1. Pingwen Zhang 张平文 School of Mathematical Sciences, Peking University January 8th, 2009 Nucleation and Boundary Layer in Diblock Copolymer SCFT Model

2. Collaborators Boundary Layer Weiquan Xu Nucleation Xiuyuan Cheng Lin Ling Weinan E An-Chang Shi

3. Outline Introduction polymer, diblock copolymer, microstructure, Gaussian Ramdom-walk Model Self-consistent Mean Field Theory (SCFT) incompressible/ compressible model Boundary layer boundary effect in compressible model Nucleation minimum energy path(MEP), string method, saddle point transition state, critical nucleus

4. Introduction: What is polymer/ soft matter? Polymer: chain molecule consisting of monomers may of different segment types and complex structure From Emppu Salonen, Helsinki University of Technology “soft matter” everywhere

5. Introduction: Copolymer Melts homopolymer: identical monomers copolymer: distinct monomers block copolymer: sequential blocks melts: one sort of molecules melts: one sort of molecules blends: sorts of molecules Electron micrographs of copolymer blends. Left: coexistence of lamellar and cylinder phases. Right: double-gyroid phase, to the [1 1 1] axis ( Cited from [1]) Diblock: Triblock: Periodic mesoscopic structure

6. NANA NBNB Introduction: Diblock Copolymer Free energy metastable stable unstable Order parameters unstable Basic system parameters Degree of polymerization N=N A +N B Composition f=N A /N Segment-segment interaction: Stability of thermodynamic phases Stable phase: global minimum Metastable phases: local minima Unstable phases: local maxima and/or saddle points

7. Scale of the system: Period of the structure ~ (Gaussian Radius of polymer chain) ~ nm Left: electron micrographs, right: mean Field approximation (Numerical solution using Spectral method). Introduction: Microstructure/ Mesoscopic Separation

8. (Cited from[1]) lamellar (L), cylindrical (C) and spherical (S) phases, and the complex gyroid (G), perforated-lamellar (PL) and double-diamond (D) phases. Take ensemble average of segment distribution: Define concentration: Introduction: Microstructure/ Mesoscopic Separation

9. Introduction: Gaussian Random-Walk Model /Edward Model (cited from [1] ) Polymer as a flexible Gaussian chain described by curve R(s) over [0,1]. l is the length of coarse-grained segment.

10. Self-consistent Mean Field Theory(SCMFT) Mean field approximation: One polymer chain in one field creating by the whole system Polymers influencing one another Criterion of the dominant field: (saddle point approximation)  (r)

11. Self-consistent Mean Field Theory(SCMFT) : Incompressible Model note: assuming short-range interaction gives interaction potential with Flory- Huggins parameter Introducing two fields, we rewrite the partition function in the form of functional integral and obtain effective Hamiltonian ( ) Partition function of the system

12. Self-consistent Mean Field Theory(SCMFT) : Incompressible Model Corresponding quantities in the Gaussian chain model: 1 st derivative of H has the form Theoretical (up) and experimental (down) equilibrium phase diagrams calculated using SCFT (cite from [3])

13. Self-consistent Mean Field Theory(SCMFT) : Compressible Model In the expression of partition function, change include additional term of “boundary potential” where we get effective Hamiltonian

14. Self-consistent Mean Field Theory(SCMFT) : Compressible Model Compare to incompressible model We see: Incompressible Compressible far from boundary

15. Real Space Computation : Numerical Result of Incompressible Model Diffusive equation of q is solved in real space with periodic boundary conditions (cubic domain). Apply Steepest Descent to SCFT iteration. cubic length. Residual less than 1e-5. Left: Gyroid, right: Cylinder, [1 1 1] axis

16. Red star : Blue star : Black line :.. Left is incompressible, right compressible (J is Leonard- Jones-shaped) Real Space Computation : 1D Numerical Result of Compressible Model

17. Red star : Blue star : Black line : Blue line : presumption for by QW [7] Layer profile fix Real Space Computation : 1D Numerical Result of Compressible Model

18. Influence of on Layer profile, fix Real Space Computation: 1D Numerical Result of Compressible Model Red star : Blue star : Black line : Blue line : presumption for by QW [7]

19. Influence of fix defined thickness of layer as follows, with the unit of Real Space Computation : 1D Numerical Result of Compressible Model

20. Influence of and with Real Space Computation : 1D Numerical Result of Compressible Model

21. Boundary effect on phase structure Right: with J on both sides Down left: with J on only left side Down right: with no J Real Space Computation : 1D Numerical Result of Compressible Model

22. Nucleation of Order-to-order Phase Transition Nucleation: the thermally active phase transition via the formation and growth of droplets of the equilibrium phase in the background of the metastable phase. Example: nucleation of C (cylinder) from disordered phase Electron micrograph obtained in experiment (Cited from [5] )

23. Snapshots from experiment (Cited from [2] ) Example: nucleation in C (cylinder) -> PL (perforated lamella) transformation Nucleation of Order-to-order Phase Transition

24. Nucleation of Order-to-order Phase Transition: Rare Event and MEP System with thermal noise described by SDE Action functional Minimum action path /minimum energy path MEP Large-deviation theory gives: Most Probable Transition Path

25. Nucleation of Order-to-order Phase Transition: Zero-temperature String Method Along MEP Using steepest descent method with a proper initial value s.t. the string connecting A and B (metastable states) will converges to the MEP as A simplified version of the method avoided calculating tangent vector of the string, giving better stability and less computational cost. Directly solve the last term moves grid points along the string according to certain monitor function. 2d example (cited from [6])

26. Nucleation of Order-to-order Phase Transition: apply to incompressible SCFT model Recall the free energy/ effective Hamiltonian and its first derivative of incompressible SCFT model Meanwhile, with the fact We have universal convexity with respect to By doing the following map numerically (convex optimization) We translate the problem in a version where string method can be applied

27. Nucleation of Order-to-order Phase Transition: Numerical Simulation of L-C Nucleation Assume orientation relationship between L and C. Simulating box is fixed, large enough to diminish influence of boundary near the saddle point. Initial string is set to a nucleus-growth-like one. We have calculated MEP of L-C nucleation at f=0.45, varying between two extremes of spinodal line and phase boundary Orientation match (cited from [2])

28. Nucleation of Order-to-order Phase Transition: Numerical Simulation of L-C Nucleation Saddle point corresponds to the critical nucleus critical nucleus at f=0.45, =11.190 slices at interface x- y- z- bound of nucleus anisotropic droplet complicated interfacial structure

29. Nucleation of Order-to-order Phase Transition: Numerical Simulation of L-C Nucleation Variation of Critical Nucleus Volume with

30. Nucleation of Order-to-order Phase Transition: Numerical Simulation of L-C Nucleation With saddle point transition state, we obtain energy barrier of the phase transition Variation of energy barrier with

31. Nucleation of Order-to-order Phase Transition: Numerical Simulation of G-C Nucleation Dynamic of the phase transition

32. Nucleation of Order-to-order Phase Transition: Numerical Simulation of G-C Nucleation Dynamic of the system (see along [1 1 1] axis)

33. Nucleation of Order-to-order Phase Transition: Numerical Simulation of G-C Nucleation Dynamic of the system (see along [-1 -1 2] axis)

34. Nucleation of Order-to-order Phase Transition: Numerical Simulation of G-C Nucleation Growth of nucleus along the MEP

35. Nucleation of Order-to-order Phase Transition: Numerical Simulation of G-C Nucleation Gyroid-cylinder interface is NOT isotropic Nucleus in growth, red line indicates the boundary of nucleus and area between green lines are interfacial area.

36. Nucleation of Order-to-order Phase Transition: Numerical Simulation of D-S Nucleation Growth of nucleus along the MEP ( slice at plain with n=[1 1 1])

37. Nucleation of Order-to-order Phase Transition: Numerical Simulation of D-S Nucleation Critical Nucleus

38. Nucleation of Order-to-order Phase Transition: Numerical Simulation of D-S Nucleation Disorder-Sphere interface is NOT isotropic, but nucleus growth is isotropic neglecting bcc-lattice-scale variation Nucleus in growth, red line indicates the boundary of nucleus and area between green lines are interfacial area. ( Left: slice at plain with n=[1 1 1], right: slice at plain with n=[-1 -1 2])

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40. Thank you for your attention!