GISC - Research topics in statistical mechanics, nonlinear dynamics and related subjects
Complex fluids, granular matter
Statistical Mechanical Modeling of Complex fluids: Certain fluids, called liquid crystals, are composed of anisotropic particles and because of that they exhibit properties common to liquids and solids. They are the main constituent blocks of several molecular and colloidal systems and have very important applications in technology and engineering, such as the design of many liquid-crystal devices like liquid-crystal displays or thermo-optics devices. We use mathematical models from Statistical Physics to study the structural and thermodynamical properties of these complex fluids and how these properties depend on the geometrical shape of the particles. Special attention is paid on the study of the effect of particle shape on the symmetries of the liquid-crystal phases dictated by a preferent orientation of the particle axes along different equivalent directions.
Experimental and Theoretical Study of vertically shaken monolayers of granular particles: When monolayers of nonoverlapping anisotropic particles are vertically shaken the orientations of particle axes produce patterns very similar to those of liquid-crystals. When the symmetries of these liquid-crystal textures are in conflict with those of the geometrical confinement, the orientational field of particles exhibits the presence of topological defects that vary in number and symmetry. We study these monolayers by performing experiments and MC simulations by selecting different particle shapes such as cylinders or prisms of different cross-sections and different cavity geometries such as circular, square or hexagonal. The theoretical study of granular systems is crucial to characterize their packing properties as a function of the particle shape and of the confining geometry producing algorithms to optimize the packing of grains. This in turn has important applications in the cereals processing technology.
Emptying of Horizontal Capillaries in Nanotechnology and Space Science
Solving the Young-Laplace equation with transversal gravity
Abstract: Determining whether a liquid remains in a horizontal capillary or flows out of it has recently been shown to be a very rich problem with important implications in nanotechnology and space science. As it happens, answering this question is equivalent to determining whether the Young-Laplace equation (a non-linear PDE) has a solution or not [https://doi.org/10.1073/pnas.1606217113].
In this project, we aim to study under which circumstances a horizontal capillary is capable of holding the liquid in it by studying the Young-Laplace equation in 1D and 2D with different boundary conditions that represent capillaries with different geometrical cross-sections. Although we anticipate that most of the work will entail the numerical solving of the equation, some analytical work could also be expected.
Candidates: Highly qualified students which enjoy numerical computational work (for example: matlab, python, fortran) and a liking for physical problems. During the development of the project, candidates are expected to understand the link between the physical problem and the Young-Laplace equation, familiarize themselves with different methods to solve PDEs numerically, and being able to organize and analyze considerable amounts of data.
FIG1: Schematic representation of the liquid front (known as 'meniscus') of an emptying cylindrical capillary showing that a tongue of length L develops when the liquid is close to flow out.
FIG2: Emptying diagram for a horizontal cylindrical capillary showing whether the liquid remains in the capillary (FILLED) or flows out of it (EMPTY) in terms of the radius of the capillary R and the contact angle, which is a measure of the affinity of the liquid for the capillary material (as indicated in parallel in the brown panel). Diagrams 1-4 represent the shape of the liquid contained in the capillary as the emptying line is approached. Points A-E show the cross-section of the (infinitely long) liquid tongue at different points of the emptying line [https://doi.org/10.1073/pnas.1606217113].
FIG3: Emptying diagram for a horizontal triangular capillary in three different orientations. Here, R is the radius of the circle circumscribing it. The dotted line represents the emptying line of a cylindrical capillary of radius R (as in FIG2) to show the striking differences resulting from the different cross-section shape (i.e. boundary conditions) [https://doi.org/10.1073/pnas.1606217113].
Figure 1: (a) Macroscopic undulations (approximate crest-to-crest distance around 10 cm) on a sand dune, due to the wind action. Copyright QT Luong/terragalleria.com (b) Nanometric undulations (approximate crest to-crest distance around 100 nm) produced on a glass surface by irradiation with ionized air at 4 keV and oblique incidence. SEM image taken from the work by Navez and coworkers [1].
Figure 2: Growth dynamics of disordered and fractal surface nanocauliflowers produced by CVD. Panels (A) to (C) are experimental top views for increasing times. The size of the white bar in (A) is 200 nm. Panels (D)–(F) show numerical results from a differential equation that models this process, for the same times as seen in the experimental images.
Nanopatterns at Surfaces and Interfaces
Who has not noticed the ripples produced by the wind onto the sand surface at the beach, or those found underwater close to the shore? Both are examples [see Fig. 1(a)] of “patterns”, structures with a certain degree of order that form in an initially inhomogeneous medium (such as sand) by the action of an external driving force, such as wind or water waves. However, the existence and the physics of very similar patterns at scales ten million times smaller is not so well known. One such example is provided by the undulations produced on the surface of a solid target by irradiation with an energetic ion beam. Fig. 1(b) shows one of the first experimental observations of this phenomenon by Navez and coworkers in 1962. Note that the wavelength of the pattern (distance between two adjoining ripple crests) is of the order of 0.1 microns! See more details and references in [1,2].
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Currently our knowledge is not yet complete on the processes by which ordered nanometric patterns [or else, disordered fractals like the surface of a thin film grown by chemical vapor deposition (CVD), shown in Fig. 2, see Ref. [3]] appear on the surfaces of solids through the previous or other experimental techniques, such as those employed e.g. in thin film production for optoelectronic or biomedical devices. In general, these are nonequilibrium systems for whose theoretical description one needs to use (and generalize) the tools of Statistical Mechanics, Nonlinear Science, and Complex Systems.
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Academic work (Bachelor or Master’s Thesis) is offered on the theoretical/computational modeling of these systems, under the supervision of faculty member Rodolfo Cuerno. Work will focus on one or several of the following areas:
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• Pattern formation at the Nanoscale. Interplay between fluctuations and order.
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• Nonequilibrium phenomena at surfaces and interfaces: statistical mechanics and nonlinear science.
• Applications to specific contexts: thin film production, Micro and Nanofluidics.
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[1] Rodolfo Cuerno, Javier Muñoz-García, Mario Castro, Raúl Gago, and Luis Vázquez, Modelos de la dinámica de las ondulaciones en la “nanoarena”. Revista Española de Física 21 (1)(2007) 65—69.
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[2] Rodolfo Cuerno, Javier Muñoz-García, Mario Castro, Raúl Gago, and Luis Vázquez, Universal non-equilibrium phenomena at submicrometric surfaces and interfaces. European Physical Journal Special Topics 146, 427-441 (2007).
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[3] Mario Castro, Rodolfo Cuerno, Matteo Nicoli, Luis Vázquez, and Josephus G Buijnsters, Universality of cauliflower-like fronts: from nanoscale thin films to macroscopic plants. New Journal of Physics 14, 103039 (2012).
Synchronization in lattices of oscillators and nonequilibrium growth processes
Synchronization is the process whereby oscillating objects end up evolving in unison due to interactions existing among them. It has been studied in lasers, qubits, the human brain, or the examples you can see on the right: flocking birds, flashing fireflies, and oscillating metronomes. While each oscillator by itself would evolve differently, as determined by its own inner structure or the effect of noise, the interactions between oscillators, if sufficiently strong to overcome such forms of disorder, may lead to synchronous motion.
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It turns out that the evolution of discrete systems of oscillators, when placed on a continuum, can be approximated by continuous equations whose dominant terms correspond to very well-known equations of nonequilibrium statistical mechanics that describe many growth processes, including the Kardar-Parisi-Zhang (KPZ) equation. Moreover the dynamics that arises is scale invariant, and, when observed at large enough space-time scales, displays universal features nonequilibrium statistical mechanics, including those of the KPZ universality class. This relation between synchronization and nonequilibrium critical behavior has been recently explored by Ricardo Gutiérrez and Rodolfo Cuerno in a series of publications listed here:
https://doi.org/10.1103/PhysRevResearch.5.0230477
https://doi.org/10.1103/PhysRevResearch.6.033324
https://doi.org/10.1103/PhysRevE.110.L052201
https://doi.org/10.1016/j.physd.2025.134552
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If you think you would like to work on a novel research topic lying at the interface of synchronization and nonequilibrium growth processes, or if you have some questions about it, please write to faculty member Ricardo Gutiérrez (see the email address below). A combination of analytical work (including the critical reassessment of exact results) and numerical analysis (simulations of systems of oscillators, characterization of critical behavior via critical exponents and scaling functions) will be required.
Figure 1: Synchronous dynamics is ubiquitous in science and technology. While its scientific study was initiated in the work of Christiaan Huygens in the 17th century, it is in the last few decades that it has received a great deal of attention in the statistical mechanics and complex systems community. Focking birds, metronomes that interact through a non-rigid common support (however you start them, they rapidly oscillate in perfect synchrony), and flashing fireflies are just some examples.
Figure 2: Phase interfaces of oscillators in a 1D dimensional lattice display dynamics found in various models of growing interfaces. It turns out that the analogies are not just phenomenological, but run much deeper: synchronizing lattices of oscillators are in the universality classes of some nonequilibrium continuum equations displaying generic scale invariance.
Fig 1. Illustrative example of a collection of Rössler (chaotic) osscilators, coupled through a network. In systems like this one can often find the emergence of synchrony.
Cluster synchronization of chaotic systems
Synchronization is a pervasive phenomena throughout many physical systems (circadian rhythms, fireflies, brain activity...), of great interest both in applications as well as from theoretical and mathematical points of view. The emergence of a global order (a collective dynamical state) from local rules (differential equations at each node with only nearest neighbor interactions) is still intriguing and inspiring scientists centuries after the initial inquiries by Christiaan Huygens.
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One of the most surprising contexts where synchronization can arise is in the coupling of chaotic systems. To put it simply, one can connect a collection of chaotic dynamical systems in an arbitrary network, and if the coupling is strong enough at some point every system will be following the same trajectory, despite the fact that chaos amplifies perturbations. The framework of the Master Stability Function (https://journals.aps.org/prl/abstract/10.1103/PhysRevLett.80.2109) was devised to quantitatively understand this, and has brought forth a revolution to the field of synchronization.
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In the decade, part of the focus has shifted to the case of cluster synchronization: understanding the conditions through which some sets of nodes in the network become synchronous while the rest of it does not. While there has already been much high-impact progress in the field (https://www.nature.com/articles/ncomms5079, https://www.nature.com/articles/s41467-024-48203-6), there is still much to be discovered. Working on this topic would entail, for the interested candidate, both analytical (reproducing calculations/proofs from the literature, and possibly produce further ones) as well as numerical work (simulating systems of coupled differential equations and its data analysis).