High-Resolution Implementation of Self-Consistent Field Theory Eric W. Cochran Chemical and Biological Engineering Iowa State University Carlos Garcia-Cervera Department of Mathematics University of California - Santa Barbara Glenn Fredrickson Materials Research Laboratory University of California - Santa Barbara Complex Fluids Design Consortium Annual Meeting January 3, 6
What do we mean by High Resolution SCFT? Any numerical implementation in which the free energy is calculated to high precision ( w ( r) ) ρ F ( )( ( ) ( ) ) w 1 nlnq N χn 3 β = d r + + r ρ A r +ρb r F numeric F F < 1 3
When is High Resolution SCFT important? χ IS = 11. χ SO = 14. χ IO = 45.8.5.45 b I = 6. b S = 5.5 b O = 7.8 1 8 6 χn 4 Q 5 / HPS Q 9 Q 9 H L H Q 3 DIS Q 5 / HPS..4.6.8 1 f LAM f I f.5.5 O O Q 3 7 D-Hex Q 14.45 χ AB = χ AC = 13. χ AC = 35. b A = b B = b C. f A.4.6.8 LAM DIS.3.5.7 Spheres f S (Im3m).8.6 f C.4. f A.3.5.5. f B.6.3 f C Spheres (Pm3m)...4.6.8 f B
Modified Diffusion Equation Largest Hurdle To Accurate Numerics Strategies Fully Spectral Express spatial quantities using the eigenfunctions of the Laplacian operator Exact solution for q(r,s) Computational cost scales as V 3 Real Space Methods / Pseudospectral Methods Approximate solution for q(r,s) Computation Cost and Numerical Stability vary widely with choice of method Proper choice numerical implementation is crucial when modeling systems subjected to stiff potentials
q(r,s) Stiffness of the MDE becomes severe in strong segregation s r, % χn = 11 s χn = 1 5-5 AB Diblock Copolymer in 1D r, %
- -4-6 -8 5-5 AB Diblock Copolymer in 1D χn = 1 3 4 Log N s Operator Splitting w - D - D ( ) D k q rs, +D s = e e e q( rs, ) s s w s
Operator Splitting + Bulirsch- Stoer Extrapolation q Rational Extrapolating Polynomial Ds Ds Ds 3 Ds 4 N s
5-5 AB Diblock Copolymer in 1D - -4-6 -8 χn = 1 4 th Order Op-split Op-split + BDF 3 4 æ ö log N s ( D s ) ç å è ø D s
4 th Order Backwards Differention Formula or 1 s q( r - æ D ö, s +D s) @Á ç Á( w ( 48 (, ) 7 (, ) 48 (, ) 1 (, 3 ))) i q s - q s - D s + q s - Ds - q s - D s çè 5 + 1 Ds k r r r r ø Fully implicit treatment of the Laplacian Adams-Bashford Explicit discretization of the source term Á= { } Fourier Transform
- -4-6 -8 5-5 AB Diblock Copolymer in 1D 4 th Order χn = 1 Op-split + BDF BDF Op-split 3 4 æ ö log N s ( D s ) ç å è ø D s
Operator Splitting BDF log ( F - F / F ) - -4-6 -8 χn = 11 χn = 1 log ( F - F / F ) - -4-6 -8 χn = 11 χn = 1-1 1.5..5 3. log N s -1 1.5..5 3. log N s Free energy of a single field configuration
Operator Splitting BDF log ( F - F / F ) - -4-6 χn = 11 χn = 1 log ( F - F / F ) - -4-6 χn = 11 χn = 1-8 1.5..5 3. log N s -8 1.5..5 3. log N s Free energy @ saddle point
log ( F - F / F ) - -4-6 Op-split χn = 1 BDF N s ~ 35 χn = 11-8 1.5..5 3. log N s
Example: Stability of Gyroid In AB diblocks at Strong Segregation F = F hex -F gyroid F = F lam -F gyroid F, kt/1 F gyroid = 37 kt @ χn = 1 1 5-5 χn = 1 6 4 1 χn = 4 8 8 6.3.3.34 f F - F F - < 1 4
1 1 8 Q 9 Q 9 8 Q 9 Q 9 χn 6 H L H χn 6 H L H 4 4 Q 5 / HPS Q 3 DIS..4.6.8 1 f Q 5 / HPS Q 5 / HPS Q 3 DIS..4.6.8 1 f Q 5 / HPS
Polymeric Nanocomposites and Confined Geometries A straightforward and flexible approach Particle-based Hamiltonian (for linear diblock copolymers): ( s) n 1 1 drα β H = ds 4R + g α= 1 ds 1 ρ Field theoretic Hamiltonian: ( d r χρa ( r) ρb ( r) + d r ρ A ( r) H A ( r) + d r ρb ( r) H B ( r) ) Particle-polymer interactions H ( r) H ( r) ( B A ) ( H( r) + w ( r) ) Cavities in the polymer density enforced through the introduction of a target particle density field ρ β = ( )( ( ) ( ) ( ) 1) N + ρ +ρ +ρ χn Pressure field enforces incompressibility 3 H d r w+ r A r B r particle r n lnq Single chain partition function ignore particle internal degrees of freedom
Polymeric Nanocomposites and Confined Geometries An example 1.8 Prescribed Particle Density Profile.6 (1D).4 ρparticle ( r). x ( 4.5 R g ) χ N = 5 H ( r) = ρ ( r) 5 Repulsive to B monomers particle 18 grid points ( ~.35 Rg / point )
Polymeric Nanocomposites and Confined Geometries 1 An example 1.8.6.4. Prescribed Particle Density Profile ρparticle ( r).8 x ( 4.5 R g ).6.4 ρ A (r) ρ B (r) ρ A (r). x χ N = 5 H ( r) = ρ ( r) 5 Repulsive to B monomers particle 18 grid points ( ~.35 Rg / point )
w w t i+ 1 i + + Semi-Implicit Seidel Pressure Field Relaxation ( ) 8 i i ( +, w ) δh w = g + g + g w + + g + g + g w ( ) i+ 1 i AA AB BB * + AA AB BB * i + w+ 3 4 15 w +, kt w -, kt -4-15 -8..4.6.8 1 r -3 χ N = 5 H ( r) = ρ ( r) 5 Repulsive to B monomers particle
w w t i+ 1 i + + Semi-Implicit Seidel Pressure Field Relaxation ( ) i i ( +, w ) δh w = g + g + g w + + g + g + g w ( ) i+ 1 i AA AB BB * + AA AB BB * i + w+ Problem: Driving force is order 1, but Dw + is order 1! Log ( dh/dw + ) -1 - -3-4 -5 Convergence of System with cavity field Is slow!! 4 Log (Iteration #)
Gaussian Thread Model near sharp interfaces should obey Ground State Dominance d ds g q rs, = R Ñ - w q rs, = Lq rs, ( ) ( ) ( ) ( ) Ly =Ly i i i f r : y R r [ ] () () For a homopolymer in a prescribed density field gnd Ly = ( f ) ' g = w+ f R g ' '' (( ) ) w+ = f + f f 8 4-4 -8 w + w + gnd..4.6.8 1 r
W W i+ 1 i + + t Semi-Implicit Seidel Pressure Field Relaxation ( ) i i ( +, ) δh W W = g + g + g W + + g + g + g W ( ) i+ 1 i AA AB BB * + AA AB BB * i + W+ gnd + = + - + W w w Log ( dh/dw + ) -1 - -3-4 -5 4 Log (Iteration #)
Polymeric Nanocomposites and Confined Geometries An experimental result can SCFT help us understand it? (Gila Stein and Ed Kramer) Polymer air interaction Polymer substrate interaction 1 layer many layers HCP spheres 111 plane (p6m D symmetry) BCC spheres 11 plane
HCP spheres (p6m D symmetry) Fm3m spheres 1 plane BCC spheres 11 plane a a a 1 a a 1 a 1 a / a 1 = 1 1 < a / a 1 < / 3 a / a 1 = / 3 = 1.155
BCC 1.16 1.1 Fmmm a 1 / a 1.8 hcp Fmmm bcc (bulk behavior) 1.4 Experiment FCC 1. 5 1 15 # Layers a 1 a
BCC 1.16 1.1 Fmmm a 1 / a 1.8 hcp Fmmm bcc (bulk behavior) 1.4 Experiment FCC 1. 5 1 15 # Layers æa ö æa ö è ø è ø F = F 1 * 1 surface thickness F + ç bulk a ç a
-.58 F nkt -.536 -.544.75 1. 1.5 a a ( ) 1 Bulk Contribution f =.1 χn = 41
.53 F nkt.58.54.5.75 1. 1.5 a a ( ) 1 Surface Contribution f A =.1 χn = 41 χ surface-a N = 4 surface =.5 R g
1.5 1..75.5.5..75 1. 1.5 1.5 1..75.5.5..75 1. 1.5 1.5 1..75.5.5 1 st Order Transition..75 1. 1.5 1.5 Decreasing Surface Contribution 1..75.5.5..75 1. 1.5 1.5 1..75.5.5..75 1. 1.5 1.5 1..75.5.5..75 1. 1.5
Acknowledgements Jeffrey Barteet - Computational Facilities Gila Stein Experimental Data Kirill Katsov Lots of good ideas Funding - NSF
Two-domain lamellae Three-domain lamellae Ia3d Double Gyroid fa fc I413 Alternating Gyriod Fddd Orthorhombic Network fb Abetz et al. CsCl Spheres (Pm3m) Bates et al. Matsushita et al. Tetragonal Cylinders Abetz et al.
Stadler et al. Stadler et al. Balsamo et al. Abetz et al.
Abetz et al. Bates et al. Bates et al. Matsushita et al.
SCFT: Beyond Diblock Copolymer Melts > Species Many candidate phases Multicomponent systems, geometrically constrained systems (slit confinement, polymer + particle systems), and strongly segregated systems Sharp interfaces Wide range of possible length scales Combined real-space / Fourier space approach of the pseudospectral implementation of SCFT offers a flexible framework with which to address these problems
Fully Spectral SCFT (Matsen and Schick, Morse) PSCFT Spectral basis guarantees defect-free single unit cell calcuations Computational effort scales as N s3. Can be problematic with -Sharp interfaces (large χn) -Multiscale systems (i.e. BCP/Homopolymer/solvent mixtures) Final field configuration depends both on initial guess and the size/shape of the simulation volume. Computational effort scales as N log N. -Sharp interfaces less expensive to model -Multiscale systems tractable Quasi-Newton convergence scheme requires an initial guess near the saddle point Solutions attainable from any initial configuration
SCFT Equations ( w w ) A B H = d r fwa ( 1 f ) wb + V lnq 4χN 1 Q = d q, s = 1 V r ( r ) Modified diffusion Equation (MDE) s ( r ) ( r ) ( r ) ( r ) q, s = q, s w q, s ; q, s= = 1 AB, Pseudospectral Solution to the MDE Vs Vs wi( r) w 1 i( ) sk r V q( r, s+ s) e I e I e q, s ( r )
Guess Evaluate Calculate Energy w A =? w B =? s ( r ) ( r ) ( r ) ( r ) q, s = q, s w q, s ; q, s= = 1 AB, Vs wi r 1 Vsk q( r, s+ s) e I e I e ( w ) A w B = ( ) + ( ) w ( r) H d r fwa 1 f wb V lnq 4χN 1 Q = d q, s = 1 V r Vs i ( r ) Relax Potentials (,,, ) (,,, ) w = f φ φ w w i+ 1 i i A A B A B w = f φ φ w w i+ 1 i i B A B A B
Explicit Euler Relaxation Define: Relaxation according to: 1 w+ = wa + w 1 w = wa w ( ) B ( ) i+ 1 i w+ = w+ + t i+ 1 i w = w t B δh w i i ( +, w ) δw + δh w ' δw i i ( +, w )
Explicit Euler Relaxation AB diblock f =.3 χn = Energy -.1 -. -.3 15 iterations Log Error -.4. -.5-1. -1.5 -. -.5-3. -3.5-4. 5 1 15 Time 5 1 15 Time
PSCFT unit cell calculations Needs Rapid convergence scheme Defect-free solutions Stress-free solutions
Semi-Implicit Siedel (SIS) Relaxation Calculate the asymptotic expansion of the MDE in the limit w i Result: δh w (, w ) δw + + ( w ) s ( r, ) = ( r, ) ( r, ) q s q s w q s AB, ( ) ( ) = g + g + g * w + g g * w AA AB BB + AA BB δh w+, = w ( gaa + gab + gbb )* w + ( gaa gbb )* w δw χn w w i+ 1 i + + i+ 1 w t Debye scattering function + ( ) i+ 1 i ( +, w ) i i ( +, w ) δh w = g + g + g w + + g + g + g w ( ) i+ 1 i AA AB BB * + AA AB BB * i + w+ i w δh w i+ 1 = w + w i t χn w χn i Cenicernos and Fredrickson, Multiscale Model. Simul.,, 45, 4
Semi-Implicit Siedel Relaxation AB diblock f =.3 χn = Energy -.1 -. -.3 SIS Euler -.4 4 6 8 Time Log Error -1 - -3 SIS Euler iterations -4-5 5 1 15 Time
PSCFT unit cell calculations Needs Rapid convergence scheme - SIS Defect-free solutions Stress-free solutions
SIS Relaxation Symmetry specified initial guess AB diblock f =.3 χn = 1 iterations Energy -.1 -. -.3 -.4 Log Error -1 - -3-4 -5-6 -7 SIS 4 6 8 Time SIS Euler Euler 5 1 15 Time
SIS Relaxation Symmetry specified initial guess (Ia3d) AB diblock f =.39 χn = 15 Energy. -.5 -.1 -.15 -. -.5 1 3 4 Time 5 iterations -1 Log Error - -3-4 -5 1 3 4 Time
PSCFT unit cell calculations Needs Rapid convergence scheme - SIS Defect-free solutions bias simulation with desired initial symmetry Stress-free solutions
With perfect symmetry, the size of the box has a large influence on the free energy -.5 Energy -.3 -.35 -.4.5 3.5 4.5 5.5 Lattice Spacing, Rg
Stress Operators within PSCFT N 1 3kT σ = δ ( r) ( r R )( R R ) ( R R ) ( discrete chain) αβ i i+ 1 i α i+ 1 i b i= 3kT = dr dr 1 a β ds b ds ds b 1 h 1 dsq 1 (,1 s ) q (, s ) h γα x α β x δβ = 3Q ( ( )) b 1 h 1 dsq 1 1 (,1 s ) k k q (, s ) h γα x α β x δβ = I I 3Q β s R ( s ) = h x Box Relaxation: h i+ 1 αβ δh dh aβ i δh = hαβ + t δ h ( 1 h ) = σ αβ aβ
Influence of stress relaxation -.1 Energy -. -.3 Start stress relaxation 1 iterations Log Error -.4-1 - -3-4 -5-6 -7 4 6 Time 5 1 Time
Simultaneous SIS relaxation, stress relaxation, and symmetry specified initial guess Energy..1. -.1 -. -.3 -.4 -.5. -.5 5 1 15 Time Energy -.1 -.15 -. -.5 1 3 4 Time
PSCFT unit cell calculations Needs Rapid convergence scheme - SIS Defect-free solutions bias simulation with desired initial symmetry Stress-free solutions Discussion