Research Topics

Field-Theoretic Simulations

Our group has pioneered the development of a new class of computer simulation methods where the degrees of freedom are fluctuating fields, rather than atom or particle coordinates. The basis for these "Field-Theoretic Simulations" (FTS) is a statistical field theory model, which is generally derived from a conventional particle-based model by means of exact mathematical transformations, such as Hubbard-Stratonovich transforms.

The FTS method has a number of advantages relative to traditional simulation techniques for studying the equilibrium structure and thermodynamic properties of complex fluids and polymers. Most importantly and unlike particle-based simulations, the method becomes more efficient as the density of the system increases, or the polymers become longer. The FTS method is also particularly convenient for studying charged polymers ("polyelectrolytes") that are characterized by long-range Coulomb interactions. In the field-theoretic framework, the problems normally associated with simulating Coulomb systems are minimized.

More information about the FTS method can be found in the following review articles:

"Field-Theoretic Computer Simulation Methods for Polymers and Complex Fluids," G.H.Fredrickson, V.Ganesan, and F. Drolet, Macromolecules 35, 16 (2002). Link
"Computational Field Theory of Polymers: Opportunities and Challenges," G.H.Fredrickson, Soft Matter 2, 1329 (2007). Link
"Recent Developments in Fully Fluctuating Field-Theoretic Simulations of Polymer Melts and Solutions," K.T.Delaney and G.H.Fredrickson, J. Phys. Chem. B, 120, 7615 (2016). Link

See also the Oxford University Press monograph:

The Equilibrium Theory of Inhomogeneous Polymers, G.H.Fredrickson (Oxford University Press, Oxford, 2006). Link

Foundations of the FTS Method

While the FTS method is a standard research tool for many projects in the Fredrickson group, we have a number of ongoing research programs that are aimed at providing an improved theoretical basis for the method, developing more robust or efficient numerical methods for implementation, or extending the capabilities of FTS.

One branch of our research involves developing methods for systematic numerical "coarse-graining" of field theory models. The idea is to conduct a very fine-grained (i.e. high spatial resolution of the fields) FTS simulation on a small system and use the results to parameterize a new model on a coarser computational grid. That new model can then be simulated to study a larger system and further reparameterize an even coarser lattice field theory model. In this way, we can bootstrap a series of simulations that will cover a broad spectrum (6 decades or more) of spatial scales, while requiring only the parameters of the finest scale model as input. This type of multi-scale simulation technique is of considerable value in a variety of polymer and complex fluid formulation problems, including systems such as blends, emulsions, and polyelectrolyte complexes, where structure can extend from nanometer to millimeter length scales.

Another branch of research seeks alternative field theory representations, such as the "coherent states" framework of quantum field theory, that can be applied to soft materials and adapted for efficient numerical simulations.

Some recent publications in the FTS Foundations area include:

"Free Energy Evaluation in Field-Theoretic Polymer Simulations," E.M.Lennon, K.Katsov, and G.H.Fredrickson, Phys. Rev. Lett. 101, 138302 (2008). Link
"Numerical Coarse-Graining of Fluid Field Theories," M.C.Villet and G.H.Fredrickson, J. Chem. Phys. 132, 034109 (2010). Link
"Field Theoretic Simulations in the Gibbs Ensemble," R.A.Riggleman and G.H.Fredrickson, J. Chem. Phys 132, 024104 (2010). Link
"Polymer Field Theory Simulations on Graphics Processing Units," K.T.Delaney and G.H.Fredrickson, Computer Physics Communications 184(9), 2102 (2013). Link
"Coherent States Formulation of Polymer Field Theory," X.Man, K.T.Delaney, M.C.Villet, H.Orland, and G.H.Fredrickson, J. Chem. Phys. 140, 024905 (2014). Link
"Efficient Field-Theoretic Simulation of Polymer Solutions," M.C.Villet and G. H. Fredrickson, J. Chem. Phys. 141, 224115 (2014). Link
"A Multi-Species Exchange Model for Fully Fluctuating Polymer Field Theory Simulations," D. Duechs, K. T. Delaney, and G. H. Fredrickson, J. Chem. Phys. 141, 174103 (2014). Link

Block Copolymer Lithography

A topic of recent emphasis in the Fredrickson group relates to the use of thin block copolymer films for patterning microelectronic devices. The self-assembled domains of block copolymers have a size and pitch that is characteristically on the 10 nm scale -- a scale that difficult and expensive to achieve by the conventional optical lithography practiced by the semiconductor industry. Before block copolymers can be widely adopted for this purpose, however, a number of challenges must be overcome. For example, we must learn how to create square lattices of cylindrical and spherical domains as opposed to hexagonal arrays. We must be able to independently vary domain/feature size and pitch, and to create individual spheres, cylinders, and lines -- not just periodic arrays. Additionally, we need to be able to minimize defects in the self-assembled structures, control placement error and line edge roughness, and manipulate the substrate and surface interactions with the individual blocks comprising the films.

In addressing these challenges, the Fredrickson group works closely with the experimentally-oriented Hawker group at UCSB, as well as with advanced lithography groups at companies such as Intel and Samsung. Our FTS and related self-consistent field theory (SCFT) simulations are used to provide insights into block copolymer assembly in thin/confined films that can be translated into experimental designs and processing protocols.

Some recent publications in the block copolymer lithography area include:

"Evolution of Block Copolymer Lithography to Highly Ordered Square Arrays," C.Tang, E.M.Lennon, G.H.Fredrickson, E.J.Kramer, and C.J. Hawker, Science 322, 429 (2008). Link
"SCFT Simulations of Thin Film Blends of Block Copolymer and Homopolymer Laterally Confined in a Square Well," S.Hur, C.J. Garcia-Cervera, E.J.Kramer, and G.H.Fredrickson, Macromolecules 42, 5861 (2009). Link
"Defectivity in Laterally Confined Lamella-Forming Diblock Copolymers: Thermodynamic and Kinetic Aspects," H. Takahashi, N. Laachi, K. T. Delaney, S. Hur, C. J. Weinheimer, D. Shykind, and G. H. Fredrickson, Macromolecules 45, 6253 (2012). Link
"Self-Consistent Field Theory Investigation of Directed Self-Assembly in Cylindrical Confinement," N. Laachi, K. T. Delaney, B. Kim, S. Hur, R. Bristol, D. Shykind, C. J. Weinheimer, and G. H. Fredrickson, J. Polym. Sci: Part B, Polym. Phys. 53, 142 (2015). Link
"Thermodynamic and Kinetic Aspects of Defectivity in Directed Self-Assembly of Cylinder-Forming Diblock Copolymers in Laterally Confining Thin Channels," B. Kim, N. Laachi, K. T. Delaney, M. Carilli, and G. H. Fredrickson, J. Appl. Polym. Sci. 131, 40790 (2014). Link
"A Landau-Peierls Analysis of Contact Hole Placement in Directed Self-Assembly of Linear Arrays of Block Copolymer Cylinders," N. Laachi, D. Shykind, and G. H. Fredrickson, Macromolecules 47, 8819 (2014). Link

Supramolecular Polymers

An emerging theme in polymer science relates to the use of specific and reversible interactions to introduce unique structure, dynamics, and function in polymeric materials. This subject reflects a marriage of the rapidly emerging field of supramolecular chemistry with traditional polymer science. Examples of the supramolecular interactions that can be employed include hydrogen bonds, metal-ligand interactions, and pi-pi stacking. By attaching such moieties to polymer chains and adjusting the strength of the reversible binding, it is possible to achieve remarkable combinations of properties. For example, one can design adhesives and coatings that are fully bonded at low temperatures and are therefore mechanically robust, i.e. strong and tough, while at elevated temperatures the bonds break -- enabling easy flow and coating of a substrate.

Our group's research is focused primarily on the largely unexplored field of heterogeneous supramolecular polymers, where dissimilar polymers are linked by reversible bonds, thereby creating complex ensembles of block and graft copolymers, as well as inhomogeneous networks and gels. The interplay between the chemical equilibria of reversible binding and physical phase separation and microphase separation processes in such systems can produce a rich and diverse range of materials with widely varying properties and potential applications. Our approach in this field has been to define some simple models that serve to illustrate this interplay and analyze the models using a combination of analytical methods and FTS and SCFT simulations.

One example of such a model is a "supramolecular diblock" model in which two homopolymers, each bearing a complementary binding group at one end, are blended together. As shown below, the product of the reversible binding reaction is a diblock copolymer. We have mapped out the phase diagram of this seemingly simple system and have found a fascinating range of phase behaviors that can be driven thermally, or with blend composition, molecular weights, or binding equilibrium constant.

Recent group publications in the area of supramolecular polymers include:

"Supramolecular Diblock Copolymers: A Field-Theoretic Model and Mean-Field Solution," E.H.Feng, W.B.Lee, and G.H.Fredrickson, Macromolecules 40, 693 (2007). Link
"Phase Morphologies in Reversibly Bonding Supramolecular Triblock Copolymer Blends," W.B.Lee, R.Elliott, K.Katsov, and G.H.Fredrickson, Macromolecules 40, 8445 (2007). Link
"Supramolecular Assembly in Telechelic Polymer Blends," R.Elliott and G.H.Fredrickson, J. Chem. Phys. 131, 144906 (2009). Link
"Field-Theoretic Model of Inhomogeneous Supramolecular Polymer Networks and Gels," A. Mohan, R. Elliott, and G. H. Fredrickson, J. Chem. Phys. 133, 174903 (2010). Link
"Macro- and Microphase Separation in Multifunctional Supramolecular Polymer Networks," Z. Mester, A. Mohan, and G. H. Fredrickson, Macromolecules 44, 9411 (2011). Link
"Phase Coexistence of Reversibly Bonded Block Copolymers: A Unit Cell Gibbs Ensemble Approach," Z. Mester, N. A. Lynd, K. T. Delaney, and G. H. Fredrickson, Macromolecules 47, 1865-1874 (2014). Link

Polyelectrolyte Complexation

Another active research area in the Fredrickson group relates to the complexation of oppositely charged polyelectrolytes in aqueous solution. Due to electrostatic attractions and the entropy of counterion release, mixtures of polycations and polyanions in water tend to phase separate into a dense phase comprising most of the polymer and a dilute "supernatant" phase containing just small ions and water. The dense phase can be a solid precipitate, but in more interesting cases can itself be a hydrated fluid phase -- a so-called "complex coacervate" Such fluid polyelectrolyte complexes are pervasive in biology, but are also utilized in a variety of food and drug encapsulation technologies, as the coacervate phase will spread easily on many surfaces and has an ultralow interfacial tension against water. An interesting biological example, which is being intensively studied by the Waite group at UCSB, is the marine "sandcastle worm" who creates its underwater habitat by means of a protein-derived coacervate glue.

The Fredrickson group is engaged in collaboration with experimentalists in the Waite, Hawker, and Hahn groups to understand the fundamental chemistry and physics of a broad range of polyelectrolyte complexation phenomena. We are using analytical theories and FTS simulations to explore a variety of systems and physical mechanisms, including the role of molecular weight asymmetries and stoichiometric imbalances in the polyelectrolyte components of the mixtures and counter-ion partitioning between the supernatant and coacervate phases. We are also investigating the principles of self-assembly in "inhomogeneous coacervates" produced by complexing oppositely charged block copolymers, as well as developing theories of interfacial structure and thermodynamics in such systems.

Recent publications related to polyelectrolyte complexes include:

"Field-Theoretic Simulation of Polyelectrolyte Complexation," Y.O.Popov, J.Lee, and G.H.Fredrickson, J. Polym. Sci. B: Polym. Phys. 45, 3223 (2007). Link
"Complex Coacervation: A Field Theoretic Simulation Study of Polyelectrolyte Complexation," J.Lee, Y.O.Popov, and G.H.Fredrickson, J. Chem. Phys. 128, 224908 (2008). Link
"Investigation of the Interfacial Tension of Complex Coacervates Using Field-Theoretic Simulations," R. A. Riggleman, R. Kumar, and G. H. Fredrickson, J. Chem. Phys. 136, 024903 (2012). Link