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Theory of Block Polymer Self-Assembly

Author(s):
Publication Date:
March 13, 2024
Copyright © 2024 American Chemical Society
eISBN:
‍9780841299221
DOI:
10.1021/acsinfocus.7e8001
Read Time:
seven to eight hours
Collection:
3
Publisher:
American Chemical Society
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This primer introduces the theory of self-assembly of block polymers, most notably self-consistent field theory (SCFT). Block polymer self-assembly is a fascinating and highly interdisciplinary topic.

 

This primer can be read at several levels, depending on what readers want to get out of it. Readers who want an overview of self-assembly in block polymer and what SCFT says about the process can read Chapters 1-3 and skip to Chapter 7 to see the open questions. If the reader is further interested in the output of SCFT calculations but not how those outputs are generated, they should read Chapter 6 as well. But if the reader wants to learn how to do the SCFT calculations themselves, Chapters 4 and 5 offer an accessible introduction to the theory and numerical methods, providing an excellent entry point into the literature. This primer includes data that the authors have computed using SCFT. All calculations use the open-source software package Polymer Self-Consistent Field (PSCF), developed by David Morse at the University of Minnesota.

 

Take breaks from reading to watch ten “Insider Q&A” videos included throughout, which offer additional insight from experts in the field, such as An-Chang Shi, Chinedum O. Osuji, Frank S. Bates, Christopher M. Bates, Glenn H. Fredrickson, and Lisa Hall. Furthermore, this primer includes multiple features to aid and enhance readers’ learning. “That’s a Wrap” summarizes key concepts at the end of each chapter, while “Read These Next” suggests references that may interest further reading. A pop-up glossary ensures readers have definitions as needed throughout the primer.

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Detailed Table of Contents
About the Series
Preface
Chapter 1
Self-Assembly in Soft Matter
1.1
What Is Self-Assembly?
1.2
What Is a Block Polymer?
1.3
Insider Q&A: Frank S. Bates and Christopher M. Bates
1.4
Insider Q&A: Glenn H. Fredrickson
1.5
Block Polymers as a Model System
1.6
Insider Q&A: An-Chang Shi
1.7
That’s a Wrap
1.8
Read These Next
Chapter 2
Polymer Thermodynamics
2.1
Enthalpy and Entropy of Polymers
2.2
Polymer Conformations
2.2.1
The Gaussian Chain Model
2.2.2
Radius of Gyration and End-to-End Distance
2.2.3
Gaussian Elasticity
2.3
Flory–Huggins Theory
2.3.1
Solution Thermodynamics of Simple Molecules
2.3.2
Extension to Homopolymer Blends and Solutions
2.3.3
Phase Separation
2.4
Mapping Experiments to Models
2.4.1
Definition of a Monomer (for Theory)
2.4.2
Degree of Polymerization
2.4.3
Statistical Segment Length
2.4.4
χ Parameter
2.4.5
Rescaling the Reference Volume
2.4.6
The N = 1 Convention
2.5
That’s a Wrap
2.6
Read This Next
Chapter 3
Microphase Separation in Diblock Copolymers
3.1
A First Look at Self-Assembly in Diblock Copolymers
3.2
A Toy Model of Self-Assembly
3.3
Predictions of Mean-Field Models of Self-Assembly
3.3.1
Conformationally Symmetric Phase Diagram
3.3.2
Conformational Asymmetry and Frank–Kasper Phases
3.4
Insider Q&A: Frank S. Bates and Christopher M. Bates
3.5
Insider Q&A: Chinedum O. Osuji
3.6
That’s a Wrap
3.7
Read This Next
Chapter 4
Self-Consistent Field Theory
4.1
A Second Look at Self-Assembly in Diblock Copolymers
4.2
What Is the Mean Field?
4.2.1
Particle-to-Field Transformation
4.2.2
Limitations of Mean-Field Theory
4.3
Insider Q&A: Lisa Hall
4.4
SCFT for Diblock Copolymers
4.4.1
Neat Melt
4.4.1.1
Chain Propagator and Normalized Partition Function
4.4.1.2
Density Fields
4.4.1.3
Self-Consistent Field Equations
4.4.1.4
Helmholtz Free Energy
4.4.2
Diblock Copolymer Blend: Canonical Ensemble
4.4.3
Diblock Copolymer Blend: Grand Canonical Ensemble
4.5
Disordered State
4.5.1
One Homopolymer
4.5.2
Homopolymer Blend (Flory–Huggins Theory)
4.5.3
Diblock Copolymer Melt
4.6
Beyond AB Diblock Copolymers
4.7
That’s a Wrap
4.8
Read This Next
Chapter 5
Ordered States: Numerical Methods
5.1
What’s Next?
5.2
Problem Setup
5.2.1
The Unit Cell
5.2.2
Plane-Wave Basis Functions and Fourier Space
5.2.3
Space-Group Symmetry
5.3
Initialization Methods
5.3.1
Level-Set Method
5.3.2
Form Factor Method
5.4
Integration of the Chain Propagators
5.4.1
Pseudospectral Method
5.4.2
Comparisons with Other Methods
5.5
Finding a Self-Consistent Solution
5.5.1
Anderson Mixing
5.5.2
Lattice Parameter Optimization
5.5.3
Other Methods
5.6
That’s a Wrap
5.7
Read This Next
Chapter 6
Ordered States: Examples
6.1
Numerical Solutions
6.2
Calculation of a Phase Boundary in the Melt State
6.3
Calculation of a Phase Boundary in a Blend
6.3.1
Common-Tangent Construction
6.3.2
Grand Canonical Ensemble
6.4
Predicting the Scattering Profile
6.5
That’s a Wrap
6.6
Read These Next
Chapter 7
Beyond SCFT and Linear Diblock Copolymers
7.1
Future Directions
7.2
Insider Q&A: Chinedum O. Osuji
7.3
Polymer Architecture
7.3.1
Panacea or Pandora’s Box?
7.3.2
Multiblock Polymers
7.3.3
Block Polymer Blends
7.4
Insider Q&A: An-Chang Shi
7.5
Solving the Inverse Problem
7.6
Fluctuation Effects and Field-Theoretic Simulations
7.7
Insider Q&A: Glenn H. Fredrickson
7.8
Insider Q&A: Chinedum O. Osuji
7.9
That’s a Wrap
7.10
Read This Next
Appendix A
The Partition Function
Bibliography
Footnotes
Glossary
Index
Reviewer quotes
Gayashani Ginige, Recent Ph.D. graduate, University of Alberta
Theory of Block Polymer Self-Assembly is a valuable resource for graduate students seeking a foundational understanding of the topic, providing clear explanations and illustrative figures to aid comprehension. The primer incorporates a well-balanced combination of text and figures to explain key concepts and principles related to the topic. It is intended to bring readers up to speed on the subject matter, offering an accessible introduction that is neither too high-level nor too detailed, making it suitable for its target audience. As a chemist who works in a laboratory about block copolymer self-assembly, I have extensively referred to the phase diagram of block copolymers. However, I have never studied how these phase diagrams are developed. This primer provides great insights into the stepwise development of phase diagrams for block copolymers and leaves chemists with new experimental pathways to explore.
Author Info
Benjamin R. Magruder
Benjamin R. Magruder is a graduate student in the Department of Chemical Engineering and Materials Science at the University of Minnesota. He received a B.S. degree in chemical engineering from the University of Washington in 2020. His thesis research uses self-consistent field theory to study the self-assembly of block polymers into bulk Frank–Kasper phases, and thin film network phases. He is also an active contributor to the PSCF software package, writing the thin-film functionality of the code and developing the most recent version of the polymer visual package. Photo: Olivia Nemec, Valera Photography
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Kevin D. Dorfman
Kevin D. Dorfman is a Distinguished McKnight University Professor in the Department of Chemical Engineering and Materials Science at the University of Minnesota. He received a B.S degree in chemical engineering from Penn State in 1999, an M.S. in chemical engineering from MIT in 2001, and a Ph.D. in chemical engineering from MIT in 2002 under the supervision of Howard Brenner. Following a 3-year HFSP postdoctoral fellowship at the Curie Institute, working with Jean-Louis Viovy, he joined the faculty at the University of Minnesota in 2006. His group focuses on polymer physics, with particular recent emphases on the self-assembly of block polymers, dynamics of block polymer micelles, and the applications of polymer physics to genome mapping technologies. His research has been recognized, most notably, by the Colburn Award of the AIChE and a Packard Fellowship in Science and Engineering. He is presently an Associate Editor of the journal Macromolecules and, with Prodromos Daoutidis, the author of the undergraduate textbook Numerical Methods with Chemical Engineering Applications, published by Cambridge University Press. Photo: Rebecca Slater, By Rebecca Studios
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