Past Industrial and Applied Mathematics Seminar

The problem of a bubble moving steadily in a Hele-Shaw cell goes back to Taylor and Saffman in 1959.  It is analogous to the well-known selection problem for Saffman-Taylor fingers in a Hele-Shaw channel.   We apply techniques in exponential asymptotics to study the bubble problem in the limit of vanishing surface tension, confirming previous numerical results, including a previously predicted surface tension scaling law.  Our analysis sheds light on the multiple tips in the shape of the bubbles along solution branches, which appear to be caused by switching on and off exponentially small wavelike contributions across Stokes lines in a conformally mapped plane. 

In biological cells, genomic DNA is complexed with proteins, forming so-called chromatin structure, and packed into the nucleus. Not only the nucleotide (A, T, G, C) sequence of DNA but also the 3D structure affects the genomic function. For example, certain regions of DNA are tightly packed with proteins (heterochromatin), which inhibits expression of genes coded there. The structure sometimes changes drastically depending on the state (e.g. cell cycle or developmental stage) of the cell. Hence, the structural dynamics of chromatin is now attracting attention in cell biology and medicine. However, it is difficult to experimentally observe the motion of the entire structure in detail. To combine and interpret data from different modes of observation (such as live imaging and electron micrograph) and predict the behavior, structural models of chromatin are needed. Although we can use molecular dynamics simulation at a microscopic level (~ kilo base-pairs) and for a short time (~ microseconds), we cannot reproduce long-term behavior of the entire nucleus. Mesoscopic models are wanted for that purpose, however hard to develop (there are fundamental difficulties).

In this seminar, I will introduce our recent theoretical/computational studies of chromatin structure, either microscopic (molecular dynamics of DNA or single nucleosomes) or abstract (polymer models and reaction-diffusion processes), toward development of such a mesoscopic model including local "states" of DNA and binding proteins.

References:

T. Kameda, A. Awazu, Y. Togashi, "Histone Tail Dynamics in Partially Disassembled Nucleosomes During Chromatin Remodeling", Front. Mol. Biosci., in press (2019).

Y. Togashi, "Modeling of Nanomachine/Micromachine Crowds: Interplay between the Internal State and Surroundings", J. Phys. Chem. B 123, 1481-1490 (2019).

E. Rolls, Y. Togashi, R. Erban, "Varying the Resolution of the Rouse Model on Temporal and Spatial Scales: Application to Multiscale Modelling of DNA Dynamics", Multiscale Model. Simul. 15, 1672-1693 (2017).

S. Shinkai, T. Nozaki, K. Maeshima, Y. Togashi, "Dynamic Nucleosome Movement Provides Structural Information of Topological Chromatin Domains in Living Human Cells", PLoS Comput. Biol. 12, e1005136 (2016).

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The dynamics of many physical systems often evolve to asymptotic states that exhibit periodic spatial and temporal variations in their properties such as density, temperature, etc. Such regular patterns look the same when moved by a basic unit and/or rotated by certain special angles. They possess both translational and rotational symmetries giving rise to discrete spatial Fourier transforms. In contrast, an aperiodic crystal displays long range spatial order but no translational symmetry. 

Recently, quasicrystals which are related to aperiodic crystals have been observed to form in diverse physical systems such as metallic alloys (atomic scale) and dendritic-, star-, and block co-polymers (molecular scale). Such quasicrystals lack the lattice symmetries of regular crystals, yet have discrete Fourier spectra. We look to understand the minimal mechanism which promotes the formation of such quasicrystalline structures using a phase field crystal model. Direct numerical simulations combined with weakly nonlinear analysis highlight the parameter values where the quasicrystals are the global minimum energy state and help determine the phase diagram. 

By locating parameter values where multiple patterned states possess the same free energy (Maxwell points), we obtain states where a patch of one type of pattern (for example, a quasicrystal) is present in the background of another (for example, the homogeneous liquid state) in the form of spatially localized dodecagonal (in 2D) and icosahedral (in 3D) quasicrystals. In two dimensions, we compute several families of spatially localized quasicrystals with dodecagonal structure and investigate their properties as a function of the system parameters. The presence of such meta-stable localized quasicrystals is significant as they may affect the dynamics of the crystallisation in soft matter.

The talk will begin with an introduction to the science of what determines the behaviour of a liquid on a on a surface and giving an overview of some of the different theories that can be used to describe the shape and structure of the liquid in the drop. These include microscopic density functional theory (DFT), which describes the liquid structure on the scale of the individual liquid molecules, and mesoscopic thin film equation (PDE) and kinetic Monte-Carlo models. A DFT based method for calculating the binding potential ?(h) for a film of liquid on a solid surface, where h is the thickness of the liquid film, will be presented. The form of ?(h) determines whether or not the liquid wets the surface. Calculating drop profiles using both DFT and also from inputting ?(h) into the mesoscopic theory and comparing quantities such as the contact angle and the shape of the drops, we find good agreement between the two methods, validating the coarse-graining. The talk will conclude with a discussion of some recent work on modelling evaporating drops with applications to inkjet printing.

One of the key unsolved challenges at the interface of physical and life sciences is to formulate comprehensive computational modeling of cells of higher organisms that is based on microscopic molecular principles of chemistry and physics. Towards addressing this problem, we have developed a unique reactive mechanochemical force-field and software, called MEDYAN (Mechanochemical Dynamics of Active Networks: http://medyan.org).  MEDYAN integrates dynamics of multiple mutually interacting phases: 1) a spatially resolved solution phase is treated using a reaction-diffusion master equation; 2) a polymeric gel phase is both chemically reactive and also undergoes complex mechanical deformations; 3) flexible membrane boundaries interact mechanically and chemically with both solution and gel phases.  In this talk, I will first outline our recent progress in simulating multi-micron scale cytosolic/cytoskeletal dynamics at 1000 seconds timescale, and also highlight the outstanding challenges in bringing about the capability for routine molecular modeling of eukaryotic cells. I will also report on MEDYAN’s applications, in particular, on developing a theory of contractility of actomyosin networks and also characterizing dissipation in cytoskeletal dynamics. With regard to the latter, we devised a new algorithm for quantifying dissipation in cytoskeletal dynamics, finding that simulation trajectories of entropy production provide deep insights into structural evolution and self-organization of actin networks, uncovering earthquake-like processes of gradual stress accumulation followed by sudden rupture and subsequent network remodeling.
 

Thin film flows of nematic liquid crystal will be considered, using the Leslie-Ericksen formulation for nematics. Our model can account for variations in substrate anchoring, which may exert a strong influence on patterns that arise in the flow. A number of simulations will be presented using an "in house" code, developed to run on a GPU. Current modeling directions involving flow over interlaced electrodes, so-called "dielectrowetting", will be discussed.

During this seminar, we will present a new mathematical model describing the transport of nitric oxide (NO) in a realistic geometrical representation of the lungs. Nitric oxide (NO) is naturally produced in the bronchial region of the lungs. It is a physiological molecule that has antimicrobial properties and allows the relaxation of muscles. It is well known that the measurement of the molar fraction of NO in the exhaled air, the so-called FeNO, allows a monitoring of asthmatic patients, since the production of this molecule in the lungs is increased in case of inflammation. However, recent clinical studies have shown that the amount of NO in the exhaled air can also be affected by « non-inflammatory » processes, such as the action of a bronchodilator or a respiratory physiotherapy session for a patient with cystic fibrosis. Using our new model, we will highlight the complex interplay between different transport phenomena in the lungs. More specifically, we will show why changes taking place in the deepest part of the lungs are expected to impact the FeNO. This gives a new light on the clinical studies mentioned below, allowing to confer a new role to the NO for the management of various pulmonary pathologies.

Tensors are higher dimensional analogues of matrices; they are used to record data with multiple changing variables. Interpreting tensor data requires finding low rank structure, and the structure depends on the application or context. Often tensors of interest define semi-algebraic sets, given by polynomial equations and inequalities. I'll give a characterization of the set of tensors of real rank two, and answer questions about statistical models using probability tensors and semi-algebraic statistics. I will also describe work on learning a path from its three-dimensional signature tensor. This talk is based on joint work with Guido Montúfar, Max Pfeffer, and Bernd Sturmfels.

When a liquid drop is deposited over a solid surface whose temperature is sufficiently above the boiling point of the liquid, the drop does not experience nucleate boiling but rather levitates over a thin layer of its own vapor. This is known as the Leidenfrost effect. Whilst highly undesirable in certain cooling applications, because of a drastic decrease of the energy transferred between the solid and the evaporating liquid due to poor heat conductivity of the vapor, this effect can be of great interest in many other processes profiting from this absence of contact with the surface that considerably reduces the friction and confers an extreme mobility on the drop. During this presentation, I hope to provide a good vision of some of the knowledge on this subject through some recent studies that we have done. First, I will present a simple fitting-parameter-free theory of the Leidenfrost effect, successfully validated with experiments, covering the full range of stable shapes, i.e., from small quasi-spherical droplets to larger puddles floating on a pocketlike vapor film. Then, I will discuss the end of life of these drops that appear either to explode or to take-off. Finally, I will show that the Leidenfrost effect can also be observed over hot baths of non-volatile liquids. The understanding of the latter situation, compare to the classical Leidenfrost effect on solid substrate, provides new insights on the phenomenon, whether it concerns levitation or its threshold.

Tendons are vital connective tissues that anchor muscle to bone to allow the transfer of forces to the skeleton. They exhibit highly non-linear viscoelastic mechanical behaviour that arises due to their complex, hierarchical microstructure, which consists of fibrous subunits made of the protein collagen. Collagen molecules aggregate to form fibrils with diameters of tens to hundreds of nanometres, which in turn assemble into larger fibres called fascicles with diameters of tens to hundreds of microns. In this talk, I will discuss the relationship between the three-dimensional organisation of the fibrils and fascicles and the macroscale mechanical behaviour of the tendon. In particular, I will show that very simple constitutive behaviour at the microscale can give rise to highly non-linear behaviour at the macroscale when combined with geometrical effects.

Likely instabilities in stochastic hyperelastic solids

L. Angela Mihai

School of Mathematics, Cardiff University, Senghennydd Road, Cardiff, CF24 4AG, UK

E-mail: @email.uk

Nonlinear elasticity has been an active topic of fundamental and applied research for several decades. However, despite numerous developments and considerable attention it has received, there are important issues that remain unresolved, and many aspects still elude us. In particular, the quantification of uncertainties in material parameters and responses resulting from incomplete information remain largely unexplored. Nowadays, it is becoming increasingly apparent that deterministic approaches, which are based on average data values, can greatly underestimate, or overestimate, mechanical properties of many materials. Thus, stochastic representations, accounting for data dispersion, are needed to improve assessment and predictions. In this talk, I will consider stochastic hyperelastic material models described by a strain-energy density where the parameters are characterised by probability distributions. These models, which are constructed through a Bayesian identification procedure, rely on the maximum entropy principle and enable the propagation of uncertainties from input data to output quantities of interest. Similar modelling approaches can be developed for other mechanical systems. To demonstrate the effect of probabilistic model parameters on large strain elastic responses, specific case studies include the classic problem of the Rivlin cube, the radial oscillatory motion of cylindrical and spherical shells, and the cavitation and finite amplitude oscillations of spheres.

Chemical reaction network (CRN) theory focusses on making claims about dynamical behaviours of reaction networks which are, as far as possible, dependent on the network structure but independent of model details such as functions chosen and parameter values. The claims are generally about the existence, nature and stability of limit sets, and the possibility of bifurcations, in models of CRNs with particular structural features. The methodologies developed can often be applied to large classes of models occurring in biology and engineering, including models whose origins are not chemical in nature. Many results have a natural algorithmic formulation. Apart from the potential for application, the results are often pleasing mathematically for their power and generality. 

This talk will concern some recent themes in CRN theory, particularly focussed on how the presence or absence of particular subnetworks ("motifs") influences allowed dynamical behaviours in ODE models of a CRN. A number of recent results take the form: "a CRN containing no subnetworks satisfying condition X cannot display behaviour of type Y"; but also, in the opposite direction, "if a CRN contains a subnetwork satisfying condition X, then some model of this CRN from class C admits behaviour of type Y". The proofs of such results draw on a variety of techniques from analysis, algebra, combinatorics, and convex geometry. I'll describe some of these results, outline their proofs, and sketch some current challenges in this area. 
 

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Abstract

The dynamical fields that emanate from self-similarly expanding ellipsoidal regions undergoing phase change (change in density, i.e., volume collapse, and change in moduli) under pre-stress, constitute the dynamic generalization of the seminal Eshelby inhomogeneity problem (as an equivalent inclusion problem), and they consist of pressure, shear, and M waves emitted by the surface of the expanding ellipsoid and yielding Rayleigh waves in the crack limit. They may constitute the model of Deep Focus Earthquakes (DFEs) occurring under very high pressures and due to phase change. Two fundamental theorems of physics govern the phenomenon, the Cauchy-Kowalewskaya theorem, which based on dimensional analysis and analytic properties alone, dictates that there is zero particle velocity in the interior, and Noether’s theorem that extremizes (minimizes for stability) the energy spent to move the boundary so that it does not become a sink (or source) of energy, and determines the self-similar shape (axes expansion speeds). The expression from Noether’s theorem indicates that the expanding region can be planar, thus breaking the symmetry of the input and the phenomenon manifests itself as a newly discovered one of a “dynamic collapse/ cavitation instability”, where very large strain energy condensed in the very thin region can escape out. In the presence of shear, the flattened very thin ellipsoid (or band) will be oriented in space so that the energy due to phase change under pre-stress is able to escape out at minimum loss condensed in the core of dislocations gliding out on the planes where the maximum configurational force (Peach-Koehler) is applied on them. Phase change occurring planarly produces in a flattened expanding ellipdoid a new defect present in the DFEs. The radiation patterns are obtained in terms of the equivalent to the phase change six eigenstrain components, which also contain effects due to planarity through the Dynamic Eshelby Tensor for the flattened ellipsoid. Some models in the literature of DFEs are evaluated and excluded on the basis of not having the energy to move the boundary of phase discontinuity. Noether’s theorem is valid in anisotropy and nonlinear elasticity, and the phenomenon is independent of scales, valid from the nano to the very large ones, and applicable in general to other dynamic phenomena of stress induced martensitic transformations, shear banding, and amorphization.

Cracks in many soft solids behave very differently to the classical picture of fracture, where cracks are long and thin, with damage localised to a crack tip. In particular, small cracks in soft solids become highly rounded — almost circular — before they start to extend. However, despite being commonplace, this is still not well understood. We use a phase-separation technique in soft, stretched solids to controllably nucleate and grow small, nascent cracks. These give insight into the soft failure process. In particular, our results suggest fracture occurs in two regimes. When a crack is large, it obeys classical linear-elastic fracture mechanics, but when it is small it grows in a new, scale-free way at a constant driving stress.

Transformation theory has long been known to be a mechanism for 
the design of metamaterials. It gives rise to the required properties of the 
material in order to direct waves in the manner desired.  This talk will 
focus on the mathematical theory underpinning the design of acoustic and 
elastodynamic metamaterials based on transformation theory and aspects of 
the experimental confirmation of these designs. In the acoustics context it 
is well-known that the governing equations are transformation invariant and 
therefore a whole range of microstructural options are available for design, 
although designing materials that can harness incoming acoustic energy in 
air is difficult due to the usual sharp impedance contrast between air and 
the metamaterial in question. In the elastodynamic context matters become 
even worse in the sense that the governing equations are not transformation 
invariant and therefore we generally require a whole new class of materials.

In the acoustics context we will describe a new microstructure that consists 
of rigid rods that is (i) closely impedance matched to air and (ii) slows 
down sound in air. This is shown to be useful in a number of configurations 
and in particular it can be employed to half the resonant frequency of the 
standard quarter-wavelength resonator (or alternatively it can half the size 
of the resonator for a specified resonant frequency) [1].

In the elastodynamics context we will show that although the equations are 
not transformation invariant one can employ the theory of waves in 
pre-stressed hyperelastic materials in order to create natural elastodynamic 
metamaterials whose inhomogeneous anisotropic material properties are 
generated naturally by an appropriate pre-stress. In particular it is shown 
that a certain class of hyperelastic materials exhibit this so-called 
“invariance property” permitting the creation of e.g. hyperelastic cloaks 
[2,3] and invariant metamaterials. This has significant consequences for the 
design of e.g. phononic media: it is a well-known and frequently exploited 
fact that pre-stress and large deformation of hyperelastic materials 
modifies the linear elastic wave speed in the deformed medium. In the 
context of periodic materials this renders materials whose dynamic 
properties are “tunable” under pre-stress and in particular this permits 
tunable band gaps in periodic media [4]. However the invariant hyperelastic 
materials described above can be employed in order to design a class of 
phononic media whose band-gaps are invariant to deformation [5]. We also 
describe the concept of an elastodynamic ground cloak created via pre-stress 
[6].

[1] Rowley, W.D., Parnell, W.J., Abrahams, I.D., Voisey, S.R. and Etaix, N. 
(2018) “Deepening subwavelength acoustic resonance via metamaterials with 
universal broadband elliptical microstructure”. Applied Physics Letters 112, 
251902.
[2] Parnell, W.J. (2012) “Nonlinear pre-stress for cloaking from antiplane 
elastic waves”. Proc Roy Soc A 468 (2138) 563-580.
[3] Norris, A.N. and Parnell, W.J. (2012) “Hyperelastic cloaking theory: 
transformation elasticity with pre-stressed solids”. Proc Roy Soc A 468 
(2146) 2881-2903
[4] Bertoldi, K. and Boyce, M.C. (2008)  “Mechanically triggered 
transformations of phononic band gaps in periodic elastomeric structures”. 
Phys Rev B 77, 052105.
[5] Zhang, P. and Parnell, W.J. (2017) “Soft phononic crystals with 
deformation-independent band gaps” Proc Roy Soc A 473, 20160865.
[6] Zhang, P. and Parnell, W.J. (2018) “Hyperelastic antiplane ground 
cloaking” J Acoust Soc America 143 (5)

This talk will concern the problem of inference when the posterior measure involves continuous models which require approximation before inference can be performed. Typically one cannot sample from the posterior distribution directly, but can at best only evaluate it, up to a normalizing constant. Therefore one must resort to computationally-intensive inference algorithms in order to construct estimators. These algorithms are typically of Monte Carlo type, and include for example Markov chain Monte Carlo, importance samplers, and sequential Monte Carlo samplers. The multilevel Monte Carlo method provides a way of optimally balancing discretization and sampling error on a hierarchy of approximation levels, such that cost is optimized. Recently this method has been applied to computationally intensive inference. This non-trivial task can be achieved in a variety of ways. This talk will review 3 primary strategies which have been successfully employed to achieve optimal (or canonical) convergence rates – in other words faster convergence than i.i.d. sampling at the finest discretization level. Some of the specific resulting algorithms, and applications, will also be presented.

A two-layer shear flow in the presence of surfactants is considered. The flow configuration comprises two superposed layers of viscous and immiscible fluids confined in a long horizontal channel, and characterised by different densities, viscosities and thicknesses. The surfactants can be insoluble, i.e. located at the interface between the two fluids only, or soluble in the lower fluid in the form of monomers (single molecules) or micelles (multi-molecule aggregates). A mathematical model is formulated, consisting of governing equations for the hydrodynamics and appropriate transport equations for the surfactant concentration at the interface, the concentration of monomers in the bulk fluid and the micelle concentration. A primary objective of this study is to investigate the effect of surfactants on the stability of the interface, and in particular surfactants in high concentrations and above the critical micelle concentration (CMC). Interfacial instabilities are induced due to the acting forces of gravity and inertia, as well as the action of Marangoni forces generated as a result of the dependence of surface tension on the interfacial surfactant concentration. The underlying physical mechanism responsible for the formation of interfacial waves will be discussed, together with the complex flow dynamics (typical nonlinear phenomena associated with interfacial flows include travelling waves, solitary pulses, quasi-periodic and chaotic dynamics).

Injection of a gas into a gas/liquid foam is known to give rise to instability phenomena on a variety of time and length scales. Macroscopically, one observes a propagating gas-filled structure that can display properties of liquid finger propagation as well as of fracture in solids. Using a discrete model, which incorporates the underlying film instability as well as viscous resistance from the moving liquid structures, we describe brittle cleavage phenomena in line with experimental observations. We find that  the dimensions of the foam sample significantly affect the speed of the  cracks as well as the pressure necessary to sustain them: cracks in wider samples travel faster at a given driving stress, but are able to avoid arrest and maintain propagation at a lower pressure (the  velocity gap becomes smaller). The system thus becomes a study case for stress concentration and the transition between discrete and continuum systems in dynamical fracture; taking into account the finite dimensions of the system improves agreement with experiment.

Many fluid flows in natural systems are highly complex, with an often beguilingly intricate and confusing detailed structure. Yet, as with many systems, a good deal of insight can be gained by testing the consequences of simple mathematical models that capture the essential physics.  We’ll tour two such problems.  In the summer melt seasons in Greenland, lakes form on the surface of the ice which have been observed to rapidly drain.  The propagation of the meltwater in the subsurface couples the elastic deformation of the ice and, crucially, the flow of water within the deformable subglacial till.  In this case the poroelastic deformation of the till plays a subtle, but crucial, role in routing the surface meltwater which spreads indefinitely, and has implications for how we think about large-scale motion in groundwater aquifers or geological carbon storage.  In contrast, when magma erupts onto the Earth’s surface it flows before rapidly cooling and crystallising.  Using analogies from the kitchen we construct, and experimentally test, a simple model of what sets the ultimate extent of magmatic intrusions on Earth and, as it turns out, on Venus.  The results are delicious!  In both these cases, we see how a simplified mathematical analysis provides insight into large scale phenomena.

Blisters form when a thin surface layer of a solid body separates/delaminates from the underlying bulk material over a finite, bounded region. It is ubiquitous in a range of industrial applications, e.g. blister test is applied to assess the strength of adhesion between thin elastic films and their solid substrates, and during natural processes, such as formation and spreading of laccoliths or retinal detachment.

We study a special case of blistering, in which a thin elastic membrane is adhered to the substrate by a thin layer of viscous fluid. In this scenario, the expansion of the newly formed blister by fluid injection occurs via a displacement flow, which peels apart the adhered surfaces through a two-way interaction between flow and deformation. If the injected fluid is less viscous than the fluid already occupying the gap, patterns of short and stubby fingers form on the propagating fluid interface in a radial geometry. This process is regulated by membrane compliance, which if increased delays the onset of fingering to higher flow rates and reduces finger amplitude. We find that the morphological features of the fingers are selected in a simple way by the local geometry of the compliant cell. In contrast, the local geometry itself is determined from a complex fluid–solid interaction, particularly in the case of rectangular blisters. Furthermore, changes to the geometry of the channel cross-section in the latter case lead to a rich variety of possible interfacial patterns. Our experiments provide a link between studies of airway reopening, Saffman-Taylor fingering and printer’s instability.   

Accurate simulation of electromagnetic scattering by ice crystals in clouds is an important problem in atmospheric physics, with single scattering results feeding directly into the radiative transfer models used to predict long-term climate behaviour. The problem is challenging for numerical simulation methods because the ice crystals in a given cloud can be extremely varied in size and shape, sometimes exhibiting fractal-like geometrical characteristics and sometimes being many hundreds or thousands of wavelengths in diameter. In this talk I will focus on the latter "high-frequency" issue, describing a hybrid numerical-asymptotic boundary element method for the simplified problem of acoustic scattering by penetrable convex polygons, where high frequency asymptotic information is used to build a numerical approximation space capable of achieving fixed accuracy of approximation with frequency-independent computational cost. 

Motion at the microscale is a fascinating field visualizing non-equilibrium behaviour of matter. Evolution has optimized the ability of microscale swimming on different length scales from tedpoles, to sperm and bacteria. A constant metabolic energy input is required to achieve active propulsion which means these systems are obeying the laws imposed by a low Reynolds number. Several strategies - including topography1 , chemotaxis or rheotaxis2 - have been used to reliably determine the path of active particles. Curiously, many of these strategies can be recognized as analogues of approaches employed by nature and are found in biological microswimmers.

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Bacteria attached to the metal caps of Janus particles3

However, natural microswimmers are not limited to being exemplary systems on the way to artificial micromotion. They certainly enable us to observe how nature overcame problems such as a lack of inertia, but natural microswimmers also offer the possibility to couple them to artificial microobjects to create biohybrid systems. Our group currently explores different coupling strategies and to create so called ‘BacteriaBots’.4

1 J. Simmchen, J. Katuri, W. E. Uspal, M. N. Popescu, M. Tasinkevych, and S. Sánchez, Nat. Commun., 2016, 7, 10598.

2 J. Katuri, W. E. Uspal, J. Simmchen, A. Miguel-López and S. Sánchez, Sci. Adv., , DOI:10.1126/sciadv.aao1755.

3 M. M. Stanton, J. Simmchen, X. Ma, A. Miguel-Lopez, S. Sánchez, Adv. Mater. Interfaces, DOI:10.1002/admi.201500505.

4 J. Bastos-Arrieta, A. Revilla-Guarinos, W. E. Uspal and J. Simmchen, Front. Robot. AI, 2018, 5, 97.