Research interests

Much of our research involves applying state-of-the-art Monte Carlo simulation methods to explore the fascinating physics that occurs in complex colloids, such as self asssembly and unusual phase behaviour. A focus is the development and application of novel simulation algorithms that allow phenomena to be revealed that are inaccessible to conventional techniques. An appealing aspects of colloidal physics is that simple models often correspond quite closely to real systems, and this allows us to collaborate meaningfully with experimentalists. Beyond colloids we are interested in computational and theoretical solutions to a wide range of problems in soft matter and statistical physics. Below are some examples of our recent work.


The chemistry or structure of some surfaces is such that water is repelled from them. The contact angle between the drop and the surface measures the size of the effect. When this angle reaches 180°, the surface is said to be dry. Using molecular Monte Carlo simulation, we have studied the approach to drying in a realistic water model. We find that close to the wall, density fluctuations, as measured via a compressibility profile, grow very large. This reveals the presence of a diverging transverse correlation length - the hallmark of a surface critical phenomenon. Our discovery of "critical drying" in water should provide a basis for interpreting and characterising experimental measurements on water near hydrophobic surface. Phys. Rev. Lett. 115, 016103 (2015) and Phys. Rev. Lett. 117, 176102 (2016)

Lock and key colloids

These are particles with complementary geometrical shapes that allow them to fit together. When immersed in a fluid of much smaller nanoparticles, they spontaneously self assemble into ordered structures such as colloidal molecules or strings. The assembly is driven by the `depletion' attraction, an entropic force that pushes the key into the lock. Using specialised algorithms and bespoke theory, we have uncovered unusual phase behaviour in indented colloids, including novel porous liquid structures. We have also clarified the fundamental principles that permit rapid and reliably assembly into ordered products. Phys. Rev. Lett. 114, 237801 (2015)

Indented colloids can be made in the laboratory and we have collaborated with the experimental group of Dirk Aarts (Oxford) to study how a monolayer of indented colloids self assembles into two-dimensional crystals on a surface. The experiments find that the indentations hinder crystallisation and our simulations give insight into the reasons for this in terms of the competeting equilibrium states and kinetic trapping. Soft Matter 11, 6089 (2015).

Cluster crystals

Soft matter systems such as star polymers and dendrimers comprise individual molecules that can overlap substantially at high concentrations. When describing such systems theoretically, it is common to dispense with the finer (atomistic) detail in favour of coarse-grained descriptions. Typically these represent each molecule in terms of an ultra-soft colloidal particle which interacts with it neighbours via a short ranged two-body effective potential. One of the interesting properties of such systems is that they form cluster crystals, in which each lattice site is occupied by multiple soft-core particles.

We have used theory and simulation to study cluster crystals in simple models of ultra-soft potentials. Special techniques were developed to determine the equilibrium properties directly from a single simulation. As the number density is increased at zero temperature, a 'cascade' of isostructural phase transitions occurs between states whose site occupancy differs by unity. For low but finite temperature, each of these transitions terminates in a critical point. We have determined the critical parameters which show interesting non-monotonic trends as a function of the potential softness.
J. Chem. Phys. 141, 094903 (2014).

Coarse graining and depletion

A key ingredient of coarse-graining strategies for complex fluids is to integrate out the degrees of freedom on small length scales to produce an effective potential between the surviving coordinates. A case in point is colloid-polymer mixtures in which the small polymers engender a depletion interaction between the colloids, which is often modelled in terms of a pair potential. Another example is star polymers, for which one replaces the full polymer by a single particle interacting via a soft pair potential. We have developed a general simulation technique for quantifying the contribution of three-body interactions to the thermodynamical properties of coarse-grained representations of complex fluids. The method is based on comparing the third virial coefficient B3 for a complex fluid with that of an approximate coarse-grained model described by a pair potential. To obtain B3 we introduce a new technique which expresses its value in terms of the measured volume-dependent asymptote of a certain structural function. The strategy is applicable to both Molecular Dynamics and Monte Carlo simulation. We have applied it to measurements of three-body effects in models of star polymers and colloid-polymer mixtures.
J. Chem. Phys. 140, 244118 (2014).

The depletion potential is central to coarse-graining strategies for colloid-polymer mixtures. We have developed accurate simulation methods for obtaining its form, which have been used to carefully check the theoretical predictions of Density Functional Theory.
J. Chem. Phys. 139, 144102 (2013).
Phys. Rev. E 84, 061136 (2011)


In a polydisperse fluid the particles are not all identical, but have a spread of sizes or interaction strengths. Polydispersity is common in colloids or polymers that form the basis of many everyday materials such as paints, fuels, personal care products and foodstuffs. Understanding its effects is thus a matter of practical as well as fundamental interest. Together with Peter Sollich, we have investigated the effects of size polydispersity on the crystalline phases of spherical particles in thermal equilibrium. In the absence of polydispersity we know from Kepler's conjecture, that such spheres can be packed to fill maximally just over 74% of space, in the face centred cubic (fcc) structure familiar from greengrocers' displays of oranges. But what is the thermodynamically optimal structure for dense spheres which have a spread of diameters? Using specialized computer simulation methods and theoretical calculations we have shown that dense polydisperse spheres demix into coexisting fcc phases, with more phases appearing as the spread of diameters increases. We managed to track up to four coexisting phases in our simulations. Each of these is fractionated ie. it contains a narrower distribution of particle sizes than is present in the system overall. Phys. Rev. Lett 104, 118302 (2010).

Monte Carlo tools

Monte Carlo has long been used as a powerful tool in statistical physics to answer questions that are too complicated for theory alone. Many techniques exist to overcome problems such as large free energy barriers, critical slowing down, rare events and optimisation problems. A large part of our work is devoted to developing new methods to overcome such problems.

A recent focus has been on depletion interactions that arise for systems in which large particles are immersed in a sea of small particles. For this we use the geometric cluster algorithm, or GCA, (see opposite) to move large particles around. The GCA is an iterative scheme that reflects particles in a pivot or a plane until there are no more overlaps. We have extended this algorithm for lock and key colloids. We have also developed statistical tools to study phase transitions in systems with large size asymmetry.

Staged insertion of a large particle into a sea of small particles

Another way of relaxing highly size-asymmetrical mixtures is to insert or delete large particles in a gradual fashion using biased sampling techniques. To do this we allow the large particle to exist in a ghost state, gradually pushing the small ones away. Doug Ashton's movie (opposite) shows the algorithm in action for a mixture of hard disks with a 20:1 size ratio.

Copyright © Nigel Wilding 2015