1. Self-Consistent Field Theory of Block Copolymers An-Chang Shi McMaster University Hamilton, Ontario, Canada shi@mcmaster.ca

2. OutlineOutline Introduction  Why block copolymers self-assemble Self-Consistent Field Theory  Theoretical framework, derivations, etc Mean-Field Approximation - SCMFT  SCMFT equations, methods of solution Gaussian Fluctuations  Stability and kinetic pathways  Nucleation of OOT

3. ReferencesReferences Doi and Edwards, The Theory of Polymer Dynamics Toshihiro Kawakatsu, Statistical Physics of Polymers Schmid, J. Phys.: Condens. Matter 10, 8105 (1998) Matsen, J. Phys.: Condens. Matter 14, R21 (2002) Fredrickson, Ganesan and Drolet, Macromolecules 35, 16, (2002) Shi, in Developments in Block Copolymer Science and Technology, Edited by Hamley (2004) More references are found in these books and papers

4. Diblock Copolymers: Complex Phase Behavior Mesoscopic separation of diblock copolymers LSCG Complex structures and phase diagrams Experiments: Hashimoto, Thomas, Lodge, Bates,... Mean-Field theory: Helfand, Whitmore, Matsen and Schick,... Fluctuations: Laradji, Shi, Noolandi, Desai, Wang,... NANA NBNB

5. Simple Model System: Diblock Copolymers Degree of polymerization: N=N A +N B Entropy: S ~ N -1 Composition: f=N A /N Segment-segment interaction:  AB =(z/2kT)(2  AB -  AA -  BB ) Enthalpy:  H ~  Current understanding is based on three parameters NANA NBNB

6. Stability of thermodynamic phases Stable phase: global minimum Metastable phases: local minima Unstable phases: local maxima and/or saddle points Sign of second-order derivatives: fluctuations Stable, Metastable and Unstable phases Free energy metastable stable unstable Order parameters unstable

7. Self-Consistent Field Theory of Polymers Functional integral approach (r)(r) Many-body interactionFluctuating field Simple theoretical framework  Chain statistics and polymer density  (r) determined by  (r)  Mean field  (r) determined self-consistently by  (r)  Flexible framework, applies to many systems

8. Self-Consistent Field Theory: Simple Example Monatomic Fluids in Canonical Ensemble  A collection of n particles in a volume V  Pairwise interaction potential The partition function can be written as,

9. Self-Consistent Field Theory: Simple Example Using the identities, The partition function can be written in the form,

10. Self-Consistent Field Theory: Simple Example Using the definition, The partition function can be written in the form,

11. Self-Consistent Field Theory: Simple Example Using the definition, The partition function can be written in the form, where the free energy functional is, Field theory model corresponds to the particle model

12. Self-Consistent Field Theory: Simple Example Transformation from particle based theory to field based theory Partition function of a single particle in a potential General theoretical framework for many systems

13. Self-Consistent Field Theory Phase behavior described by free energy functional Fluctuations in an ordered state

14. Self-Consistent Field Theory: Mean-Field Approximation Saddle-point approximation: Ignore higher-order terms leads to, Conditions for saddle-point are: Self-consistent mean field equations

15. Self-Consistent Field Theory: Mean-Field Approximation Using the relation: Conditions for saddle-point are: For a given potential, these equations are solved self-consistently

16. Self-Consistent Field Theory of Polymers Standard Model or Edwards Model  Chain statistics modeled by Gaussian chains  Interactions modeled by Flory-Huggins Parameters  Hard-core interaction modeled by incompressibility condition Weiner MeasureFlory-Huggins Interaction Incompressibility

17. Self-Consistent Field Theory of Polymers Standard Model or Edwards Model  Chain statistics modeled by Gaussian chains

18. Self-Consistent Field Theory of Polymers Standard Model or Edwards Model  Flory-Huggins monomer-monomer interaction. Model of short range interactions

19. Self-Consistent Field Theory of Polymers Standard Model or Edwards Model  Hard-core interaction approximated by incompressibility

20. Self-Consistent Field Theory of Polymers Using the identity, The partition function can be written as, Free energy functional

21. Self-Consistent Field Theory of Polymers Single Chain Partition function in a field Propagator representation Definition of propagator (Green Function)

22. Self-Consistent Field Theory Chain statistics specified by Q(r,t|r 0 ) Probability of finding t -th monomer at r, given the end at r 0 (0,r 0 ) (t,r) End-integrated propagators They are solutions of the modified diffusion equation with,

23. Self-Consistent Field Theory of Polymers Free Energy Expansion

24. Self-Consistent Field Theory of Polymers Free Energy Expansion Correlation functions: Cumulant Correlation functions:

25. Self-Consistent Field Theory Cumulant Correlation Functions Computed in the zeroth-order solutions

26. Self-Consistent Field Theory Free energy expansion

27. Self-Consistent Field Theory: Mean-field approximation Motivation and justification The functional integral can be evaluated by the largest integrand Control parameter:

28. Self-Consistent Mean Field Theory: SCMFT Technically saddle points determined by F (1) =0, leading to: Coupled nonlinear equations Lagrangian multiplier  to ensure: Propagators are solutions of: Single chain partition function is:

29. Self-Consistent Mean Field Theory (SCMFT) Mean Field Free Energy Density (per chain) Within SCMFT, the phase behavior is controlled by the parameters  N, f  and   SCMFT is controlled by the parameters,  N, f  and     N characterizes the degree of segregation  f  and   characterize chain structure  Constant shifts in   do not change the solution

30. Self-Consistent Mean Field Theory: SCMFT Five coupled nonlinear equations for five variables

31. Self-Consistent Mean Field Theory (SCMFT) For a give set of control parameters, the SCMFT equations must be solved to obtain the density profile and free energy Phase diagram can be constructed from the solutions Methods of solving SCMFT equations  Exact solutions – rarely possible  Approximate solutions  Weak segregation limit (WSL) theory  Strong segregation limit (SSL) theory  Numerical techniques  Real space method  Reciprocal space method

32. SCMFT: Exact Solution Exact solutions are hard to come by! Trivial case: Homogeneous phase Homogeneous phase is always a solution!

33. SCMFT: Weak Segregation Theory Homogeneous phase as the zeroth-order solution Solve the SCMFT equation for  in terms of  Similar to Landau Theory – more later Leibler 1980

34. SCMFT: Strong Segregation Theory A and B segments are completely separated and the chains are strongly stretched Interaction and stretching energies treated separately Extremely valuable tool to understand structures Semenov 1985

35. Microphase Separation: Selection of Length Scale Another view: spinodal point of the disordered phase (Leibler 1980) Chain Connectivity Order-disorder transition (ODT) occurs at finite q 0 : Microscopic Phase Separation! NANA NBNB

36. Phase Separation: Polymer Blends Spinodal point of the disordered phase – Polymer Blend Critical point occurs at zero q: Macroscopic Phase Separation! NANA NBNB

37. SCMFT: Analogy with Quantum Mechanics Modified diffusion equation is Schrödinger equation with imaginary time! Many ideas and techniques in QM can be applied to polymers!

38. SCMFT: Numeric Solutions SCMFT equations are solved self-consistently Recent progresses include fast Fourier algorithm Real Space Method  The space is discretized in to grids  An initial guess of the mean fields    The modified diffusion equations are solved to obtain the propagators, and the results are used to compute the densities  The fields for the next iterations are obtained using a linear mixture of the new and old fields  The iteration is repeated until the solution becomes self- consistent

39. SCMFT: Numeric Solutions SCMFT equations are solved self-consistently The SCMFT equations can be cast in these Fourier components Reciprocal-Space Method  The functions are periodic  For an ordered phase, the reciprocal lattice vectors are completely specified by the space group of the structure  The plane waves corresponding to the reciprocal lattice vectors can be used as basis functions

40. SCMFT: Reciprocal-Space Method In Fourier Space:

41. SCMFT: Reciprocal-Space Method Construct eigenvalues and eigenfunctions The propagators are solved in terms of the eigenfunctions

42. SCMFT: Reciprocal-Space Method The single chain partition function and density are Differential equations become algebra ones

43. SCMFT: Reciprocal-Space Method Further simplification: Point group symmetry SCMFT equations in terms of the expansion coefficients New Basis Functions for equivalent lattice vectors

44. Diblock Copolymers: SCMFT Phase Diagram Controlling parameters:

45. Diblock Copolymers: Polydispersity Effect Perturbation Theory:

46. Diblock Copolymers: Polydispersity Effect Mechanism of stabilizing non-lamellar phases Interfacial and entropic contributions to free energy Hex phase,  N = 15, f = 0.35 Full free energy

47. Non-Centrosymmetric Structures Lamellar Structures from ABC and ac block copolymer blends

48. Comparison with Recent Experiments  2 = 0.18  2 = 0.4 ABC: polystyrene-b-polybutediene-b-poly(tert-butyl methacrylate) TEM Staining: A-light grey, B-black, C-white T. Goldacker, V. Abetz, R. Stadler, I. Erukhimovich and L. Leibler, Nature 398, 137 (1999). L. Leibler, C. Gay and I. Erukhimovich, Europhys. Lett. 46, 549 (1999). Pure polymeric NCS Structure is still illusive

49. Phase Diagram:  Z 2 –  2 plane Z 3 /Z 2 = 1.5 NCS Structure stable at  2 =0.4 Wickham and Shi, Macromolecules, 2001

50. ABCD Tetrablock Melt Phase Diagram with NCS phase: Possibility of pure NCS Experiments: Matsushita and coworkers (2003) Karim, Wickham and Shi, Macromolecules, 2004

51. Self-Consistent Field Theory Phase behavior described by free energy functional F( ,  ) Expansion around an ordered state

52. Gaussian Fluctuations Lowest order free energy cost of fluctuations Linear Hermitian operator

53. Fluctuation Modes Eigenfunction representation Nature of eigenmodes  (r) ? eigenmode eigenvalue Free energy cost

54. Fluctuation Modes: Simple Example Meanfield solution:  1 -  2 =a 0 Fluctuation modes:

55. Fluctuation Modes The operator C(r,r’) quantifies free energy cost of fluctuations The ordered phase is unstable if the smallest eigenvalue of C(r,r’) is negative Scattering function = Fourier transform of C(r,r’) Fluctuation modes classified by eigenfunctions of C(r,r’) Elastic moduli can be obtained from C(r,r’)  Direct calculation of C(r,r’) impossible! New techniques needed

56. Symmetry and Fluctuation Modes Ordered structure: invariant under symmetry operations  are simultaneous eigenfunctions of O and C   can be constructed and catalogued using symmetries

57. Symmetry and Fluctuation Modes Discrete translation symmetry Fluctuation modes classified by Bloch waves q and q+G degenerate u nk (r+R)= u nk (r), k within the 1st Brillouin zone

58. Symmetry and Fluctuation Modes Point group symmetry: (rotation, mirror reflection, inversion) Fluctuation modes classified by Bloch waves with a wave vector k within the irreducible BZ

59. Nature of Fluctuation Modes Space group symmetry ensures: Eigenvalues n ( k ) form a band structure n ( k ) labeled by wave vector k within the 1st BZ Eigenmodes are Bloch functions  nk (r)=e ikr u nk (r) u nk (r+R)= u nk (r) periodic functions Fluctuation modes = Eigenmodes n ( k ) of Fluctuation modes determined by

60. Band Structure of Fluctuation Modes      Example: Cylindrical structure First BZ: hexagonal prism

61. ApplicationsApplications Fluctuation modes provide: n ( k ) = free energy cost of fluctuation modes n ( k ) < 0: unstable phase  min nk [ n ( k )] = 0: spinodal line Most unstable mode: structural relations Scattering functions Elastic moduli General symmetry argument provides: A powerful technique for classifying the anisotropic fluctuation modes in an ordered structure

62. Spinodal Decomposition vs. Nucleation min nk [ n (k)] < 0min nk [ n (k)] > 0 Free energy metastable stable unstable Order parameters Saddle-point

63. Weak Segregation Theory Revisited Model of systems with short wavelength instability * Microphases of diblock copolymers * Nematic to smectic-C transition in liquid crystals * Weakly anisotropic antiferromagnets * Onset of Rayleigh-Benard convection * Pion condensates in neutron stars Studied extensively * Mean field approximation * Hartree approximation for fluctuations Theory of weak crystallization (Landau-Brasovsky)

64. Microphase Separation: Selection of Length Scale Another view: spinodal point of the disordered phase (Leibler 1980) Chain Connectivity Order-disorder transition (ODT) occurs at finite q 0 : Microscopic Phase Separation! NANA NBNB

65. Systems with Competing Interactions Competing interactions & modulated structures Interactions of long range  Dipolar interactions  Nonlocal interactions associated with: polarization, magnetization, elastic strain  Polymer connectivity and/or entropy effects Coupled order parameters  Membranes, amphiphilic monolayers, liquid crystal films Spontaneous selection of primary length scales Fluctuation spectrum has a minimum at q=q 0

66. Systems with Competing Interactions Modulated phases are ubiquitous in nature Ripple phase in Lipids Rayleigh-Bernard Instability SuperconducterTuring patterns

67. Systems with Competing Interactions Phases and phase transitions Ferromagnetic film Ferrofluid Stripes Bubbles

68. Landau-Brasovsky Theory Free energy expansion

69. Mean Field Theory: Phase Diagram Mean field equation Mean field phase diagram  Kats et al. 1993, Podneks and Hamley 1996, Shi 1999

70. Gaussian Fluctuations Fluctuation modes determined by:

71. Space group symmetry ensures: Eigenvalues n ( k ) form a band structure n ( k ) labeled by wave vector k within the 1st BZ Eigenmodes are Bloch functions  nk (r)=e ikr u nk (r) u nk (r+R)= u nk (r) periodic functions Nature of Fluctuation Modes Fluctuation modes = Eigenmodes n ( k ) of Shi. Yeung, Laradji, Desai, Noolandi

72. Anisotropic Fluctuation Modes Fluctuation modes determined by: Fluctuation modes as Bloch functions:

73. Band Structure of Fluctuation Modes      Example: Cylindrical structure First BZ: hexagonal prism

74. Stability of the Ordered Phases  Spinodal lines determined by

75. Fluctuation Modes of Lamellar Phase Fluctuation modes from perturbation theory For k 2 =(k-G) 2 q0q0 q 0 /2 kxkx kzkz Most unstable modes at two rings defined by: * Hexagonal symmetry in x-z plane * Infinity degenerate in x-y plane * Two layer periodicity in z-direction

76. Scattering Functions of Lamellar Phase kxkx kzkz * Hexagonal symmetry in x-z plane * Infinity degenerate in x-y plane qzqz qyqy qxqx qxqx

77. Application: Structural Relation Lamellar to hexagonal phase Effect of the most unstable mode a=0a=1a=2 

78. Application: Structural Relation Hexagonal to lamellar phase Effect of the most unstable mode a=0a=0.5a=1 

79. Application: Structural Relation Hexagonal to spherical phase Effect of the most unstable mode a=0a=0.25a=0.75 

80. Applications: Block Copolymers Hexagonally packed cylinder to sphere transition Good agreement with experimentally observed epitaxies Reference: Ryu, Vigild, Lodge PRL 81, 5354, 1998

81. NucleationNucleation Decay of a metastable phase into a stable, equilibrium phase via nucleation Sota et al., Macromolecules (2003)

82. Nucleation rate = F(R) R RCRC ( e.g. nucleation of liquid from supersaturated vapour via thermal fluctuations)  = liquid/vapour interfacial free-energy R (quasi-equilibrium) attempt frequency Droplet free-energy: drop shrinks drop grows Classical Nucleation Theory

83. (micro)structured phases in/out of droplet account for length scales symmetry of microstructure leads to: anisotropic interfacial free-energies anisotropic droplet shapes complicated interfacial structure needs to be modified Key quantity:  Nucleation at order-order transitions: challenges

84. Landau-Brazovskii field-theory single-mode, slowly varying envelope approximations Amplitude (phase-field) model for interfaces between eg. lamellae and cylinders profile for planar interface between e.g. lamellae and cylinders Anisotropic interfacial free-energy:  ( ,  ) Non-spherical droplet shape Critical droplet size and nucleation barrier Wulff construction modified classical nucleation theory Outline of the Theoretical Method

85. Order parameter: Can be derived from the many-chain Edwards Hamiltonian for diblock copolymers S(q) q q* F local  0 Admits equilibrium microphase solutions (lamellae, cylinders, spheres, gyroid) Landau-Brazovskii Model

86. Model and Assumptions Assume separation of length-scales: microstructure period < droplet interfacial width << droplet size Weak segregation: single mode approximation to order parameter : reciprocal lattice vectors : spatially-varying amplitudes amplitude-only model: spheres: cylinders: lamellae:

87. G1G1 G2G2 G3G3 Write density as (eg. for lamellar/cylinder transition): Assume separation of length-scales: microstructure period < droplet interfacial width << droplet size Amplitude-only model: Single-mode Approximation

88. MethodMethod 1)Use amplitude model to compute the interfacial free-energy  ( ,  ) for an interface of arbitrary orientation ( ,  ) between coexisting, epitaxial cylinder and sphere phases. 2) Compute droplet shape using Wulff construction droplet size >> interface width (near coexistence) non-spherical droplet shape 3) Compute nucleation barrier, critical droplet size using classical nucleation theory

89. Theory predicts that bcc symmetry leads to:  isotropic interfacial free-energy  spherical droplet Mean-field critical behaviour Sota et al. (2003) Interfacial free-energy BCC phase nucleating from disorder

90. Interfacial Free Energy: Cylinders in lamellae  Lowest when: cylinders perpendicular to interface or lamellae parallel to surface  Two fold symmetry in x-y plane Wickham et al., J. Chem. Phys. 118, 10293 (2003).

91. Disorder –to- bcc sphere Sphere –to- hex cylinder Lamellar -to- hex cylinder Disorder -to- hex cylinder (Disorder -to- lamellar: Fredrickson & Binder, Hohenberg & Swift) All these phases can be studied near the mean-field critical point within the single Fourier mode approximation Work near phase coexistence lines Order-order and disorder-order transitions studied

92. Given a fixed droplet volume, what droplet shape minimizes the surface free-energy? (Wulff, 1901) Constrained minimization of Lagrange multiplier Leads to a method to calculate droplet shape from Wulff Construction

93. Nucleation barrierDroplet Radius f A = 0.4 N = 1000 Consider barriers in the range: Critical radius ~ 25 cubic lattice spacings Suggests observed droplets have grown beyond critical size, but kept initial shape BCC phase nucleating from disorder

94. Shape elongated along cylinder axis Aspect ratio ~ 1.45 Critical diameter ~ 40 cylinders across Droplet Shape high  low  Hex cylinders nucleating from BCC phase

95. Droplet is lens-shaped, flattened along the cylinder axis Aspect ratio ~ 1/4 at f A = 0.45 Diameter of critical droplet ~ 30 cylinders at f A = 0.45 Aspect ratio -> 0 as f A -> ½  = 0 o  = 90 o Droplet shape high  low  [Wickham, Shi, Wang, J. Chem. Phys. (2003).] Hex cylinders nucleating from lamellar phase

96. (PS-PI) (f PI ~ 0.2)/ homopolymer (PS) blend Cylinder/disorder coexistence region Koizumi et al. Macromolecules (1994) Aspect ratio ~ 0.29 Diameter ~ 20 cylinders But a different system! ExperimentsExperiments

97. Sota et al. Macromolecules, (2003). Metastable disorder –to- metastable cylinder (in our model) Droplet is lens-shaped, flattened along the cylinder axis Aspect ratio ~ 0.17 at f A = 0.4 Diameter of critical droplet ~ 60 cylinders at f A = 0.4 Aspect ratio -> 0 as f A -> ½ A =0.4 f A =0.4 Hex cylinders nucleating from disorder

98. Discussions and Conclusions Self-consistent field theory provides a useful theoretical framework for polymeric systems SCFT is capable of studying the ordered phases in block copolymer systems SCFT provides a general framework for any statistical mechanical systems More challenges ahead: complex structures, phase transition kinetics, rod-coil copolymers, associating polymers, micelles, membranes, …