I am a Senior Postdoctoral Research Associate at the University of Bristol (UK), working on disordered systems and soft matter. I study active systems, deeply supercooled liquids, vitrification, gelation and crystallisation. I am also interested in cross-disciplinary problems of data reduction and visualization.
The Vicsek model is one of the simplest models for active matter. It displays interesting features, such as swarming.
Large scale simulations are often needed in order to provide firm statements on the statistical properties of this kind of models. However, for a pedagogical and illustrative purpose it may be useful to have an elementary code to play with. For this purpose, I have written a relatively simple Python code which implements the model with a few clever tricks to make simulations of a few thousands of agents possible on a standard laptop.
We follow Gregoire and Chaté in the formalism: point-wise particles move synchronously at constant speed v in discretised time of steps Δt=1. The particles have an orientation described by an angle θ which evolves taking into account all particles k within a given radius of interaction
For the neighbourhood calculations, cell-lists would be ideal, but they are too complex for the kind of elementary code that we want to write. What we are going to use is the
kd-tree quick nearest neighbour lookup algorithm as implemented in
Scipy and some clever sparse matrix manipulation. For visualisation, we employ
matplotlib, so that the resulting code is just 60 lines with only very popular libraries.
import numpy as np import scipy as sp from scipy import sparse from scipy.spatial import cKDTree import matplotlib.pyplot as plt from matplotlib.animation import FuncAnimation L = 32.0 rho = 3.0 N = int(rho*L**2) print(" N",N) r0 = 1.0 deltat = 1.0 factor =0.5 v0 = r0/deltat*factor iterations = 10000 eta = 0.15 pos = np.random.uniform(0,L,size=(N,2)) orient = np.random.uniform(-np.pi, np.pi,size=N) fig, ax= plt.subplots(figsize=(6,6)) qv = ax.quiver(pos[:,0], pos[:,1], np.cos(orient), np.sin(orient), orient, clim=[-np.pi, np.pi]) def animate(i): print(i) global orient tree = cKDTree(pos,boxsize=[L,L]) dist = tree.sparse_distance_matrix(tree, max_distance=r0,output_type='coo_matrix') #important 3 lines: we evaluate a quantity for every column j data = np.exp(orient[dist.col]*1j) # construct a new sparse marix with entries in the same places ij of the dist matrix neigh = sparse.coo_matrix((data,(dist.row,dist.col)), shape=dist.get_shape()) # and sum along the columns (sum over j) S = np.squeeze(np.asarray(neigh.tocsr().sum(axis=1))) orient = np.angle(S)+eta*np.random.uniform(-np.pi, np.pi, size=N) cos, sin= np.cos(orient), np.sin(orient) pos[:,0] += cos*v0 pos[:,1] += sin*v0 pos[pos>L] -= L pos[pos<0] += L qv.set_offsets(pos) qv.set_UVC(cos, sin,orient) return qv, FuncAnimation(fig,animate,np.arange(1, 200),interval=1, blit=True) plt.show()
The result is the following animation (the colour indicates the orientation):
Last May, James Drewitt from the School of Earth Science here in Bristol asked me to have a look at his data on ver high pressure and temperature gallium. Used to my idealised particles in box, I thought that it would be interesting to look understand what information can reasonably emerge in this more realistic setting.
It turns out that gallium is a liquid with a number of noticeable physical characteristics: a melting point just above room temperature at ambient pressure conditions, high thermal conductivity and a strong tendency to undercooling, i.e. to remain disordered well below its melting temperature (which is even enhanced with respect to the bulk behaviour when small droplets of gallium as considered). To my surprise, it also appears that many of the tools that I have employed to study the structure of simple liquids are useful to understand how extreme pressure and temperatures affect this metallic liquid.
We have shown, for example, that as the pressure increases the liquid shows a preference to form local motifs of radically different nature in somewhat similar proportions, as opposed to what would happen in purely repulsive systems. This is interesting, as this competition between different forms of local order provides a mechanism for the enhanced stability in supercooled conditions. We also have found out that simplistic approaches to the modelling of the three dimensional structure of the liquid (such as naive Reverse Monte Carlo methods) overlook these changes in structure and are strongly biased by their initial guesses.
The reference to the full work, that combines new experimental evidence with a detailed numerical simulation analysis, is
Hard colloidal spheres present a certain degree of local ordering that has been in the past described in different (related) ways: one can identify an increased role played by tetrahedral arrangements, or focus on icosahedral or partially icosahedral structures.
Expanding on a previous work, James Hallett (now in Oxford) has produced earlier this year a detailed analysis of very high density experiments where we show that increased local ordering can be described in terms of the number of interlacing pentagonal rings formed by neighbouring particles. This provides a finer description of the changes that high densities impose on the local structure and on the geometric constraints that are satisfied (or not) by the microscopic reordering.
The full work is available on the Journal of Statistical Mechanics.
Computing high order correlations in liquids is not easy. Josh Robinson – with the help of Paddy Royall, Roland Roth and myself – has shown earlier in 2019 that with employing mostly geometrical principles one can accurately estimate the free energy of different motifs in a simple hard sphere fluid, see here 10.1103/PhysRevLett.122.068004 .
In more detailed paper we now show how this approach can be connected to classical liquid state theory and seen as an extension of so-called scaled-particle theory, where one computes the work of insertion of solutes in a fluid in order to estimate their free energy (see, for example, the Widom insertion method.)
Our approach allows us to write down a potential of mean force for interactions between a subset of n particles and a fluid, generalising previous methods and opening a way to accurate measurement of free energy barriers in the formation of local structural inhomogeneities in fluids, as in the formation of crystalline precursors.
The full article reference is:
Joshua F. Robinson , Francesco Turci, Roland Roth, and C. Patrick Royall, Many-body correlations from integral geometry, Physical Review E 100, 062126 (2019)
During the summer of a few years ago Eric Brillaux, a student of the Ecole Normale Superieure de Lyon, visited Bristol for a summer project. Thinking of something moderately ambitious that could (in theory) be achieved in a few months, we started to explore how simple crystals respond to oscillatory shear.
The original motivation of the work was rather speculative: I was wondering if it could be possible to transform mechanically an ordered system (a crystal) into an amorphous system (a glass) whilst preserving some degree of local order. To this purpose, the ideal model to consider was an atomistic binary mixture whose supercooled liquid state presents local structural motifs that are identical to the repeated units of its crystalline state (as we have previously shown, see here) .
In particular, we considered an oscillatory shear protocol. This is interesting not only because it mirrors more closely actual experimental methods, but also because it allows the system to behave either reversibly or irreversibly: if the oscillations are small, the crystal survives; if the deformations are large, amorphisation takes place.
In our article just out on Soft Matter, we discuss how this dynamical transition between the reversible and irreversible regime takes place in actual three dimensional crystals and how it depends on the crystal composition. For example, single-component crystals can transform structurally, from face-centred cubic structures to more body-centred cubic structures before becoming disordered. Instead, crystals of two components either transform reversibly or become amorphous at a critical oscillation amplitude.
The dynamics is also rather interesting: for instance, the growth of amorphous regions follows a coarsening pattern that is reminiscent of spinodal decomposition in non-driven systems.
In the end, the amorphous states that we obtain are not as rich in local structural motifs as I hoped, but the dynamical transition in itself has appeared to be very intriguing!
More details in the original article
Eric Brillaux, Francesco Turci, Soft Matter, 2019, doi 10.1039/C8SM01950A
Supercooled liquids become more and more viscous as their temperature is reduced. The increased viscosity corresponds to an enormous increase in the characteristic time for the relaxation of density fluctuations. What is often puzzling is that, differently from many other physical phenomena, this dramatic change in the correlations in time appears to be weakly reflected in conventional measures of spatial correlations. These are typically so-called pair or two-body correlations, measuring how likely it is to find randomly chosen pairs of particles at particular characteristic distances.
The lack of strong correlations between two-body spatial correlation and the emergent, enormous relaxation times of supercooled liquids suggests that more complex, eventually multi-body correlations may be at play.
Thanks to the work of a very gifted PhD student in Paddy Royall‘s group, Joshua F. Robinson, we have obtained a first theoretical insight on the origin of such emergent correlations in a reference model for supercooled liquids, i.e. hard spheres, which are often employed to understand the behaviour of colloidal particles and as a basis to develop approximate theories of liquids.
We rooted our work in a geometric approach to treat the free energy of thermal hard spheres developed by Roland Roth (a co-author of our work) termed morphometric theory and this has allowed us to study the free energy of a certain number of thermal structural motifs of hard spheres immersed in an effective medium and predict with a high degree of precision their respective populations. Furthermore, the approach that we have used has revealed that it is possible to follow local deformations of the motifs and compute the free energy barriers between them.
The work appeared as an Editor’s Suggestion in Physical Review Letters:
Colloidal hard spheres at high volume fractions (beyond ~0.49) can crystallise: up to to 0.54, they coexist with a fluid phase, and at even higher densities they completely crystallise. The way the hard sphere fluid transforms into the solid is called crystal nucleation: due to thermal fluctuations, every now and then locally denser and more ordered regions appear and disappear; occasionally, these are large enough to further grow irreversibly, and form a crystalline region.
Nucleation is a generic process: in hard spheres, it should present its simplest traits, as it can be driven only by entropic forces. Yet, nucleation rates in colloidal hard spheres are rather different from what can be predicted from theory and numerical simulations. In particular, the discrepancy between the two increases of many orders of magnitude with decreasing the volume fraction from, for example, 0.55 to 0.53. Simulations normally consider idealised hard spheres in an ideal solvent. What could possibly go wrong?
Nick Wood, a talented PhD student here in Bristol, has taken care of some potential origins of the discrepancy analysing the effect of sedimentation on local order. Together with John Russo and Paddy Royall, we have considered in our recent paper how the flow induced by sedimentation may, via hydrodynamics interactions, transform the structure of the liquid, compared to the case in absence of sedimentation, and how such changes would impact on the nucleation barriers. The result is that some structural signatures clearly vary as a function of the density mismatch between colloids and solvent and that this leads to an estimated correction of the rates in the right direction, but by an amount that is not sufficent to address the entire discrepancy.
The complete article can be found here:
Nicholas Wood, John Russo, Francesco Turci and C. Patrick Royall J. Chem. Phys. 149, 204506 (2018); https://doi.org/10.1063/1.5050397
Computer simulations are very powerful: in the case of molecular dynamics, we can model the positions and velocities of atoms or molecules and observe the emergence of pattern and structures in situ, following each individual atom in its trajectory.
However, when we study supercooled liquids or glasses, it is hard to probe in computer simulations very low temperatures or very tightly packed systems, unless we opt for indirect and clever routes to glean some information on the low temperature behaviour. It would be great if one could directly take a very cold (or, similarly, very dense) liquid at equilibrium and see how the constituent particles are arranged.
This is precisely what James Hallett has managed to do during his stay in Bristol using super-resolution microscopy: this method can access the coordinates of equilibrium themal packings so dense that a direct simulation would never do. Thanks to James’s clever imaging, we have then carefully analysed the individual coordinates and trajectories of dense repulsive colloids and managed to clearly show how the local environment of these densely packed equilibrium systems changes as the density is increased.
We have found some notable features: as we take denser samples, the liquid becomes gradually richer in regions where particles are arranged into five-fold symmetric structures; those regions display reduced mobility compared to other regions of the sample; randomly selected domains of the system become more and more “similar to each other” as the density is increased, accompanied by the decrease in the number of distinct configurations the liquid can take, a quantity related to its so-called configurational entropy.
This work has just appeared in Nature Communications. Full text here:
Together with CP Royall, S Tatsumi, J Russo and PhD student J Robinson we have just published on the Journal of Physics: Condensed Matter a handy review on recent approaches to the exploration of very stable glasses in experiments and simulations.
We cover a variety of topics, including vapor deposited glasses in experiments, importance sampling in trajectory space, random pinning, representing distinct attempts to address the following question: is glassiness linked to some kind of novel thermodynamic transition in very low temperature liquids?
Find out more here:
Supercooled liquids present dynamical heterogeneities at low temperatures: on a certain length and timescale, some areas are very mobile (active) while others are much more solid-like (inactive). This feature is often interpreted as the signature of the fact that the liquid, when supercooled, starts exploring different metastable regions of the free energy landscape.
A possible route to illustrate this effect is through large deviations of structural-dynamical obserables, as we first did in the case of a canonical atomistic model for glassformers, the Kob-Andersen mixture. A main observation of that work was that dynamical heterogeneities correspond to a first order phase transition in a (reweighted) space of possible steady states between high energy trajectories that are rich in structure and low energy trajectories that are poor in structure. Moreover, such a transition has a strong temperature dependence, so that the structure-rich trajectories become more and more likely to be observed as the temperature decreases.
Now, we have published a follow-up work on the European Physical Journal E, where we show that the same mechanism is at play in another model glass-former (the Wahnström mixture), showing that while the overall qualitative picture may be general, the details depend on the nature of the interactions between the constituents. Moreover, we also show that configurations extracted from the structure-rich trajectories have much larger yield stresses than the normal supercooled liquid: the emerging rigidity of glasses appears to be strongly related to the structural-dynamical transition that we have highlighted.
More information in
F Turci, T Speck and C P Royall, Structural-dynamical transition in the Wahnström mixture Eur. Phys. J. E , 41: 54 (2018) https://doi.org/10.1140/epje/i2018-11662-3