High-Resolution Implementation of Self-Consistent Field Theory
|
|
- Willis Eric Lawson
- 5 years ago
- Views:
Transcription
1 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
2 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
3 When is High Resolution SCFT important? χ IS = 11. χ SO = 14. χ IO = b I = 6. b S = 5.5 b O = χn 4 Q 5 / HPS Q 9 Q 9 H L H Q 3 DIS Q 5 / HPS f LAM f I f.5.5 O O Q 3 7 D-Hex Q χ AB = χ AC = 13. χ AC = 35. b A = b B = b C. f A LAM DIS Spheres f S (Im3m).8.6 f C.4. f A f B.6.3 f C Spheres (Pm3m) f B
4 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
5 q(r,s) Stiffness of the MDE becomes severe in strong segregation s r, % χn = 11 s χn = AB Diblock Copolymer in 1D r, %
6 AB Diblock Copolymer in 1D χn = Log N s Operator Splitting w - D - D ( ) D k q rs, +D s = e e e q( rs, ) s s w s
7 Operator Splitting + Bulirsch- Stoer Extrapolation q Rational Extrapolating Polynomial Ds Ds Ds 3 Ds 4 N s
8 5-5 AB Diblock Copolymer in 1D 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
9 4 th Order Backwards Differention Formula or 1 s q( r - æ D ö, s +D ç Á( w ( 48 (, ) 7 (, ) 48 (, ) 1 (, 3 ))) i q s - q s - D s + q s - Ds - q s - D s çè Ds k r r r r ø Fully implicit treatment of the Laplacian Adams-Bashford Explicit discretization of the source term Á= { } Fourier Transform
10 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
11 Operator Splitting BDF log ( F - F / F ) χn = 11 χn = 1 log ( F - F / F ) χn = 11 χn = log N s log N s Free energy of a single field configuration
12 Operator Splitting BDF log ( F - F / F ) χn = 11 χn = 1 log ( F - F / F ) χn = 11 χn = log N s log N s Free saddle point
13 log ( F - F / F ) Op-split χn = 1 BDF N s ~ 35 χn = log N s
14 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 χn = χn = χn = f F - F F - < 1 4
15 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 f Q 5 / HPS Q 5 / HPS Q 3 DIS f Q 5 / HPS
16 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
17 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 )
18 Polymeric Nanocomposites and Confined Geometries 1 An example 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 )
19 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 w +, kt w -, kt r -3 χ N = 5 H ( r) = ρ ( r) 5 Repulsive to B monomers particle
20 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 + ) Convergence of System with cavity field Is slow!! 4 Log (Iteration #)
21 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 w + w + gnd r
22 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 + ) Log (Iteration #)
23 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
24 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
25 BCC Fmmm a 1 / a 1.8 hcp Fmmm bcc (bulk behavior) 1.4 Experiment FCC # Layers a 1 a
26 BCC Fmmm a 1 / a 1.8 hcp Fmmm bcc (bulk behavior) 1.4 Experiment FCC # Layers æa ö æa ö è ø è ø F = F 1 * 1 surface thickness F + ç bulk a ç a
27 -.58 F nkt a a ( ) 1 Bulk Contribution f =.1 χn = 41
28 .53 F nkt a a ( ) 1 Surface Contribution f A =.1 χn = 41 χ surface-a N = 4 surface =.5 R g
29 st Order Transition Decreasing Surface Contribution
30 Acknowledgements Jeffrey Barteet - Computational Facilities Gila Stein Experimental Data Kirill Katsov Lots of good ideas Funding - NSF
31 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.
32 Stadler et al. Stadler et al. Balsamo et al. Abetz et al.
33 Abetz et al. Bates et al. Bates et al. Matsushita et al.
34 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
35 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
36 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 )
37 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
38 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 )
39 Explicit Euler Relaxation AB diblock f =.3 χn = Energy iterations Log Error Time Time
40 PSCFT unit cell calculations Needs Rapid convergence scheme Defect-free solutions Stress-free solutions
41 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
42 Semi-Implicit Siedel Relaxation AB diblock f =.3 χn = Energy SIS Euler Time Log Error SIS Euler iterations Time
43 PSCFT unit cell calculations Needs Rapid convergence scheme - SIS Defect-free solutions Stress-free solutions
44 SIS Relaxation Symmetry specified initial guess AB diblock f =.3 χn = 1 iterations Energy Log Error SIS Time SIS Euler Euler Time
45 SIS Relaxation Symmetry specified initial guess (Ia3d) AB diblock f =.39 χn = 15 Energy Time 5 iterations -1 Log Error Time
46 PSCFT unit cell calculations Needs Rapid convergence scheme - SIS Defect-free solutions bias simulation with desired initial symmetry Stress-free solutions
47 With perfect symmetry, the size of the box has a large influence on the free energy -.5 Energy Lattice Spacing, Rg
48 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
49 Influence of stress relaxation -.1 Energy Start stress relaxation 1 iterations Log Error Time 5 1 Time
50 Simultaneous SIS relaxation, stress relaxation, and symmetry specified initial guess Energy Time Energy Time
51 PSCFT unit cell calculations Needs Rapid convergence scheme - SIS Defect-free solutions bias simulation with desired initial symmetry Stress-free solutions Discussion
Self-Assembly on the Sphere: A Route to Functional Colloids
Self-Assembly on the Sphere: A Route to Functional Colloids Tanya L. Chantawansri Glenn H. Fredrickson, Hector D. Ceniceros, and Carlos J. García-Cervera January 23, 2007 CFDC Annual Meeting 2007 Contents
More informationField-based Simulations for Block Copolymer Lithography (Self-Assembly of Diblock Copolymer Thin Films in Square Confinement)
Field-based Simulations for Block Copolymer Lithography (Self-Assembly of Diblock Copolymer Thin Films in Square Confinement) Su-Mi Hur Glenn H. Fredrickson Complex Fluids Design Consortium Annual Meeting
More informationarxiv: v1 [cond-mat.soft] 11 Oct 2014
Phase diagram of diblock copolymer melt in dimension d=5 M. Dziecielski, 1 K. Lewandowski, 1 and M. Banaszak 1, 1 Faculty of Physics, A. Mickiewicz University ul. Umultowska 85, 61-614 Poznan, Poland (Dated:
More informationStudy of Block Copolymer Lithography using SCFT: New Patterns and Methodology
Study of Block Copolymer Lithography using SCFT: New Patterns and Methodology Su-Mi Hur Glenn Fredrickson Complex Fluids Design Consortium Annual Meeting Monday, February 2, 2009 Materials Research Laboratory
More informationComparison of Pseudo-Spectral Algorithms for. Field-Theoretic Simulations of Polymers
Comparison of Pseudo-Spectral Algorithms for Field-Theoretic Simulations of Polymers Debra J. Audus,, Kris T. Delaney, Hector D. Ceniceros, and Glenn H. Fredrickson,,, Materials Research Laboratory, University
More informationPHYSICAL REVIEW E 69,
Morphology and phase diagram of complex block copolymers: ABC linear triblock copolymers Ping Tang, Feng Qiu,* Hongdong Zhang, and Yuliang Yang Department of Macromolecular Science, The Key Laboratory
More informationEfficient Order-Adaptive Methods for Polymer Self-Consistent Field Theory
Efficient Order-Adaptive Methods for Polymer Self-Consistent Field Theory Hector D. Ceniceros November 3, 218 Abstract A highly accurate and memory-efficient approach for the solution of polymer selfconsistent
More informationSymmetry Breaking in Block Copolymer Thin Films
Experimets Symmetry Breakig i Block Copolymer Thi Films Eric Cochra, Gila E. Stei, 2 Kirill Katsov, 2 Gle H. Fredrickso, 2 Edward Kramer 2 Chemical & Biological Egieerig Iowa State Uiversity 2 Materials
More informationCoupled flow-polymer dynamics via statistical field theory: modeling and computation
Coupled flow-polymer dynamics via statistical field theory: modeling and computation Hector D. Ceniceros 1 Glenn H. Fredrickson 2 George O. Mohler 3, Abstract Field-theoretic models, which replace interactions
More informationSelf-consistent field theory simulations of block copolymer assembly on a sphere
Self-consistent field theory simulations of block copolymer assembly on a sphere Tanya L. Chantawansri, 1 August W. Bosse, 2 Alexander Hexemer, 3, Hector D. Ceniceros, 4 Carlos J. García-Cervera, 4 Edward
More informationChapter 2. Block copolymers. a b c
Chapter 2 Block copolymers In this thesis, the lamellar orientation in thin films of a symmetric diblock copolymer polystyrene-polymethylmethacylate P(S-b-MMA) under competing effects of surface interactions
More informationFluctuations in polymer blends
Fluctuations in polymer blends Dominik Düchs and Friederike Schmid Fakultät für Physik, Universität Bielefeld, Postfach 100131, 33501 Bielefeld E-mail: {schmid, duechs}@physik.uni-bielefeld.de We have
More informationDiblock Copolymer Melt in Spherical Unit Cells of Higher Dimensionalities
Vol. 11 (01 ACTA PHYSICA POLONICA A No. 3 Diblock Copolymer Melt in Spherical Unit Cells of Higher Dimensionalities M. Banaszak, A. Koper, P. Knychaªa and K. Lewandowski Faculty of Physics, A. Mickiewicz
More informationMy path: BS, Rose-Hulman PhD, UIUC Postdoc, Sandia. Map from Wikimedia Commons
CoarseGrained Modeling of Ionomers and SaltDoped Block Copolymers Lisa Hall H.C. Slip Slider Assistant Professor William G. Lowrie Department of Chemical and Biomolecular Engineering The Ohio State University
More informationSpherical phases with tunable steric interactions formed in
pubs.acs.org/macroletters σ Phase Formed in Conformationally Asymmetric AB-Type Block Copolymers Nan Xie, Weihua Li,*, Feng Qiu, and An-Chang Shi State Key Laboratory of Molecular Engineering of Polymers,
More informationBlock copolymer microstructures in the intermediate-segregation regime
Block copolymer microstructures in the intermediate-segregation regime M. W. Matsen a) and F. S. Bates Department of Chemical Engineering and Materials Science, University of Minnesota, Minneapolis, Minnesota
More informationPhase Diagram of Diblock Copolymer Melt in Dimension d = 5
COMPUTTIONL METHODS IN SCIENCE ND TECHNOLOGY 7(-2), 7-23 (2) Phase Diagram o Diblock Copolymer Melt in Dimension d = Faculty o Physics,. Mickiewicz University ul. Umultowska 8, - Poznań, Poland e-mail:
More informationMean field theory and the spinodal lines
11 Chapter 2 Mean field theory and the spinodal lines In Chapter 1, we discussed the physical properties of reversible gel and the gelation of associating polymers. At the macroscopic level, reversible
More informationChapter 2 Polymer Physics Concentrated Solutions and Melts
Chapter 2 Polymer Physics Concentrated Solutions and Melts Chapter 1 discussed the statistical thermodynamics of an isolated polymer chain in a solvent. The conformation of an isolated polymer coil in
More informationNumerical methods for the Navier- Stokes equations
Numerical methods for the Navier- Stokes equations Hans Petter Langtangen 1,2 1 Center for Biomedical Computing, Simula Research Laboratory 2 Department of Informatics, University of Oslo Dec 6, 2012 Note:
More informationF O R SOCI AL WORK RESE ARCH
7 TH EUROPE AN CONFERENCE F O R SOCI AL WORK RESE ARCH C h a l l e n g e s i n s o c i a l w o r k r e s e a r c h c o n f l i c t s, b a r r i e r s a n d p o s s i b i l i t i e s i n r e l a t i o n
More informationTheory and Simulation of Multiphase Polymer Systems
arxiv:1001.1265v1 [cond-mat.soft] 8 Jan 2010 Chapter 3 Theory and Simulation of Multiphase Polymer Systems Friederike Schmid Institute of Physics, Johannes-Gutenberg Universität Mainz, Germany I Introduction
More informationSupratelechelics: thermoreversible bonding in difunctional polymer blends
Supratelechelics: thermoreversible bonding in difunctional polymer blends Richard Elliott Won Bo Lee Glenn Fredrickson Complex Fluids Design Consortium Annual Meeting MRL, UCSB 02/02/09 Supramolecular
More informationExample: Uniaxial Deformation. With Axi-symmetric sample cross-section dl l 0 l x. α x since α x α y α z = 1 Rewriting ΔS α ) explicitly in terms of α
Eample: Uniaial Deformation y α With Ai-symmetric sample cross-section l dl l 0 l, d Deform along, α = α = l0 l0 = α, α y = α z = Poisson contraction in lateral directions α since α α y α z = Rewriting
More informationPHASE TRANSITIONS IN SOFT MATTER SYSTEMS
OUTLINE: Topic D. PHASE TRANSITIONS IN SOFT MATTER SYSTEMS Definition of a phase Classification of phase transitions Thermodynamics of mixing (gases, polymers, etc.) Mean-field approaches in the spirit
More informationSelf-Assembled Morphologies of a Diblock Copolymer Melt Confined in a Cylindrical Nanopore
8492 Macromolecules 2006, 39, 8492-8498 Self-Assembled Morphologies of a Diblock Copolymer Melt Confined in a Cylindrical Nanopore Weihua Li and Robert A. Wickham* Department of Physics, St. Francis XaVier
More informationEffect of Architecture on the Phase Behavior of AB-Type Block Copolymer Melts M. W. Matsen*
pubs.acs.org/ Effect of Architecture on the Phase Behavior of AB-Type Block Copolymer Melts M. W. Matsen* School of Mathematical and Physical Sciences, University of Reading, Whiteknights, Reading RG6
More informationSelf Consistent Field Theory Study of Tetrablock Terpolymer Phase Behavior
Self Consistent Field Theory Study of Tetrablock Terpolymer Phase Behavior A THESIS SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL OF THE UNIVERSITY OF MINNESOTA BY Lynn M Wolf IN PARTIAL FULFILLMENT
More informationA Multi-Fluid Model of Membrane Formation by Phase-Inversion
A Multi-Fluid Model of Membrane Formation by Phase-Inversion Douglas R. Tree 1 and Glenn Fredrickson 1,2 1 Materials Research Laboratory 2 Departments of Chemical Engineering and Materials University of
More informationA Numerical Investigation of Laser Heating Including the Phase Change Process in Relation to Laser Drilling
A Numerical Investigation of Laser Heating Including the Phase Change Process in Relation to Laser Drilling I.Z. Naqavi, E. Savory & R.J. Martinuzzi Advanced Fluid Mechanics Research Group Department of
More informationComputer simulation of copolymer phase behavior
JOURNAL OF CHEMICAL PHYSICS VOLUME 117, NUMBER 22 8 DECEMBER 2002 Computer simulation of copolymer phase behavior Andrew J. Schultz, Carol K. Hall, a) and Jan Genzer Department of Chemical Engineering,
More informationTheoretical Study of Phase Behavior of Frustrated ABC Linear Triblock Copolymers
pubs.acs.org/macromolecules Theoretical Study of Phase Behavior of Frustrated ABC Linear Triblock Copolymers Meijiao Liu, Weihua Li,* and Feng Qiu State Key Laboratory of Molecular Engineering of Polymers,
More informationFinite Difference Solution of the Heat Equation
Finite Difference Solution of the Heat Equation Adam Powell 22.091 March 13 15, 2002 In example 4.3 (p. 10) of his lecture notes for March 11, Rodolfo Rosales gives the constant-density heat equation as:
More informationMultilevel Preconditioning of Graph-Laplacians: Polynomial Approximation of the Pivot Blocks Inverses
Multilevel Preconditioning of Graph-Laplacians: Polynomial Approximation of the Pivot Blocks Inverses P. Boyanova 1, I. Georgiev 34, S. Margenov, L. Zikatanov 5 1 Uppsala University, Box 337, 751 05 Uppsala,
More informationIndex. higher order methods, 52 nonlinear, 36 with variable coefficients, 34 Burgers equation, 234 BVP, see boundary value problems
Index A-conjugate directions, 83 A-stability, 171 A( )-stability, 171 absolute error, 243 absolute stability, 149 for systems of equations, 154 absorbing boundary conditions, 228 Adams Bashforth methods,
More informationDesign appropriate modeling for determining optimal friction reduction with surface textures
Georgia Institute of Technology Marquette University Milwaukee School of Engineering North Carolina A&T State University Purdue University University of California, Merced University of Illinois, Urbana-Champaign
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 9 Initial Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at Urbana-Champaign
More informationConfined Self-Assembly of Block Copolymers
Confined Self-Assembly of Block Copolymes An-Chang Shi Depatment of Physics & Astonomy McMaste Univesity Hamilton, Ontaio Canada Collaboatos: Bin Yu and Baohui Li, Nankai Univesity Peng Chen and Haojun
More informationSelf Organization. Order. Homogeneous state. Structurally ordered state. Structurally ordered state. Order. Disorder
Muthukumar, M., Ober, C.K. and Thomas, E.L., "Competing Interactions and Levels of Ordering in Self-Organizing Materials," Science, 277, 1225-1237 (1997). Self Organization Homogeneous state Order Disorder
More informationChapter 7. Entanglements
Chapter 7. Entanglements The upturn in zero shear rate viscosity versus molecular weight that is prominent on a log-log plot is attributed to the onset of entanglements between chains since it usually
More informationCS 450 Numerical Analysis. Chapter 9: Initial Value Problems for Ordinary Differential Equations
Lecture slides based on the textbook Scientific Computing: An Introductory Survey by Michael T. Heath, copyright c 2018 by the Society for Industrial and Applied Mathematics. http://www.siam.org/books/cl80
More informationThe standard Gaussian model for block copolymer melts
INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS: CONDENSED MATTER J. Phys.: Condens. Matter 14 (2002) R21 R47 PII: S0953-8984(02)17948-3 TOPICAL REVIEW The standard Gaussian model for block copolymer
More informationNUMERICAL METHODS FOR ENGINEERING APPLICATION
NUMERICAL METHODS FOR ENGINEERING APPLICATION Second Edition JOEL H. FERZIGER A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto
More informationMOLECULAR DYNAMICS SIMULATIONS OF IONIC COPOLYMERS
COMPUTATIONAL METHODS IN SCIENCE AND TECHNOLOGY 6, 15-24 (2000) MOLECULAR DYNAMICS SIMULATIONS OF IONIC COPOLYMERS MICHAŁ BANASZAK Macromolecular Physics Laboratory Institute of Physics, Adam Mickiewicz
More informationEnhancing the Potential of Block Copolymer Lithography with Polymer Self-Consistent Field Theory Simulations
8290 Macromolecules 2010, 43, 8290 8295 DOI: 10.1021/ma101360f Enhancing the Potential of Block Copolymer Lithography with Polymer Self-Consistent Field Theory Simulations Rafal A. Mickiewicz,, Joel K.
More informationBlock Copolymers in Electric Fields: A Comparison of Single-Mode and Self-Consistent-Field Approximations
Macromolecules 2006, 9, 289-29 289 Block Copolymers in Electric Fields: A Comparison of Single-Mode and Self-Consistent-Field Approximations Yoav Tsori* Department of Chemical Engineering, Ben Gurion UniVersity,
More informationMonica Olvera de la Cruz Northwestern University Department of Materials Science and Engineering 2220 Campus Drive Evanston, IL 60202
Introduction to Biomaterials and Biomolecular Self Assembly Monica Olvera de la Cruz orthwestern University Department of Materials Science and Engineering 0 Campus Drive Evanston, IL 600 Outline Introduction
More informationCE 530 Molecular Simulation
1 CE 530 Molecular Simulation Lecture 14 Molecular Models David A. Kofke Department of Chemical Engineering SUNY Buffalo kofke@eng.buffalo.edu 2 Review Monte Carlo ensemble averaging, no dynamics easy
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 9
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 9 Initial Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More informationChap. 2. Polymers Introduction. - Polymers: synthetic materials <--> natural materials
Chap. 2. Polymers 2.1. Introduction - Polymers: synthetic materials natural materials no gas phase, not simple liquid (much more viscous), not perfectly crystalline, etc 2.3. Polymer Chain Conformation
More informationSurface physics, Bravais lattice
Surface physics, Bravais lattice 1. Structure of the solid surface characterized by the (Bravais) lattice + space + point group lattice describes also the symmetry of the solid material vector directions
More informationQuantum Condensed Matter Physics Lecture 4
Quantum Condensed Matter Physics Lecture 4 David Ritchie QCMP Lent/Easter 2019 http://www.sp.phy.cam.ac.uk/drp2/home 4.1 Quantum Condensed Matter Physics 1. Classical and Semi-classical models for electrons
More informationBenchmarking a Self-Consistent Field Theory for Small Amphiphilic Molecules
This document is the accepted manuscript version of a published article. Published by The Royal Society of Chemistry in the journal "Soft Matter" issue 38, DOI: 10.1039/C2SM26352A Benchmarking a Self-Consistent
More informationNumerical Methods School of Mechanical Engineering Chung-Ang University
Part 5 Chapter 19 Numerical Differentiation Prof. Hae-Jin Choi hjchoi@cau.ac.kr 1 Chapter Objectives l Understanding the application of high-accuracy numerical differentiation formulas for equispaced data.
More informationSynopsis of Numerical Linear Algebra
Synopsis of Numerical Linear Algebra Eric de Sturler Department of Mathematics, Virginia Tech sturler@vt.edu http://www.math.vt.edu/people/sturler Iterative Methods for Linear Systems: Basics to Research
More informationMicrophase separation in multiblock copolymer melts Smirnova, YG; ten Brinke, G; Erukhimovich, IY; Smirnova, Yuliya G.; Erukhimovich, Igor Ya.
University of Groningen Microphase separation in multiblock copolymer melts Smirnova, YG; ten Brinke, G; Erukhimovich, IY; Smirnova, Yuliya G.; Erukhimovich, Igor Ya. Published in: Journal of Chemical
More informationA theoretical study for nanoparticle partitioning in the lamellae of diblock copolymers
THE JOURNAL OF CHEMICAL PHYSICS 128, 074901 2008 A theoretical study for nanoparticle partitioning in the lamellae of diblock copolymers Jiezhu Jin and Jianzhong Wu a Department of Chemical and Environmental
More informationNumerical Methods for ODEs. Lectures for PSU Summer Programs Xiantao Li
Numerical Methods for ODEs Lectures for PSU Summer Programs Xiantao Li Outline Introduction Some Challenges Numerical methods for ODEs Stiff ODEs Accuracy Constrained dynamics Stability Coarse-graining
More informationEffects of polydispersity on the order-disorder transition of diblock copolymer melts
Eur. Phys. J. E 27, 323 333 (2008) DOI 10.1140/epje/i2008-10383-6 THE EUROPEAN PHYSICAL JOURNAL E Effects of polydispersity on the order-disorder transition of diblock copolymer melts T.M. Beardsley a
More informationThe Intermaterial Dividing Surface (IMDS)
The Intermaterial Dividing Surface (IMDS) Can think of the microdomain structure as comprised of a set of surfaces that define the changeover in composition from Block A to Block B The IMDS in an AB diblock
More informationPolymer Solution Thermodynamics:
Polymer Solution Thermodynamics: 3. Dilute Solutions with Volume Interactions Brownian particle Polymer coil Self-Avoiding Walk Models While the Gaussian coil model is useful for describing polymer solutions
More informationT i t l e o f t h e w o r k : L a M a r e a Y o k o h a m a. A r t i s t : M a r i a n o P e n s o t t i ( P l a y w r i g h t, D i r e c t o r )
v e r. E N G O u t l i n e T i t l e o f t h e w o r k : L a M a r e a Y o k o h a m a A r t i s t : M a r i a n o P e n s o t t i ( P l a y w r i g h t, D i r e c t o r ) C o n t e n t s : T h i s w o
More informationPhysical interpretation of coarse-grained bead-spring models of complex fluids. Kirill Titievsky
Physical interpretation of coarse-grained bead-spring models of complex fluids by Kirill Titievsky Submitted to the Department of Chemical Engineering in partial fulfillment of the requirements for the
More information2017 SOCAMS Poster List
2017 SOCAMS Poster List Jason Dark UC Irvine A Chemical Master Equation Toolkit The direct solution of the chemical master equation (CME), when paired with approximation techniques like the finite state
More informationKinetic Pathways of Lamellae to Gyroid Transition in Weakly Segregated Diblock Copolymers
pubs.acs.org/macromolecules Kinetic Pathways of Lamellae to Gyroid Transition in Weakly Segregated Diblock Copolymers Nan Ji, Ping Tang,* and Feng Qiu* State Key Laboratory of Molecular Engineering of
More informationApplicable Simulation Methods for Directed Self-Assembly -Advantages and Disadvantages of These Methods
Review Applicable Simulation Methods for Directed Self-Assembly -Advantages and Disadvantages of These Methods Hiroshi Morita Journal of Photopolymer Science and Technology Volume 26, Number 6 (2013) 801
More information... 3, , = a (1) 3 3 a 2 = a (2) The reciprocal lattice vectors are defined by the condition a b = 2πδ ij, which gives
PHZ646: Fall 013 Problem set # 4: Crystal Structure due Monday, 10/14 at the time of the class Instructor: D. L. Maslov maslov@phys.ufl.edu 39-0513 Rm. 114 Office hours: TR 3 pm-4 pm Please help your instructor
More informationApplied Numerical Analysis
Applied Numerical Analysis Using MATLAB Second Edition Laurene V. Fausett Texas A&M University-Commerce PEARSON Prentice Hall Upper Saddle River, NJ 07458 Contents Preface xi 1 Foundations 1 1.1 Introductory
More informationSupporting Information for: Rapid Ordering in. Wet Brush Block Copolymer/Homopolymer
Supporting Information for: Rapid Ordering in Wet Brush Block Copolymer/Homopolymer Ternary Blends Gregory S. Doerk* and Kevin G. Yager Center for Functional Nanomaterials, Brookhaven National Laboratory,
More informationP. W. Anderson [Science 1995, 267, 1615]
The deepest and most interesting unsolved problem in solid state theory is probably the theory of the nature of glass and the glass transition. This could be the next breakthrough in the coming decade.
More informationNumerical Oscillations and how to avoid them
Numerical Oscillations and how to avoid them Willem Hundsdorfer Talk for CWI Scientific Meeting, based on work with Anna Mozartova (CWI, RBS) & Marc Spijker (Leiden Univ.) For details: see thesis of A.
More informationPartitioned Methods for Multifield Problems
C Partitioned Methods for Multifield Problems Joachim Rang, 6.7.2016 6.7.2016 Joachim Rang Partitioned Methods for Multifield Problems Seite 1 C One-dimensional piston problem fixed wall Fluid flexible
More informationAnderson acceleration for time-dependent problems
for time-dependent s PROBLEM SETTNG : 1D FLOW FGURE : One-dimensional flow in a tube. incompressible inviscid gravity neglected conservative form PROBLEM SETTNG : 1D FLOW gu t + gu2 x ( ) g gu t + + 1
More informationBindel, Fall 2011 Intro to Scientific Computing (CS 3220) Week 12: Monday, Apr 18. HW 7 is posted, and will be due in class on 4/25.
Logistics Week 12: Monday, Apr 18 HW 6 is due at 11:59 tonight. HW 7 is posted, and will be due in class on 4/25. The prelim is graded. An analysis and rubric are on CMS. Problem du jour For implicit methods
More informationTAU Solver Improvement [Implicit methods]
TAU Solver Improvement [Implicit methods] Richard Dwight Megadesign 23-24 May 2007 Folie 1 > Vortrag > Autor Outline Motivation (convergence acceleration to steady state, fast unsteady) Implicit methods
More informationModule 6: Implicit Runge-Kutta Methods Lecture 17: Derivation of Implicit Runge-Kutta Methods(Contd.) The Lecture Contains:
The Lecture Contains: We continue with the details about the derivation of the two stage implicit Runge- Kutta methods. A brief description of semi-explicit Runge-Kutta methods is also given. Finally,
More informationMaterials at Equilibrium. G. Ceder Fall 2001 COURSE 3.20: THERMODYNAMICS OF MATERIALS. FINAL EXAM, Dec 18, 2001
NAME: COURSE 3.20: THERMODYNAMICS OF MATERIALS FINAL EXAM, Dec 18, 2001 PROBLEM 1 (15 POINTS) PROBLEM 2 (15 POINTS) PROBLEM 3 (10 POINTS) PROBLEM 4 (20 POINTS) PROBLEM 5 (14 POINTS) PROBLEM 6 (14 POINTS)
More informationMagnetic wells in dimension three
Magnetic wells in dimension three Yuri A. Kordyukov joint with Bernard Helffer & Nicolas Raymond & San Vũ Ngọc Magnetic Fields and Semiclassical Analysis Rennes, May 21, 2015 Yuri A. Kordyukov (Ufa) Magnetic
More informationNumerical Analysis. A Comprehensive Introduction. H. R. Schwarz University of Zürich Switzerland. with a contribution by
Numerical Analysis A Comprehensive Introduction H. R. Schwarz University of Zürich Switzerland with a contribution by J. Waldvogel Swiss Federal Institute of Technology, Zürich JOHN WILEY & SONS Chichester
More informationA Fluctuating Immersed Boundary Method for Brownian Suspensions of Rigid Particles
A Fluctuating Immersed Boundary Method for Brownian Suspensions of Rigid Particles Aleksandar Donev Courant Institute, New York University APS DFD Meeting San Francisco, CA Nov 23rd 2014 A. Donev (CIMS)
More informationKinetic relaxation models for reacting gas mixtures
Kinetic relaxation models for reacting gas mixtures M. Groppi Department of Mathematics and Computer Science University of Parma - ITALY Main collaborators: Giampiero Spiga, Giuseppe Stracquadanio, Univ.
More informationIntroduction to multiscale modeling and simulation. Explicit methods for ODEs : forward Euler. y n+1 = y n + tf(y n ) dy dt = f(y), y(0) = y 0
Introduction to multiscale modeling and simulation Lecture 5 Numerical methods for ODEs, SDEs and PDEs The need for multiscale methods Two generic frameworks for multiscale computation Explicit methods
More informationSupporting Online Material. Directed Assembly of Block Copolymer Blends into Non-regular Device Oriented Structures
Supporting Online Material Directed Assembly of Block Copolymer Blends into Non-regular Device Oriented Structures Mark P. Stoykovich, 1 Marcus Müller, 2 Sang Ouk Kim, 1* Harun H. Solak, 3 Erik W. Edwards,
More informationIntroduction to Numerical Analysis
J. Stoer R. Bulirsch Introduction to Numerical Analysis Second Edition Translated by R. Bartels, W. Gautschi, and C. Witzgall With 35 Illustrations Springer Contents Preface to the Second Edition Preface
More informationON THE PHASE DIAGRAM FOR MICROPHASE SEPARATION OF DIBLOCK COPOLYMERS: AN APPROACH VIA A NONLOCAL CAHN HILLIARD FUNCTIONAL
SIAM J. APPL. MATH. Vol. 69, No. 6, pp. 171 1738 c 009 Society for Industrial and Applied Mathematics ON THE PHASE DIAGRAM FOR MICROPHASE SEPARATION OF DIBLOCK COPOLYMERS: AN APPROACH VIA A NONLOCAL CAHN
More informationStructural Changes of Diblock Copolymer Melts Due to an External Electric Field: A Self-Consistent-Field Theory Study
5766 Macromolecules 2005, 38, 5766-5773 Structural Changes of Diblock Copolymer Melts Due to an External Electric Field: A Self-Consistent-Field Theory Study Chin-Yet Lin* and M. Schick Department of Physics,
More informationString method for the Cahn-Hilliard dynamics
String method for the Cahn-Hilliard dynamics Tiejun Li School of Mathematical Sciences Peking University tieli@pku.edu.cn Joint work with Wei Zhang and Pingwen Zhang Outline Background Problem Set-up Algorithms
More informationHyeyoung Shin a, Tod A. Pascal ab, William A. Goddard III abc*, and Hyungjun Kim a* Korea
The Scaled Effective Solvent Method for Predicting the Equilibrium Ensemble of Structures with Analysis of Thermodynamic Properties of Amorphous Polyethylene Glycol-Water Mixtures Hyeyoung Shin a, Tod
More informationFast Fourier Transform Solvers and Preconditioners for Quadratic Spline Collocation
Fast Fourier Transform Solvers and Preconditioners for Quadratic Spline Collocation Christina C. Christara and Kit Sun Ng Department of Computer Science University of Toronto Toronto, Ontario M5S 3G4,
More informationComputation Fluid Dynamics
Computation Fluid Dynamics CFD I Jitesh Gajjar Maths Dept Manchester University Computation Fluid Dynamics p.1/189 Garbage In, Garbage Out We will begin with a discussion of errors. Useful to understand
More informationProject: Vibration of Diatomic Molecules
Project: Vibration of Diatomic Molecules Objective: Find the vibration energies and the corresponding vibrational wavefunctions of diatomic molecules H 2 and I 2 using the Morse potential. equired Numerical
More informationStructure of Crystalline Solids
Structure of Crystalline Solids Solids- Effect of IMF s on Phase Kinetic energy overcome by intermolecular forces C 60 molecule llotropes of Carbon Network-Covalent solid Molecular solid Does not flow
More informationCHAPTER 5: Linear Multistep Methods
CHAPTER 5: Linear Multistep Methods Multistep: use information from many steps Higher order possible with fewer function evaluations than with RK. Convenient error estimates. Changing stepsize or order
More informationLecture 20: Spinodals and Binodals; Continuous Phase Transitions; Introduction to Statistical Mechanics
Lecture 20: 11.28.05 Spinodals and Binodals; Continuous Phase Transitions; Introduction to Statistical Mechanics Today: LAST TIME: DEFINING METASTABLE AND UNSTABLE REGIONS ON PHASE DIAGRAMS...2 Conditions
More informationFaster Kinetics: Accelerate Your Finite-Rate Combustion Simulation with GPUs
Faster Kinetics: Accelerate Your Finite-Rate Combustion Simulation with GPUs Christopher P. Stone, Ph.D. Computational Science and Engineering, LLC Kyle Niemeyer, Ph.D. Oregon State University 2 Outline
More informationGAME PHYSICS SECOND EDITION. дяййтаййг 1 *
GAME PHYSICS SECOND EDITION DAVID H. EBERLY дяййтаййг 1 * К AMSTERDAM BOSTON HEIDELBERG LONDON NEW YORK OXFORD PARIS SAN DIEGO SAN FRANCISCO SINGAPORE SYDNEY TOKYO MORGAN ELSEVIER Morgan Kaufmann Publishers
More informationPhysics 212: Statistical mechanics II Lecture XI
Physics 212: Statistical mechanics II Lecture XI The main result of the last lecture was a calculation of the averaged magnetization in mean-field theory in Fourier space when the spin at the origin is
More informationProject Mentor(s): Dr. Evelyn Sander and Dr. Thomas Wanner
STABILITY OF EQUILIBRIA IN ONE DIMENSION FOR DIBLOCK COPOLYMER EQUATION Olga Stulov Department of Mathematics, Department of Electrical and Computer Engineering State University of New York at New Paltz
More informationÄ is a basis for V Ä W. Say xi
Groups Fields Vector paces Homework #3 (2014-2015 Answers Q1: Tensor products: concrete examples Let V W be two-dimensional vector spaces with bases { 1 2} v v { } w w o { vi wj} 1 2 Ä is a basis for V
More informationFinding small factors of integers. Speed of the number-field sieve. D. J. Bernstein University of Illinois at Chicago
The number-field sieve Finding small factors of integers Speed of the number-field sieve D. J. Bernstein University of Illinois at Chicago Prelude: finding denominators 87366 22322444 in R. Easily compute
More information