Abstracts

Wednesday

Design criteria for two length-scale phase separation of block co-polymers

Daniel Read, Merin Joseph and Alastair Rucklidge (University of Leeds)

A candidate soft matter system for quasicrystalline order is phase segregated block copolymers, and indeed several observations of quasicrystalline block copolymers exist in the literature. Current ideas for design criteria for such behaviour in block copolymers are based on highly coarse-grained models, which led to the understanding that quasicrystals might be favoured when instability towards phase segregation occurs at two lengthscales simultaneously (or near-simultaneously). However, these coarse-grained models do not account for the detail of polymer architecture and chemistry: what sizes, shapes and connectivity of polymeric blocks, and what chemical interactions, are required to favour quasicrystals over periodic patterns? We use a well-established theory - the polymeric "Random Phase Approximation" (RPA) - as a means to predict the onset of instability towards phase separation, and (crucially) the effect that the polymer structure on this instability. We search for particular structures which favour the "two lengthscale" instability. This talk will proceed in two halves... in the first half, one of us (DR) will discuss the background RPA theory and how to quickly make predictions for a given polymer structure. In the second half another of us (MJ) will show specific design results for two candidate structures: a two component A(BA)_n polymer and three component ABC star.

Entropic Colloidal (Quasi)Crystallization Pathways

Michael Engel (Friedrich-Alexander University Erlangen-Nürnberg)

The transformation from a fluid to a (quasi)crystal can be complicated and does not necessarily proceed in one single step. Multistep crystallization pathways are known for many chemical and physical systems but are generally not well understood because they are difficult to analyze in experiment. Here, we discuss simulations of simple hard particle systems that order in multiple steps and have as a final state a crystal that is highly complex (many particles per unit cell) or is a quasicrystal (QC). In particular, we report an error-and-repair mechanism for the formation of a dodecagonal QC from hard tetrahedra. We apply pattern recognition and automatically extract tilings for phason strain analysis to follow the evolution of quasiperiodic order. We observe that initially QCs with high phason strain crystallize rapidly before converting to QCs with minimal phason disorder. We also directly demonstrate the thermodynamic stability of the hard tetrahedron QC.

Mean field theory and fluctuation effects in block copolymers

Bart Vorselaars (University of Lincoln)

A mixture of polymers made of two or more covalently bonded blocks can show a rich behaviour of structures. Even for the prototypical diblock copolymer melt a complex phase diagram is observed. Various theoretical techniques have been developed, though the self-consistent mean-field approach has been very successful in predicting such phase diagrams. The mean-field approach does lead to quantitative deviations, in particular near critical points. In this talk I will discuss field theoretical simulations to model compositional fluctuations and the effect on the phase behaviour in block copolymer melts and ternary blends.

Thursday

Morning Talks

Efficient methods for computing the stationary states of phase field crystal models

Kai Jiang (Xiangtan University)

In this talk, we will present efficient methods for computing the stationary states of phase field crystal models. These approaches consist two parts: (1) an efficient and accurate numerical method is proposed to evaluate ordered structures, including quasicrystals and periodic crystals; (ii) a class of novel methods are developed to compute the stationary states by combining numerical PDE methods and the modern optimization approaches. The efficiency and robustness of the proposed methods will be demonstrated by some numerical experiments. Meanwhile, some new physical phenomena will be presented.

Modeling the structural properties and dynamics of polycrystals

Ken Elder (Oakland University)

Material processes, such as solidification, sintering or thermal annealing, often produce complex microstructures. Modeling the formation of such structures is important as the microstructures often strongly influence material properties. For example in polycrystals the yield stress, magnetic coercivity and thermal conductivity can change by orders of magnitude as the average polycrystalline grain size changes. In this talk I would like to discuss modeling two-dimensional polycrystalline structures where out of plane deformations are often important, using phase field crystal or it's related complex amplitude formulation. Connections between this approach and the discrete dislocation dynamics (DDD) approach will be discussed.

Quasicrystals with two length scales: tilings and density functional theory

Alastair Rucklidge (University of Leeds)

Since their discovery, there has been a strong connection between quasicrystalline structures and aperiodic tilings. The inflation rule for constructing the aperiodic Penrose tiling guarantees ten-fold rotation symmetry in the power spectrum and tilings based on squares and equilateral triangles can have twelve-fold symmetry, matching the ten-fold or twelve-fold symmetry in the diffraction spectrum of metallic and soft-matter quasicrystals. What is less clear is: given an aperiodic tiling, are there systems of particles that will assemble to match the tiling? Here we examine two tilings based on combinations of small and large equilateral triangles along with a rectangular tile whose edges match the two triangles. One of the two is the recently discovered Bronze-mean tiling, the other is a new variant on this. We use the diffraction spectrum of the two tilings to design particles with two length-scale soft interaction potentials, and show that the particles can assemble into structures that are local minima of their free energies and that match the two tilings.

Ordered phases of block copolymers

An-Chang Shi (McMasters University)

The observation of ordered phases in hard-condensed matter systems such as metallic alloys has a long history in materials physics. In recent years, intricate periodic and aperiodic order has emerged in a host of soft matter systems including supramolecular assemblies, surfactants and block copolymers. The occurrence of complex phases in these diverse systems underscores the universality of emergent order in condensed matter. The richness of the phase behavior is exemplified by block copolymers. In particular, recent experimental and theoretical studies have revealed that non-classical ordered phases, such as the Frank-Kasper phases and quasicrystals, could emerge from block copolymers as equilibrium or metastable morphologies. We have examined the occurrence of complex spherical packing phases in block copolymer systems using the self-consistent field theory. Our study reveals that one key mechanism of forming complex spherical phases is the conformational asymmetry of the blocks. Furthermore, we have predicted that the segregation of different polymeric species in block copolymer blends provides another mechanism to stabilize spherical packing phases with very different sized-spherical domains. In my presentation, I will summarize recent theoretical and experimental progresses on this fascinating topic and discuss possible future research directions.

Afternoon Talks

Going from microscale to mesoscale in soft matter: consequences of the approximations

Priya Subramanian (University of Oxford)

Density functional theory (DFT) is a microscopic theory that allows us to describe the evolution of the density profile in a soft matter system using interparticle interactions and the underlying thermodynamic conditions as input. Starting from DFT, a sequence of approximations leads to a phase field crystal (PFC) model of a soft matter system which retains information at the atomic length scales but one that operates over diffusive timescales. In this talk, we trace the sequence of approximations and their consequences for predicting the phase behaviour by comparing predictions from both DFT and PFC for a simple model of a two-dimensional fluid undergoing crystallisation.

A spatially non-uniform phase with no long-range order: Analytical solutions in the Phase-Field Crystal model

Gyula Toth (Loughborough University)

The theoretical understanding of the fundamental nature of structural glasses and the glass transition has been a long standing problem in condensed matter physics. Due to the lack of analytical solutions, however, not even the (mathematical) existence of disordered solid phases in various theories has been proved yet, and the access to these structures is limited to approximations. In this talk, we prove the existence and calculate the physical properties of a spatially non-uniform phase with no long-range order in the Phase-Field Crystal (PFC) model. The developed mathematical framework for finding disordered solutions of partial differential equations relies on the "concentration of measure phenomenon". We show that the PFC free energy density functional has infinitely many spatially non-uniform extrema with no long-range order, and these extrema are the realisations of a coloured spatial Gaussian random field with correlation function $C(r)\propto \sin(r)/r$. In the vicinity of the critical point the free energy density of the configurations is unique, and the phase is metastable in the thermodynamic limit. Since the developed mathematical method eliminates all uncertainties used in traditional methods of describing glassy states, and the general mathematical idea is not limited to continuum models, our results have important implications on all models of the glass transition.

Mean field approaches to alternative mechanisms for the stabilization of quasicrystals

Michael Schmeideberg (Friedrich-Alexander-Universität Erlangen-Nürnberg)

In many successful mean field approaches quasicrystals are obtained if two or more length scales are preferred by the interaction potential (see, e.g., [1-3]). Usually the length scales are chosen in Fourier space such that the structure with the desired rotational symmetry is supported. Here we study two different approaches for the stabilization of quasicrystals: First, we consider quasicrystals and other complex patterns that self-assemble in systems consisting of patchy colloids, i.e., particles that interact with preferred binding angles [4,5]. To construct a phase field crystal model for patchy colloids, we append an approach with a density and an orientation field that originally was developed for liquid crystals [6]. Second, we want to study a hard-core soft-shoulder potential (cf. [7]) with a Density Functional Theory. The hard-core is implemented by using a variant of the Fundamental Meassure Theory [8] that probably is the best mean field approach to hard particles. Finally, we shortly comment on how neural networks can be used to detect defects in quasicrystalline patterns.

The work presented in this contribution is done together with Ali Döner, Anja Gemeinhardt, Markus Hoffmann, Miriam Martinsons, and Robert Weigel.

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Growth Modes of Quasicrystals

Cristian Vasile Achim (Aalto University)

The growth of quasicrystals, i.e., aperiodic structures with long-range order, seeded from the melt is investigated using a dynamical phase field crystal model. Depending on the thermodynamic conditions, two different growth modes are detected, namely defect-free growth of the stable quasicrystal and a mode dominated by phasonic flips which are incorporated as local defects into the grown structure such that random tiling-like ordering emerges.

Friday

Structural crossover in fluids exhibiting two length scales: Repercussions for quasicrystal formation

Bob Evans (University of Bristol)

Liquid state theory and Monte Carlo simulations predict that for binary mixtures in which the species are sufficiently different in size the presence of two different length scales should lead to a novel structural crossover line in the mixture phase diagram : the three pair correlation functions gij (r) decay in an exponentially damped oscillatory fashion with a common (short) wavelength on one side and with a long wavelength on the other. Particle-resolved experiments in Bristol on binary hard-sphere-like colloidal suspensions confirm the existence of the crossover line and its location in the phase diagram [1]. For certain one-component models of ‘soft particles’ interacting via pair potentials that exhibit two (suitably chosen) competitive length scales, we also expect structural crossover in the asymptotic decay, r →∞, of g(r). Using classical DFT we show that for the two-dimensional model introduced by K. Barkan et. al. [Phys. Rev. Lett. 113, 098304 (2014)] crossover occurs as the density is increased in the liquid phase; the ratio of the two wavelengths is ≈ 1.932. Following the locus in the phase diagram of the crossover line leads directly to the region where quasicrystals are found. Identifying and following such a crossover line towards higher densities where the solid phase(s) occur might be a strategy for finding quasicrystals in a variety of systems. We also show how the pole analysis of the asymptotic decay of equilibrium fluid correlations is intimately connected with the nonequilibrium growth or decay of small-amplitude density fluctuations in a bulk fluid.

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[2] M. C. Walters, P. Subramanian, A. J. Archer, and R. Evans, Phys. Rev. E, 98, 012606 (2018).

A general strategy for finding soft matter quasicrystals

Alberto Scacchi (Loughborough University)

Quasicrystals (QCs) were discovered in 1982 [1] and attracted the Nobel Prize for Chemistry in 2011. A regular crystal is formed by atoms or molecules which are ordered in a structure having rotational and translational symmetries. These structures have a discrete X-ray diffraction pattern, or in other words, they have a periodicity in real space. In contrast to regular crystals, QCs have rotation symmetries on average, but no translation symmetries. The vast majority of QCs discovered so far are metallic alloys with at least two components. Recently, other types of QCs have been discovered in nano-particle [2] and soft matter [3] systems. There are still fundamental open questions relating to the formation and the stability of monodispersed QCs, although there has been important progress from studies of model systems in two [4, 5, 6] and three [7, 8] dimensions.

This progress has come by using classical Density Functional Theory (DFT) applied to model systems. It has been found that certain features in the static structure factor (or equivalently in the dispersion relation) are crucial. In binary mixtures the system is far more complex, but we discuss what features of this general picture still hold.

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[3] T. Dotera, Quasicrystals in Soft Matter, Isr. J. Chem. 51, 1197 (2011).

[4] A. J. Archer, A. M. Rucklidge and E. Knobloch, Soft-core particles freezing to form a quasicrystal and a crystal-liquid phase, Phys. Rev. E 92, 012324 (2015).

[5] K. Barkan, H. Diamant and R. Lifshitz, Stability of quasicrystals composed of soft isotropic particles, Phys. Rev. B 83, 172201 (2011).

[6] T. Dotera, T. Oshiro and P. Ziherl, Mosaic two-lengthscale quasicrystals, Nature 506, 208–211 (2014).

[7] P. Subramanian, A. J. Archer, E. Knobloch and A. M. Rucklidge, Three-Dimensional Icosahedral Phase Field Quasicrystal, Phys. Rev. Lett. 117, 075501 (2016).

[8] P. Subramanian, A. J. Archer, E. Knobloch and A. M. Rucklidge, Spatially localized quasicrystalline structures, New Journal of Physics, 20 (12), 122002 (2018).

Block copolymers in confinements and under external fields

Andrei Zvelindovsky (University of Lincoln)

Block copolymers (BCPs) are long chain molecules consisting of several chemically different blocks. Due to the chemical nature of the bond between blocks they do not macrophase, but form various structures on the nano-scale. Due to the intrinsic complexity of the systems, which have a very large physical parameter space, their experimental study is a much elaborated task. With the advances of computers, computational methods become a crucial component in the BCP research and the advances materials design. In our contribution we discuss computer simulation results for BCP systems and their relation to experimental data. Computer simulation results presented are based on two models: a Ginzburg-Landau type description and on self-consistent field theory (SCFT) for polymers. The Ginzburg-Landau model used is a basis for Cell Dynamics simulation (CDS). In this talk we focus on two topics – confinements and external fields (electric, shear). In real practice BCP are often found in thin films (of the thickness of several structural domains), and most recently - in nano-pores. Confined structures are found to be very different from the bulk ones. Confinement has a profound influence on the BCP structure. In thin films non-bulk structures are formed in the layers next to the confining surfaces. In this way some such structures as perforated lamellae can be formed. In cylindrical pores helical and toroid structures are formed in various combinations. In spherical confinement the observed structures are reminiscent of knitting ball, onion, perforated spherical layer, virus-like morphology and others. Manipulation by the external electric or flow fields is a way of the nanostructure alignment. Kinetics of this process can be different depending on the field strength. In the case of the electric field lamellae orientation is found to depend on the strength of the electric field and the temperature. Using CDS can serve as a first part of the simulation tandem together with SCFT in a computer-aided design of novel nanostructured materials.