Self-assembling block copolymer systems involving competing lenght scales Nap, Rikkert Jan

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1 University of Groningen Self-assembling block copolymer systems involving competing lenght scales Nap, Rikkert Jan IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2003 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Nap, R. J. (2003). Self-assembling block copolymer systems involving competing lenght scales s.n. Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date:

2 Self-Assembling Block Copolymer Systems Involving Competing Length Scales R.J. Nap

3 Self-Assembling Block Copolymer Systems Involving Competing Length Scales R.J. Nap Ph.D. thesis University of Groningen, the Netherlands September 2003 MSC Ph.D.-thesis series ISSN ISBN

4 Rijksuniversiteit Groningen Self-Assembling Block Copolymer Systems Involving Competing Length Scales Proefschrift ter verkrijging van het doctoraat in de Wiskunde en Natuurwetenschappen aan de Rijksuniversiteit Groningen op gezag van de Rector Magnificus, dr. F. Zwarts, in het openbaar te verdedigen op maandag 29 september 2003 om uur door Rikkert Jan Nap geboren op 21 oktober 1974 te Emmen

5 promotor: Prof. Dr. G. ten Brinke beoordelingscommissie: Prof. Dr. I.Ya. Erukhimovich Prof. Dr. J.G.E.M. Fraaije Prof. Dr. J.J.M. Slot ISBN:

6 It is a capital mistake to theorize before you have all the evidence. It biases the judgement. Arthur Conan Doyle : A study in scarlet (1888)

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8 Contents List of symbols and abbreviations xi 1 Introduction Block copolymers Hierarchically structured materials Self-organized comb-shaped supramolecules Responsive materials Physics of block copolymers melts Basic concepts Statistical mechanics of block copolymers Outline of this thesis Microphase separation involving two length scales Introduction Competing length scales Phase stability and structure factor Spinodal of diblock copolymers Spinodal of comb copolymers Spinodal of linear-comb and linear-alternating copolymers Bifurcation and Classification Asymptotic analysis of the phase boundaries Limiting spinodal curve Conclusion WSL of block copolymers Introduction

9 viii Contents 3.2 Weak-segregation theory of linear block copolymers Weak-segregation theory of non-linear block copolymers Correlation functions Correlation functions, diagrams Chemical correlation functions Correlation functions, topological equivalent diagrams Free energy expansion Spatial symmetries Minimization of free energy First-harmonic approximation Second-harmonic approximation Selected phase diagrams Conclusion Phase behavior of linear-comb copolymer in WSL Introduction Phase behavior of comb block copolymer Phase behavior of linear-comb block copolymer Critical points of linear-comb copolymers Selected phase diagrams Second-harmonic approximation and two length scale problem Conclusion SCFT of linear block copolymers Introduction Self-consistent field theory Partition function Q Partial differential equation Diblock copolymers Alternating copolymers Linear-alternating block copolymers Conclusion Phase behavior of linear-alternating copolymers in SCFT Introduction Phase behavior of diblock copolymers Phase behavior of alternating block copolymers

10 Contents ix 6.4 Phase behavior of linear-alternating block copolymers Phase diagrams for n = Phase diagrams for n = Conclusion Epilogue Introduction Objectives, results and suggestions for future research Conclusion Bibliography 153 A q=q* approximation 169 B Scaling of amplitudes A 2 O(A 2 1 ) 173 C Symmetry and spatial structures 175 D Coefficients of free energy expansion 181 E Differential equation for the propagator 187 F Numerical methods 191 Summary 197 Samenvatting 203 Dankwoord 209

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12 List of symbols and abbreviations χ Flory-Huggins thermodynamical interaction parameter N total degree of polymerization of a polymer a statistical length of one polymer segment A m -b-(a-g-b) n linear-comb block copolymer A m -b-(b-b-a) n linear-alternating block copolymer d number of monomers between consecutive branched points n number of branched points or repeat units n s number of side chains m length of linear A-block in units of d f, f A total volume fraction of (A-) monomers f l volume fraction of linear A-block D, D domain spacing F, F free energy Z configurational partition function Q, G ideal configurational partition function P various distribution functions S (q) = Γ 1 2 (q) structure factor Γ n coefficients of Landau free energy expansion r(s) contour curve r, x, y vector in real space q, Q, G vector in reciprocal space q inverse domain spacing in WSL R G (N) radius of gyration of a chain of length N x = q 2 R G (N) 2 dimensionless length y = q 2 R G (d) 2 dimensionless length

13 g α1 α n (r 1,, r n ) g i1 i n (r 1,, r n ) ψ φ α (r) w α (r) ξ(r) ϕ i (r) λ i Γ i jk σ α (s) H n q n q(r, s) G(r i, s i ; r f, s f ) 1 p6mm p4mm Im 3m Fm 3m Pm 3m Ia 3d ODT WSL ISL SSL WST SCFT PS PI PDP P4VP chemical correlation function segment correlation function order parameter, concentration profile monomer density function external fields coupled to the density function external field enforcing the imcompressibility constraint basisfunction: eigenfunctions of Laplace operator eigenvalues of Laplace operator integral over product of three basisfunctions function describing whether monomer s is of type α harmonic: set of vectors length of vectors belonging to harmonic set H n end-segments distribution function chain-propagator lamallar structure hexagonal ordered cilinders tetragonal ordered cilinders body centered cubical structure face centered cubical structure simple cubical structure gyroid: complex cubical structure order-disorder transition weak-segregation limit intermediate-segregation limit strong-segregation limit weak-segregation theory self-consistent field theory polystyrene polyisoprene pentadecyl phenol poly(4-vinylpyridine)