Force chains are networks of stresses that propagate in complex, distinctive patterns across disordered media. They have been successfully quantified in granular materials and have been a useful concept to rationalise the flow behaviour of such systems. They can be visualised in granular experiments for example with the help of photo-elastic polymers.
Granular and colloidal materials mainly differ due to the scale at play: in granular systems, the constituents (particles) are too large for thermal fluctuations to play a role, while in soft materials, it is precisely those fluctuations that govern the interesting spontaneous processes, such as self-assembly. This distinction can be quantified by adimensional numbers, for example the Péclet number for sedimentation is
where is the particle radius. This means that typically particles well above the micron size immersed in a fluid at ambient temperature will lead to a granular response.
How do force chains change when we move from the granular to the colloidal scale? I was involved in devising a means to characterise and model an interesting experiment performed by J Dong (supervisor CP Royall) in collaboration with M Faers from Bayer and RL Jack from Cambridge. Jun imaged the contact regions between the droplets of an emulsion gel, during its formation, with droplet sizes of about 3 microns in diameter that are well inside the colloidal regime.
I constructed a model using an effective interaction potential, calibrated on the pair-wise static correlations, and then studied the evolution of the gel network as it forms. The experiments are able to identify short chains of a few particles where the compressive stresses are concentrated. These appear to start the formation of a gel backbone, but the experiments are limited in the accessible time window.
In the simulations, I can track the process and its evolution and compare the statistics of the forces and the length of the chains with the experiments. One can even access very late times and find that the compressive chains are well inside the backbone of the arms of the formating gel, as illustrated below.
Several questions remain open: the effective interaction potential appears to point to the existence of very attractive forces at very short range, which are very difficult to measure directly; the arrested structure of the gel can be affected by kinetic pathways modified by — for example — hydrodynamic interactions; the repulsive forces of the emulsion droplet are large and hard to calibrate precisely. Yet we have demonstrated that an effective, particle-based model can capture the essence of the force distributions in gels, including the shape of the force-chain length distributions.
Yushi Yang has recently completed his PhD project in Bristol focusing on the collective behaviour of a specific living organism: Zebrafish, a small, semi-transparent fish coming from tropical fresh waters employed extensively in biology as a model organism. Quantitative studies of the collective behaviour of animals have been performed in many different contexts, from midges to starlings and fish are not excluded. However, Yushi has been one of the first to concretely realise an experimental setup allowing for the three-dimensional tracking of the trajectories of a conspicuous number of individuals (about 50).
Together we looked into some simple models making physical sense of the emerging patterns of behaviour. This approach has serious epistemological challenges: what justifies the reduction of the social behaviour of an animal to a minimal physical model? what do we actually learn in the process of reduction of complexity? what predictive power is associated with such results?
In our approach, we pursued the identification of quantitative variables describing the evolving physical features of groups of fish: their average orientation, their relative distances, and their speed. We actually did not attempt to measure any specific form of interactions between the fish: these are certainly taking place by means that can be complex (vision, hydrodynamic feedback, maybe some form of signalling) but, in our description, they only appear as effective terms. In fact, what we seek is a small number of physical properties that allow us to organise the data: for example, we identify a scaled persistence length that appears to control the degree of polarisation of groups of fish of different ages: incoherent groups of older fish and well-coordinated groups of younger fish. It is indeed a key result of Yushi’s work that the behaviour of zebrafish changes markedly with age and that non-trivial correlations appear to be present only for the younger groups. We are able to map these different behaviours on a single master curve and model it with a simple physical model dominated by alignment interactions and delay (or inertia) in reorientation.
While the different physical ingredients of the simplified model do not have an immediate biological interpretation, they allow us to narrow down the spectrum of relevant variables controlling collective behaviour and justify further research aimed at providing a causal, biological link between the behaviour of the group and the phenotypical characteristics of the individuals.
This research has been published in PLoS Computational Biology
When my 12-month old daughter started pushing her baby walker around the apartment, I noticed how often she ends up stuck against tables, walls, and obstacles in general, as she struggles (or simply does not even try) to turn the walker around. I could not avoid thinking that that was a transparent example of the typical behaviour of self-propelled agents (or particles). As they travel in a given direction and their rotational diffusion is low, they typically get stuck against hard barriers and obstacles, leading to their accumulation.
However, some self-propelled systems do more than just accumulate. For example, so-called active Brownian particles (self-propelled repulsive spheres) can display phase coexistence between a dense and a dilute phase originating from a bulk phase transition known as motility-induced phase separation. This transition is similar to vapour-liquid phase transitions in equilibrium fluids, and this similarity motivated our recent work in Physical Review Letters. Vapour-liquid coexistence is, in fact, a prerequisite for commonplace surface phase transitions in ordinary liquids. These are the transitions associated with the change in contact angle between a droplet and a surface: from spread out droplets that cover a cloth (wetting it) to almost perfectly spherical water droplets on Lotus leaves (keeping the surface dry). Whether wetting or drying occurs depends on the interaction between surface and fluid atoms/molecules: attractive interactions promote wetting while repulsive ones favour drying. Tuning the strength of these interactions can transform droplets from partially wetting/drying a surface to completelywetting /drying it.
Active systems appear to undergo only complete wetting, as impenetrable walls lead to particle accumulation as much as attractive forces. To go beyond this regime, we explore the effect of finite strength repulsive barriers. Our hypothesis was that tuning the barrier strength would have revealed the competition between two mechanisms: the propensity to accumulate to hard walls and the tendency of potential barriers to reduce the density locally.
This set-up allows us to realise a connection between equilibrium wetting and the behaviour of active systems. Such connection takes the form of a symmetry transition in the structure of the density profiles and highlights the role of density fluctuations. It reveals marked differences between two and three dimensional systems, with the former more prone to gas bubble formation while the latter appear more dramatically sensitive to the barrier strength, with a sharp transition from partial to complete wetting.
The article (which also Featured as an Editor’s suggestion in Physics) can be found here:
Soft matter is a broad field, ranging from colloidal particles to micelles, from proteins to cellular tissues. As I mentioned in a previous post, a physics-based approach to characterising the morphology of soft biological matter can be very insightful. It provides simple, geometrical and structural metrics to identify variation in tissues. We recently demonstrated this approach in a rich article in Nature’s Bone Research, a journal dedicated to the quantitative study of bone properties in different species.
A few years ago, Erika Kague, from Physiology at the University of Bristol, asked me to think about her problem. By inducing genetic mutations, she can generate individual zebrafish (Danio rerio) – a common model organism in biology – with various malformations. These she can investigate to understand the effect of bone mineral density on the insurgence of osteoporosis. The issue is that one may want to rapidly, systematically and quantify the amount of malformation.
To do that, I worked with talented PhD student Yushi Yang to automatise a computer vision workflow of segmentation of three-dimensional images, detection of relevant anchor points and determination of several geometrical and structural metrics. Some of these quantities have been inspired by thermal soft matter analogues, such as the porosity of gels; others rely on graph theory metrics. The entire manually tagged database can be augmented using deep learning U-nets that Stephen Cross (Wolfson Bioimaging Facility, Bristol) optimised for our use case.
The result is a detailed characterisation of which genes promote certain kinds of changes in the bone structure. The data reveal a delicate balance between too little and too much bone mineral density to minimise the chances of osteoporosis.
Self-organisation has many forms, many of which have been studied for systems in equilibrium or metastable equilibrium, as in crystal formation or in gelation. The striking feature of these phenomena is the emergence of complex patterns of aggregation just from elementary interactions among the constituents. These are driven by an imbalance in the thermodynamic potentials for example the chemical potential. However, in recent years it has been shown that this spontaneous organisation is not the prerogative of equilibrium for (passive) systems: completely out of equilibrium systems such as bacterial colonies or self-propelled colloids present a similar behaviour, even if thermodynamic concepts such as a chemical potential are difficult to generalise to these systems. An interesting example is the amoeba-like crawling crystals that Abraham Mauleon-Amieva, here in Bristol has studied in his PhD and which present an interesting competition between electrostatic and active forces, with multiple mesh-phases, see Phys. Rev. E 102, 032609.
An important question in this area is to understand whether the phase diagrams of similar active systems can be understood exclusively invoking effective equilibrium concepts. A possible route is, for example, to think that the active forces lead to collisions and these can be effectively coarse-grained into suitable attractive effective two-body interactions. It would such an effective attraction to favour aggregation and hence explain (in an effective picture) the observed motility-induced self organisation.
In a recent article published in Physical Review letters with Nigel B Wilding we explore these ideas for an elementary model for active matter in three dimensions, active Brownian particles. Through the characterisation of the phase diagram, its phase separations and single phase fluid region we show that the system indeed shares many similarities with passive systems with short ranged interactions in 3d: a metastable liquid-gas phase separation, a crystalline phase, several pre-critical lines. However, a quantitative analysis of the effective interactions shows that it is not possible to explain the motility-induced phase separation only in terms of effective twobody interactions: multibody effects involving large numbers of particles are important, and can be quantified using information-theoretic tools.
Liquids are normally considered to be thermodynamically stable. However, rapidly cooled liquids attain a so-called metastable state — the supercooled or undercooled liquid. As we decrease the temperature, these liquids become more and more viscous and structural relaxation becomes slower and slower. One could naively infer that, as the slowing down proceeds, nothing happens in such systems since the motion of the constituents (atoms or molecules) is so severely hindered. Actually, the mystery of supercooled liquids resides precisely in the origin of such slowing down. Ultimately, this form of dynamic arrest leads to the formation of an amorphous solid, i.e. a solid that is not crystalline: we call this a glass.
This longstanding open problem in thermodynamics and statistical mechanics has prompted several theoretical approaches. These are inspired by different facets characterising glassy physics: glasses present a wide and heterogeneous distribution of relaxation timescales; glasses have signatures of reordering on very small lengths; glass-forming liquids display a reduction of the number of distinct configurations that the constituents can attain. Several conflicting theories have emerged over time, attempting to provide a unified picture.
We have just published a Perspective (a Featured Article in the Journal of Chemical Physics) retracing the connections existing between the theory of dynamical phase transitions and the structural and thermodynamical approaches to the glass transitions. We show that the theory of dynamical phase transitions allows us to identify metastable states in an operative sense. The glassy phenomenology can then be re-interpreted in an extended phase space. Here dynamical transitions between metastable states are coupled to structural features and configurational entropy reduction. This suggests a close relationship between microscopic structural arrangement, mesoscopic kinetic rules and thermodynamic phase transitions.
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)
r0 = 1.0
deltat = 1.0
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])
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
anim = FuncAnimation(fig,animate,np.arange(1, 200),interval=1, blit=True)
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.