamics otherwise

10. summary I • outline: • soft generic coarse-grained models for multi-component polymer melts • coarse-grained model that incorporates the relevant interactions: • chain connectivity/molecular architecture • low compressibility of melt / excluded volume • repulsion between unlike species • invariant degree of polymerization controls fluctuation effects • experimentally large values of are conveniently described • with soft interactions • large time and length scales: structure formation, • phase separation kinetics, self-assembly • free energy of self-assembled structure • kinetics (single-chain and collective) Development and Analysis of Multiscale Methods Minneapolis, Nov 3, 2008

11. free energy of self-assembled structures relevance: properties of coarse- macroscopic behavior grained model examples: • free energy of morphologies accurate location of 1st order phase transitions phase diagram input for coarser models (phase field) • interface and surface wetting behavior (Young’s equation) free energies nucleation • defect free energies free energies of intermediates kinetics of structure formation

12. self-assembly vs. crystallization order parameter: Fourier mode of composition fluctuation Fourier mode of density fluctuation ideal ordered state: SCFT solution ideal crystal (T=0) ideal disordered state: homogeneous melt ideal gas in ordered phase, composition fluctuates in ordered state, particles vibrate around around reference state (SCFT solution), ideal lattice positions but molecules diffuse (liquid) no simple reference state for self- Einstein crystal is reference state assembled morphology use thermodynamic integration wrt to uniform, harmonic coupling of particles to ideal position (Frenkel & Ladd, Wilding & Bruce)

13. self-assembly vs. crystallization order parameter: Fourier mode of composition fluctuation Fourier mode of density fluctuation ideal ordered state: SCFT solution ideal crystal (T=0) ideal disordered state: homogeneous melt ideal gas in ordered phase, composition fluctuates in ordered state, particles vibrate around around reference state (SCFT solution), ideal lattice positions but molecules diffuse (liquid) no simple reference state for self- Einstein crystal is reference state assembled morphology use thermodynamic integration wrt to uniform, harmonic coupling of particles free energy per molecule (ex vol) N kBT to ideal position relevant free energy differences 10-3 kBT (Frenkel & Ladd, Wilding & Bruce) absolute free energy must be measured with a relative accuracy of 10-5

14. self-assembly vs. crystallization order parameter: Fourier mode of composition fluctuation Fourier mode of density fluctuation ideal ordered state: SCFT solution ideal crystal (T=0) ideal disordered state: homogeneous melt ideal gas in ordered phase, composition fluctuates in ordered state, particles vibrate around around reference state (SCFT solution), ideal lattice positions but molecules diffuse (liquid) no simple reference state for self- Einstein crystal is reference state assembled morphology use thermodynamic integration wrt to uniform, harmonic coupling of particles free energy per molecule (ex vol) N kBT to ideal position relevant free energy differences 10-3 kBT (Frenkel & Ladd, Wilding & Bruce) absolute free energy must be measured with a relative accuracy of 10-5 measure free energy differences betweendisordered and ordered phase (10-3 relative accuracy needed)

15. self-assembly vs. crystallization order parameter: Fourier mode of composition fluctuation Fourier mode of density fluctuation ideal ordered state: SCFT solution ideal crystal (T=0) Ideal disordered state: homogeneous melt ideal gas in ordered phase, composition fluctuates in ordered state, particles vibrate around around reference state (SCFT solution), ideal lattice positions but molecules diffuse (liquid) no simple reference state for self- Einstein crystal is reference state assembled morphology use thermodynamic integration wrt to uniform, harmonic coupling of particles to ideal position (Frenkel & Ladd, Wilding & Bruce) PRE 51, R3795 (1995) see also Grochola, JCP 120, 2122 (2004)

16. calculating free energy differences optimal choice of external field (Sheu et al): structure does not change along 2nd branch SCFT: Müller, Daoulas, JCP 128, 024903 (2008)

17. TDI vs expanded ensemble/replica exchange • only replica exchange is • impractical because one • would need several 100 • configurations • at initial stage, where weights h • are unknown (DF~104kBT), • replica exchange guarantees • more uniform sampling • expanded ensemble technique • is useful because it provides • an error estimate

18. TDI vs expanded ensemble/replica exchange no kinetic barrier, ie no phase transition roughly equal probability

19. first-order fluctuation-induced ODT cNODT<14 at fixed spacing cNODT=13.65(10) hysteresis Pike, Detcheverry, Müller, de Pablo, submitted (2008) soft, off-lattice model: measure chemical potential m via inserting method in NpT- ensemble two structures will coexist, if they have same p and m Lennon, Katsov, Fredrickson, PRL 101, 138302 (2008) Einstein-integration for fluctuations of lattice-based density fields around SCFT

20. further applications: T-junctions 0.19(2) 0.21 SCF theory: Duque, Katsov, Schick, JCP 117, 10315 (2002)

21. further applications: rupture of lamellar ordering 0.01(3)

22. further applications: stalks in solvent-free membrane models with Yuki Norizoe

23. summary II • outline: • soft generic coarse-grained models for multi-component polymer melts • free energy of self-assembled structures • general computational scheme to calculate free-energies • of self-assembled structures relies on reversibly converting • one structure into another via an external ordering field • does not rely on soft interactions • (alternative: measure p and msimulateneously) • does not require a field-theoretic formulation • accurate and suitable for parallel computers • metastablity: observed structures may depend on kinetics of structure formation • What is the coarse-grained parameter that parameterizes dynamic properties? • kinetics (single-chain and collective) Development and Analysis of Multiscale Methods Minneapolis, Nov 3, 2008

24. ordering kinetics in thin films: lamellar-forming copolymer on stripe pattern length set by bulk lamellar spacing, L0 time set by time it takes to diffuse L0 experiment: PS-PMMA diblock L0=48nm, D0=42nm DPS =6.8 10-12cm2/s tPS=0.56s DPMMA=9.5 10-15cm2/s tPMMA=404s Mw=100 000, Mwe(PS)= 35 000 observation: stripe period matches L0 hexagonal order @ 3h, registration @ 6h SCMF simulation: L0=1.786Re DPMMA=DPS/10=3.3 10-5Re2/MCS tloc=L02/6D=16 100 MCS match time scale via PS: 1s = 287.5 MCS 1h = 1 035 000 MCS match time scale via PMMA: 1s = 40 MCS 1h = 143 000 MCS cN=36.7, kN=50, N=32=15+17, LN=-3, npoly=44 000 10 stripes with period 1.7Re=0.95L0, system size: 1.2Re * (17Re)2=32nm * (0.457mm)2 Edwards, Stokovich, Müller, Solak, de Pablo, Nealey, J. Polym. Sci B 43, 3444 (2005)

25. ordering kinetics: SCMF simulations PS-PMMA interface (green) 10000 MCS PS-rich regions (red)

26. ordering kinetics: SCMF simulations

27. ordering kinetics: SCMF simulations match time scale via PMMA: 1s = 40 MCS, 1h = 143 000 MC (slow component dictates the ordering kinetics) • time scale to fast (no entanglement effects) • defects anneal out from substrate to surface hexagonal surface morph. morphology no lateral diffusion of defects 100 500 1000 1500 5000 10000 20000=8min =0.66 t

28. towards a more realistic dynamics: slip-links single chain theory (ensemble of independent chains) natural dynamics is Rouse-like, entanglements are not predicted but have to be introduced ``by hand’’ also: softer interactions in coarse-grained models do not guarantee non-crossability idea: restrict lateral motion by tethering to chain contour anchor points aj are fixed in space; attachment points r(sj) hop from one segment to another Likhtman, Macro 38, 6128 (2005)

29. towards a more realistic dynamics: slip-links single chain theory (ensemble of independent chains) natural dynamics is Rouse-like, entanglements are not predicted but have to be introduced ``by hand’’ also: softer interactions in coarse-grained models do not guarantee non-crossability idea: restrict lateral motion by tethering to chain contour anchor points aj are fixed in space; attachment points r(sj) hop from one segment to another

30. towards a more realistic dynamics: slip-links • comparison of the simulation results with predictions of the tube model • yields entanglement lengths, depends on quantity because of approximations • invoked in the tube model (e.g., constraint release). For N=128 and Nsl=32 • one obtains: self-diffusion coefficient: Ne ≈ 3.5 • early mean-square displacements: Ne ≈ 7 • dynamic structure factor: Ne ≈ 12 • single-chain theory entanglements cannot be predicted but are input parameter • 1) concept of packing length: • purely dynamic relation between and • 2) primitive path analysis using the • multi-chain configurations (SL) (SS)

31. towards a more realistic dynamics: slip-links single-chain dynamics in the lamellar phase without and with slip-links entangled dynamics in spatially inhomogeneous systems lamellar phase: cN=80, N=128 Rouse: dynamics parallel and perpendicular decouple, parallel dynamics unaltered Reptation: coupling of directions and significant slowing down of parallel and perpendicular dynamics Müller, Daoulas, JCP 129, 164906 (2008)

32. towards a more realistic dynamics: slip-links structure formation with Rouse-dynamics, slip-links and slithering snake-dynamics

33. towards a more realistic dynamics: flow idea: describe over-damped motion in a polymer melt by Brownian dynamics Langevin-thermostat with respect to local velocity field determine self-consistently from particle displacements in a short time interval Dt , average over all particles and time interval T (self-consistent Brownian dynamics) replace non-bonded interactions by self-consistent external fluctuating field, W, which is frequently updated (quasi-instantaneous field approximation) hydrodynamic velocity field must not fluctuate (dense systems and large T) limited to quasi-stationary flows Miao, Guo, Zuckermann (1996), Doyle, Shaqfeh, Gast (1997) Saphiannikova, Prymitsyn, Crosgrove (1998), Narayanan, Prymitsyn, Ganesan (2004)

34. flow of a melt over brush of identical chains • flow through channels coated with a • brush/network • reduction of friction • motion of block-copolymer saturated • interfaces in AB homopolymer-diblock • copolymer blends (mechanical strength • of interfaces) • hydrodynamic boundary condition • at brush-melt interface • how does a polymer brush • respond to shear flow? • Couette and Poiseuille flows • and consistency of Navier • slip boundary condition

35. velocity profiles Poiseuille and Couette flow

36. velocity profiles inversion of flow direction inside the brush

37. inversion of flow direction inside the brushtumbling motion of grafted chains v top of the brush is dragged along by the flow, vb>0 for large x average velocity of brush vanishes (grafted chains) vb<0 close to grafting surface confirmed by MD simulation of bead-spring model tumbling motion of isolated, grafted chains in shear flow Doyle, Ladoux, Viovy, 2000, Gerashchenko, Steinberg, 2006, Delgado-Buscaliono 2006, Winkler, 2006 hydrodynamic interactions not important, non-Gaussian distribution of orientations

38. summary III • outline: • soft generic coarse-grained models for multi-component polymer melts • free energy of self-assembled structures • kinetics (single-chain and collective) • explicit dynamics of molecules • (field-theoretic descriptions require Onsager coefficients) • soft potentials do not enforce non-crossability and result in Rouse dynamics • slip-link model can be utilized to describe the entangled dynamics in melt • entanglement length/number of slip-links is an input parameter • basic characteristics of hydrodynamic flow in dense systems can be • captured by self-consistent Brownian dynamics • (only quasi-stationary flows and no hydrodynamic interactions of solvent) Development and Analysis of Multiscale Methods Minneapolis, Nov 3, 2008

39. SCMF simulations vs SCF theory ensemble of independent molecules in independent molecules in static, real fluctuating, real fields representing quasi- fields, self-consistently determined from instantaneous interactions with surrounding average densities result: explicit many body configuration spatial density distribution (including intermolecular correlations) statics: QIF-approximation is controlled by MF-approximation controlled by a small parameter Ginzburg-parameter that depends on discretization DLand N that is an invariant of the system correlations and fluctuations no correlations or fluctuations dynamics: evolution of explicit molecular time evolution of collective densities conformations (Rouse-like dynamics) non-local Onsager-coeffizient required dynamic asymmetries and ``freezing’’ molecular conformations are assumed of one component feasible to be in equilibrium with external fields

40. structure formation of amphiphilic molecules amphiphilic molecules: two, incompatible portions covalently linked into one molecule, e.g., block copolymers or biological lipids no macroscopic phase separation but self-assembly into spatially structured, periodic microphases universality: systems with very different molecular interactions exhibit common behavior (e.g., biological lipids in aqueous solution,high molecular weight amphiphilic polymers in water, diblock copolymer in a melt) use coarse-grained models that only incorporate the relevant interactions: connectivity along the molecule and repulsion between the two blocks 1-100 nanometer(s)

41. free energy difference via TDI

42. SMC: Brownian dynamics as smart MC simulation Rossky, Doll, Friedman, 1978 idea: uses forces to construct trial displacements Dr SMC or force bias MC allow for a much larger time step (factor 100) than Brownian dynamics Rouse-like dynamics via SMCsimulations Müller, Daoulas, JCP 129, 164906 (2008)

43. orientation distributions of tumbling chains v ? static tilt vs cyclic motion brush does not act like a static, porous medium Gerashchenko, Steinberg, 2006, Delgado-Buscaliono 2006, Winkler, 2006