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Self-Consistent Field Theory of Block Copolymers An-Chang Shi McMaster University Hamilton, Ontario, Canada shi@mcmaster.ca
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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
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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
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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
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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
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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
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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
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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,
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Self-Consistent Field Theory: Simple Example Using the identities, The partition function can be written in the form,
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Self-Consistent Field Theory: Simple Example Using the definition, The partition function can be written in the form,
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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
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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
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Self-Consistent Field Theory Phase behavior described by free energy functional Fluctuations in an ordered state
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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
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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
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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
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Self-Consistent Field Theory of Polymers Standard Model or Edwards Model Chain statistics modeled by Gaussian chains
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Self-Consistent Field Theory of Polymers Standard Model or Edwards Model Flory-Huggins monomer-monomer interaction. Model of short range interactions
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Self-Consistent Field Theory of Polymers Standard Model or Edwards Model Hard-core interaction approximated by incompressibility
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Self-Consistent Field Theory of Polymers Using the identity, The partition function can be written as, Free energy functional
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Self-Consistent Field Theory of Polymers Single Chain Partition function in a field Propagator representation Definition of propagator (Green Function)
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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,
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Self-Consistent Field Theory of Polymers Free Energy Expansion
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Self-Consistent Field Theory of Polymers Free Energy Expansion Correlation functions: Cumulant Correlation functions:
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Self-Consistent Field Theory Cumulant Correlation Functions Computed in the zeroth-order solutions
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Self-Consistent Field Theory Free energy expansion
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Self-Consistent Field Theory: Mean-field approximation Motivation and justification The functional integral can be evaluated by the largest integrand Control parameter:
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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:
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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
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Self-Consistent Mean Field Theory: SCMFT Five coupled nonlinear equations for five variables
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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
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SCMFT: Exact Solution Exact solutions are hard to come by! Trivial case: Homogeneous phase Homogeneous phase is always a solution!
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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
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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
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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
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Phase Separation: Polymer Blends Spinodal point of the disordered phase – Polymer Blend Critical point occurs at zero q: Macroscopic Phase Separation! NANA NBNB
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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!
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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
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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
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SCMFT: Reciprocal-Space Method In Fourier Space:
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SCMFT: Reciprocal-Space Method Construct eigenvalues and eigenfunctions The propagators are solved in terms of the eigenfunctions
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SCMFT: Reciprocal-Space Method The single chain partition function and density are Differential equations become algebra ones
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SCMFT: Reciprocal-Space Method Further simplification: Point group symmetry SCMFT equations in terms of the expansion coefficients New Basis Functions for equivalent lattice vectors
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Diblock Copolymers: SCMFT Phase Diagram Controlling parameters:
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Diblock Copolymers: Polydispersity Effect Perturbation Theory:
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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
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Non-Centrosymmetric Structures Lamellar Structures from ABC and ac block copolymer blends
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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
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Phase Diagram: Z 2 – 2 plane Z 3 /Z 2 = 1.5 NCS Structure stable at 2 =0.4 Wickham and Shi, Macromolecules, 2001
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ABCD Tetrablock Melt Phase Diagram with NCS phase: Possibility of pure NCS Experiments: Matsushita and coworkers (2003) Karim, Wickham and Shi, Macromolecules, 2004
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Self-Consistent Field Theory Phase behavior described by free energy functional F( , ) Expansion around an ordered state
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Gaussian Fluctuations Lowest order free energy cost of fluctuations Linear Hermitian operator
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Fluctuation Modes Eigenfunction representation Nature of eigenmodes (r) ? eigenmode eigenvalue Free energy cost
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Fluctuation Modes: Simple Example Meanfield solution: 1 - 2 =a 0 Fluctuation modes:
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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
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Symmetry and Fluctuation Modes Ordered structure: invariant under symmetry operations are simultaneous eigenfunctions of O and C can be constructed and catalogued using symmetries
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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
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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
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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
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Band Structure of Fluctuation Modes Example: Cylindrical structure First BZ: hexagonal prism
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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
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Spinodal Decomposition vs. Nucleation min nk [ n (k)] < 0min nk [ n (k)] > 0 Free energy metastable stable unstable Order parameters Saddle-point
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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)
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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
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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
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Systems with Competing Interactions Modulated phases are ubiquitous in nature Ripple phase in Lipids Rayleigh-Bernard Instability SuperconducterTuring patterns
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Systems with Competing Interactions Phases and phase transitions Ferromagnetic film Ferrofluid Stripes Bubbles
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Landau-Brasovsky Theory Free energy expansion
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Mean Field Theory: Phase Diagram Mean field equation Mean field phase diagram Kats et al. 1993, Podneks and Hamley 1996, Shi 1999
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Gaussian Fluctuations Fluctuation modes determined by:
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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
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Anisotropic Fluctuation Modes Fluctuation modes determined by: Fluctuation modes as Bloch functions:
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Band Structure of Fluctuation Modes Example: Cylindrical structure First BZ: hexagonal prism
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Stability of the Ordered Phases Spinodal lines determined by
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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
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Scattering Functions of Lamellar Phase kxkx kzkz * Hexagonal symmetry in x-z plane * Infinity degenerate in x-y plane qzqz qyqy qxqx qxqx
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Application: Structural Relation Lamellar to hexagonal phase Effect of the most unstable mode a=0a=1a=2
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Application: Structural Relation Hexagonal to lamellar phase Effect of the most unstable mode a=0a=0.5a=1
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Application: Structural Relation Hexagonal to spherical phase Effect of the most unstable mode a=0a=0.25a=0.75
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Applications: Block Copolymers Hexagonally packed cylinder to sphere transition Good agreement with experimentally observed epitaxies Reference: Ryu, Vigild, Lodge PRL 81, 5354, 1998
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NucleationNucleation Decay of a metastable phase into a stable, equilibrium phase via nucleation Sota et al., Macromolecules (2003)
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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
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(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
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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
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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
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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:
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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
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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
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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
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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).
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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
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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
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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
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Shape elongated along cylinder axis Aspect ratio ~ 1.45 Critical diameter ~ 40 cylinders across Droplet Shape high low Hex cylinders nucleating from BCC phase
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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
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(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
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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
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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, …
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