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

Pingwen Zhang张平文

School of Mathematical Sciences, Peking UniversityJanuary 8th, 2009

Nucleation and Boundary Layer in Diblock Copolymer SCFT Model

Collaborators

• Boundary Layer Weiquan Xu

• Nucleation Xiuyuan Cheng Lin Ling Weinan E An-Chang Shi

Outline

• Introduction polymer, diblock copolymer, microstructure, Gaussian Ramdom-walk Model

• Self-consistent Mean Field Theory (SCFT)incompressible/ compressible model

• Boundary layerboundary effect in compressible model

• Nucleationminimum energy path(MEP), string method, saddle point transition state, critical nucleus

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

Introduction: Copolymer Melts

• homopolymer: identical monomers• copolymer: distinct monomers• block copolymer: sequential blocks• melts: one sort of moleculesmelts: 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

NA NB

Introduction: Diblock Copolymer

Free energy

metastable

stable

unstable

Order parameters

unstable

Basic system parameters

• Degree of polymerization N=NA+NB

• Compositionf=NA/N

• Segment-segment interaction:

Stability of thermodynamic phases

• Stable phase: global minimum

• Metastable phases: local minima

• Unstable phases: local maxima and/or saddle points

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

(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

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.

Self-consistent Mean Field Theory(SCMFT)

Mean field approximation:

One polymer chain One polymer chain in one field creating in one field creating by the whole systemby the whole system

Polymers Polymers influencing one influencing one anotheranother

Criterion of the dominant field: (saddle point approximation)

(r)

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

Self-consistent Mean Field Theory(SCMFT) : Incompressible Model

Corresponding quantities in the Gaussian chain model:

1st derivative of H has the form

Theoretical (up) and experimental (down) equilibrium phase diagrams calculated using SCFT (cite from [3])

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

Self-consistent Mean Field Theory(SCMFT) : Compressible Model

Compare to incompressible model

We see:

IncompressibleIncompressibleCompressibleCompressible

far from boundary

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

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

Red star : Blue star : Black line : Blue line : presumption for by QW [7]

• Layer profile fix


• 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]

Influence of fix

defined thickness of layer as follows, with the unit of


• Influence of and with


• Boundary effect on phase structure

Right: with J on both sidesDown left: with J on only left sideDown right: with no J


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] )

Snapshots from experiment (Cited from [2] )

Example: nucleation in C (cylinder) -> PL (perforated lamella) transformation

Nucleation of Order-to-order Phase Transition

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

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])

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

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])

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 interfacex- y- z- bound of nucleus

• anisotropic droplet

• complicated interfacial structure


Variation of Critical Nucleus Volume with


With saddle point transition state, we obtain energy barrier of the phase transition

Variation of energy barrier with

Nucleation of Order-to-order Phase Transition: Numerical Simulation of G-C Nucleation

Dynamic of the phase transition


Dynamic of the system (see along [1 1 1] axis)


Dynamic of the system (see along [-1 -1 2] axis)


Growth of nucleus along the MEP


Gyroid-cylinder interface is NOT isotropic

Nucleus in growth, red line indicates the boundary of nucleus and

area between green lines are interfacial area.

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])


Critical Nucleus


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])

[1] Phase Behavior of Ordered Diblock Copolymer Blends: Effect of Compositional Heterogeneity, Macromolecules 1996, 29, 4494-4507)

[2] Robert A. Wickham & An-Chang Shi, J. Chem. Phys., (2003) 22,118

[3] M.W.Matsen, M. Schick, Stable and Unstable Phases of a Diblock Copolymer Melt, PRL (1994)

[4] Fredrickson, The Equilibrium Theory of Inhomogeneous Polymers,(2006),CH5

[5] S Koizumi, H Hasegawa, T Hashimoto, Macromolecules, (1994), 27, 6532

[6] Weinan E., Weiqing Ren and Eric Vanden-Eijndenc,Simplified and improved string method for computing the minimum energy paths in barrier-crossing events, J. Chem. Phys.,126, 164103 2007

[7] Dong Meng and Qiang Wang, J. Chem. Phys. 126, 234902 (2007)

Reference

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