Lattice-Boltzmann Simulations of Colloidal Particlesat Fluid Interfaces

Jens Harting

Department of Applied Physics, Eindhoven University of TechnologyP.O. Box 513, NL-5600MB Eindhoven, The Netherlands

andFaculty of Science and Technology, Mesa+ Institute, University of Twente

P.O. Box 217, NL-7500AE Enschede, The Netherlands

E-mail: j.harting@tue.nl

Particle-stabilized fluid interfaces are very common in industrial applications as they can befound for example in the food, cosmetics, or oil industries. However, until recently our un-derstanding of these systems was mostly based on experiments, highly simplified theoreticalcalculations and only very few numerical approaches. The lack of well established and widelyapplied simulation codes is not surprising since computer simulations of particle stabilized fluidinterfaces are a highy complex task. Suitable algorithms need to be able to treat the hydrody-namics of the involved solvents, the dynamics of the suspended colloidal particles, as well asthe interactions between all those constituents at the same time. Furthermore, the simulationof large scale 3D emulsions requires highly efficient and massively parallel implementationsand access to state of the art supercomputers. In these lecture notes we summarize the relevantdetails of our own simulation method for particle stabilized interfaces which is based on a com-bined Lattice-Boltzmann and molecular dynamics solver. We provide an overview on importantimplementation details and review a number of recent applications from our group with the aimto demonstrate the specific features of particle stabilized fluid interfaces.

1 Introduction

Particle stabilized emulsions have a high potential for various purposes with industrial ap-plications, such as cosmetics, improved low-fat food products, ice cream, drug delivery,or tertiary oil recovery1–3. While amphiphilic surfactant molecules are traditionally em-ployed as emulsification agents, their effects can be mimicked or supplemented by theuse of colloidal particles. These may be a cheaper or less toxic alternative to surfactants,but most importantly they may be customized to include additional desirable properties.Examples include ferromagnetic particles4, 5, particles with different interfacial propertieson different parts of their surface (for example Janus particles)6–10, or nonspherical parti-cles11–18, where the geometric anisotropy has an impact on their stabilization properties.Those chemically or geometrically anisotropic particles will under certain conditions de-form the surrounding fluid-fluid interface. This leads to capillary interactions which canbe used to tune the attraction or repulsion of adsorbed colloidal particles and as such mightfind applications such as the formation of new soft and highly tunable materials17, 10.

The energy differences involved in the adsorption of colloidal particles at a fluid in-terface are generally orders of magnitude larger than thermal fluctuations. Therefore,this adsorption process is practically irreversible19, 18 and Ostwald ripening can be fullyblocked20–24. In this manner, particles allow for long-term stabilization of an emulsion15, 25.

However, colloidal particles stabilize fluid interfaces kinetically and not thermodynam-ically. They reduce the interfacial free energy by their presence: maintaining a fluid-fluid

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interface is energetically expensive, and it is favourable to replace it with particle-fluid in-terface. This is exactly what happens when a particle adsorps to such a fluid-fluid interface.In contrast, due to their amphiphilic ineractions with the involved fluids, surfactants reducethe interfacial tension directly, which also reduces the total interfacial free energy. Theseeffects are described in some detail in Frijters at al.26. Note that emulsions are in generalnot necessarily thermodynamically stable, as the energy cost of retaining the remaining in-terfacial area is still larger than the gain in entropy when the fluids mix completely. Thereare several ways to improve the stabilization properties, for example by using anisotropicparticles with an additional rotational degree of freedom15.

Particle-stabilized emulsions can present in various forms, classified by the shapesand sizes of their fluid domains. The form that most resembles a traditional, surfactant-stabilized emulsion is the “Pickering emulsion”27, 28, which consists of particle-covereddroplets of one fluid suspended in another fluid. A more recent discovery is the bicon-tinuous interfacially jammed emulsion gel, or “bijel”, which is characterized by two largecontinuous fluid domains that are intertwined and are only stable because of the particlespresent at their interfaces. This form was first predicted by numerical simulations29, andshortly thereafter confirmed experimentally30, 31. Parameters that affect the final state of anemulsion include the ratio between the two fluid components, the volume fraction of theparticles, and their wettability. These parameters have been studied numerically by variousauthors15, 25, 32–35.

Computer simulations of particle stabilized emulsions require a solver for the dynam-ics of the involved fluid species combined with an algorithm to track the suspended parti-cles and their interactions. Here we use the Lattice-Boltzmann (LB) method for the fluidcomponents, coupled to solid particles whose inter-particle interactions are simulated bymolecular dynamics. This approach has proven very successful during recent years dueto its ease of implementing multiphase flows, inherent parallelism allowing to harness thepower of state of the art supercomputers, and straightforward methods to couple the fluid-and particle solvers. The simulation method is briefly explained in the following section 2.Within the scope of this lecture we then review three distinct applications of the methodwhere we studied fundamental properties of particle stabilized fluid interfaces before weconclude in Sec. 4. The current lecture nodes represent a shortened collection of severaloriginal articles36, 26, 37, 38, 18, 15, 25 and the interested reader is referred to those for more de-tails.

2 Simulation Method

2.1 The Lattice-Boltzmann method

For the simulation of the fluids we apply the Lattice-Boltzmann method which is based onthe discrete form of the Boltzmann equation:39

f ci (x + ci∆t, t+ ∆t) = f ci (x, t) + Ωci (x, t), (1)

where f ci (x, t) is the single-particle distribution function for a fluid component c withdiscrete lattice velocity ci at time t located at lattice position x. Numerous discretizationsin space and time are possible, but the most popular one in three dimensions is the so-called

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D3Q19 lattice. The distance between lattice nodes is given by the lattice constant ∆x andnineteen directions are allowed for the velocity. ∆t is the timestep and

Ωci (x, t) = −fci (x, t)− f eq

i (ρc(x, t),uc(x, t))

(τ c/∆t)(2)

is the Bhatnagar-Gross-Krook (BGK) collision operator40. The fluid density is defined as

ρc(x, t) = ρ0

∑

i

f ci (x, t), (3)

where ρ0 is a unit mass factor. τ c denotes the relaxation time for component c and

f eqi (ρc,uc)=ζiρ

c

[1+

ci ·ucc2s

+(ci ·uc)2

2c4s−(uc ·uc)

2c2s+

(ci ·uc)3

6c6s−(uc ·uc) (ci ·uc)

2c4s

](4)

is a discretized third order expansion of the Maxwell-Boltzmann distribution function.

cs =1√3

∆x

∆t(5)

is the speed of sound,

uc =∑

i

f ci (x, t)ci/ρc(x, t) (6)

is the fluid velocity and ζi denotes a coefficient depending on the direction: ζ0 = 1/3 forthe zero velocity, ζ1,...,6 = 1/18 for the six nearest neighbors and ζ7,...,18 = 1/36 for thenext nearest neighbors in diagonal direction. The kinematic viscosity can be calculated as

νc = c2s∆t

(τ c

∆t− 1

2

). (7)

In the following we choose ∆x = ∆t = ρ0 = 1 for simplicity, but a conversion to SI unitsis trivial41. In all simulations the relaxation time is set to τ c ≡ 1.

2.2 A multicomponent Lattice-Boltzmann method

Several extensions for the Lattice-Boltzmann method have been developed to simulatemulticomponent and multiphase fluids42–46. The article Liu et al.47 provides an extensiveoverview. Here, we use the multicomponent pseudopotential method introduced by Shanand Chen42. Every species has its own set of distribution functions following Eq. (1). Amean field force

Fc(x, t) = −Ψc(x, t)∑

c′

gcc′∑

x′

Ψc′(x′, t)(x′ − x) (8)

is calculated locally and acts between the components. The summation includes the dif-ferent fluid species c′ and x′, the nearest neighbors of lattice positions x. gcc′ is a phe-nomenological coupling constant between the species and Ψc(x, t) is a monotonous weightfunction representing an effective mass. For the results presented here, the form

Ψc(x, t) ≡ Ψ(ρc(x, t)) = 1− e−ρc(x,t) (9)

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is used. To incorporate Fc(x, t) in f eqi we define

∆uc(x, t) =τ cFc(x, t)

ρc(x, t). (10)

The macroscopic velocity included in f eqi is shifted by ∆uc as

uc(x, t) =

∑i f

ci (x, t)ci

ρc(x, t)−∆uc(x, t). (11)

As we are interested in immiscible fluids we choose a positive value for gcc′ which leadsto a repulsive interaction and thus to the emergence of surface tension. A typical range ofthe coupling parameter is 0.08 ≤ gcc′ ≤ 0.14.

2.3 Colloidal particles

We couple a traditional molecular dynamics algorithm to the Lattice-Boltzmann solver inorder to compute the interactions and trajectories of the suspended particles. The particletrajectories follow Newton’s equations of motion

F = mupar, D = Jωpar, (12)

which are integrated using a classical leap frog integrator. F and D are the force and torqueacting on the particle with massm and moment of inertia J . upar and ωpar are the velocityand the rotation vector of the particle.

The particles are discretized on the Lattice-Boltzmann lattice. They are coupled toboth fluid species by a modified bounce-back boundary condition as pioneered by Laddand Aidun33, 48–52. The lattice Boltzmann equation then becomes

f ci (x + ci, t+ 1) = f ci (x + ci, t) + Ωci (x + ci, t) + C. (13)

C depends linearly on the local particle velocity, i is defined in a way that ci = −ciis fulfilled. A change of the fluid momentum due to a particle leads to a change of theparticle momentum in order to keep the total momentum conserved:

F(x, t) =(2f ci (x + ci, t) + C

)ci. (14)

If the particle moves, some lattice nodes become free and others become occupied. Thefluid on the newly occupied nodes is deleted and its momentum is transferred to the particleas

F(x, t) = −∑

c

ρc(x, t)uc(x, t). (15)

A newly freed node (located at x) is filled with the average density of theNFN neighboringfluid lattice nodes xiFN for each component c33,

ρc(x, t) ≡ 1

NFN

∑

iFN

ρc(x + ciFN, t). (16)

Hydrodynamics leads to a lubrication force between the particles. This force is reproducedautomatically by the simulation for sufficiently large particle separations. If the distancebetween the particles so small that there is no lattice node between them anymore, this is

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supposed to fail. Therefore, if the smallest distance between two identical spheres withradius R is smaller than a critical value ∆c = 2

3 a lubrication correction is introduced as52

Fij =3πµR2

2rij(rij(ui − uj))

(1

rij − 2R− 1

∆c

). (17)

Here, µ is the dynamic viscosity, rij a unit vector pointing from one particle center tothe other one and ui is the velocity of particle i. These corrections can be generalizedto nonspherical particles in several ways. A simple, but less accurate approach is basedon a scaling of the potentials following Berne and Pechukas53, 54, 35. By appropriately de-termining the distance between particle surfaces and also including tangential forces, theaccuracy can be improved substantially55. For very small particle distances, the divergingnature of the lubrication correction can cause the simulation to become unstable. Sincethis case only happens a very few times even in very dense systems, we introduce a Hertzpotential56 which has the following shape for two identical spheres with radius R:

φH = KH(2rp − r)5/2 for r < 2rp. (18)

Here, r is the distance between particle centers. For larger distances φH vanishes. KH isa force constant and is chosen to be KH = 100 in all our simulations.

The Shan-Chen forces also act between a node in the outer shell of a particle and itsneighboring node outside of the particle. This leads to an unphysical increase of the fluiddensity around the particle. In order to avoid this effect, the lattice nodes in the outer shellof the particle are filled with a virtual fluid with a fluid density corresponding to the averageof the value in the neighboring free nodes for each fluid component:

ρcvirt(x, t) = ρc(x, t). (19)

This can be used to control the wettability of the particle surface. We define the parameter∆ρ named “particle color”. Positive values of ∆ρ are added to the “red” fluid component,

ρrvirt = ρr + ∆ρ, (20)

and negative values are added to the “blue” component,

ρbvirt = ρb + |∆ρ|. (21)

This leads to an approximately linear relation between ∆ρ and the three-phase contactangle θp35.

2.4 Implementation

The locality of the Lattice-Boltzmann algorithm makes the implementation of massivelyparallel codes straightforward57. Generally, a regular, orthogonal grid is used, and thecollision operator and boundary implementations are based on local operations so that ateach lattice node only information from its own location is required. The computationallymost demanding parts of a typical Lattice-Boltzmann code are “streaming” and “collision”routines. In practice, the number of floating point operations required at every timestep isnot the limiting factor, but the random access in memory required for the streaming andeventual non-local force computations makes the algorithm a memory-bound numericalmethod. This implies a number of issues for the implementation of highly efficient codes,

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1

2

4

8

16

32

64

128

256

1 2 4 8 16 32 64 128 256

speedup n

orm

aliz

ed to 1

024 c

ore

s

number of cores [1024]

LB3D v6.5.4ideal

(a)

1

2

4

8

16

32

64

128

256

1 2 4 8 16 32 64 128 256

speedup n

orm

aliz

ed to 1

024 c

ore

snumber of cores [1024]

LB3D v6.5.4ideal

(b)

Figure 1: Strong scaling of LB3D on JUQUEEN. (a) relates to a system with only one fluid component, (b) toone with two fluid species and suspended particles37, 35.

be it on shared-memory multicore nodes, distributed memory clusters, or accelerator cardssuch as graphical processing units (GPUs) or Xeon Phi cards.

Several well established and highly scalable Lattice-Boltzmann implementations exist,which have all demonstrated excellent scaling on hundreds of thousand CPUs. To namea few, Ludwig58, LB3D59, walBerla60, MUPHY61, and Taxila LBM62 are all codes whichhave been used for numerous high quality scientific publications. Most of these codes areable to handle solid objects suspended in fluids. The first three can even combine this withmultiple fluid components or phases.

Our own implementation, LB3D, is written in Fortran 90 and parallelized using MPI.Long-running simulations on massively parallel architectures require parallel I/O strategiesand checkpoint and restart facilities. LB3D uses the parallel HDF5 library for I/O which hasproven to be highly robust and performant on all supercomputers we had access to. LB3Dhas been shown to scale almost linearly on up to 262,144 cores on the European Blue Genesystems Jugene and Juqueen based at the Julich Supercomputing Centre in Germany36, 47.However, such excellent scaling required some optimizations of the code: Initially, LB3Dshowed only low efficiency in strong scaling beyond 65 536, which could be related toa mismatch of the network topology of the domain decomposition in the code and thenetwork actually employed for point-to-point communication. The Blue Gene/P providesdirect links only between direct neighbors in a three-dimensional torus, so a mismatch cancause severe performance losses. Allowing MPI to reorder process ranks and manuallychoose a domain decomposition based on the known hardware topology, efficiency canbe brought close to ideal. The importance of these implementation details is depicted bystrong scaling measurements based on a system of 1 0242×2 048 lattice sites carrying onlyone fluid species (Fig. 1a) and a similarly sized system containing two fluid species and4 112 895 uniformly distributed particles with a diameter of ten lattice units (Fig. 1b).

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Wall

Wall

Fluid 1

Fluid 2

F < 0

F > 0

Rh

θ

Figure 2: Equilibrium state of a spherical particle at a fluid-fluid interface. The contact angle, θ = cos−1(h/R),where h is the height of the particle centre of mass above the interface and R the particle radius, is defined withrespect to Fluid 1. The force, F , on the particle acts to detach it into the wetting (positive force) or non-wetting(negative force) fluid (Figure reprinted from Davies et al.18).

3 Applications

3.1 Detachment energies of spheroidal particles from fluid-fluid interfaces

The detachment energy of a single particle from a fluid-fluid interface plays a crucial rolein our understanding of particle-stabilised emulsions. Several authors have studied thedetachment of particles in the past, but to the best of our knowledge, none of these extendedtheir treatment to the case of anisotropic spheroidal particles, which we focused on in arecent paper by Davies et al.18. This section gives a summary of the results presented inthat publication.

Previous detachment energy studies focussing on free energy differences between anequilibrated particle at an interface63–70 and in the bulk revealed a crucial dependence onparticle shape: prolate and oblate spheroidal particles attach to interfaces more stronglybecause they reduce the interface area more than spherical particles for a given particlevolume.71–75 For a particle already adsorbed at an interface to detach itself, the particlemust deform the interface and overcome the interface’s resistive force: there is a free-energy barrier and an associated activation energy.

We develop a simple thermodynamic model for the detachment energy of spheroidalparticles from fluid-fluid interfaces as a function of contact angle and aspect ratio only, andhighlight the implications of our simplifications. Assuming that the effect of gravity canbe neglected, the surface free energy of a particle at an interface (Fig. 2) is given by

E = σ12A12 + σp1Ap1 + σp2Ap2, (22)

where Aij is the area and σij is the surface-energy of the i, j interface wherei, j = 1: fluid 1, 2: fluid 2, p: particle.76 We neglect line-tension since it is relevantonly for nano-sized particles.72 The surface area of the particle is Ap = Ap1 + Ap2. Thefree energy of a system in which the particle is fully immersed in either fluid 1 or fluid2 is given by Ei = σ12A12 + σpiApi where i = 1, 2. Taking the free energy differencebetween a spherical particle at an interface (Fig. 2) and a spherical particle immersed inthe bulk fluid yields the detachment energy73, 74

E = πR2σ12(1− | cos θ|)2. (23)

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Equilibrium Resistance Detachment

Figure 3: Snapshots of a prolate spheroidal particle of aspect ratio α = 2 under the influence of an externalforce detaching from an interface. For each snapshot, we run a simulation with the particle fixed and measure theresistive force from the interface on the particle (Figures reprinted from Davies et al.18).

For neutrally wetting micron-sized particles at an interface with surface-tension σ12 =50 mN m−1, the detachment energy is much larger than the thermal energy, E/kBT ∼107, and particles irreversibly attach to the interface. For nano-sized particles at the sameinterface with very large or small contact angles, E ∼ kBT , and particles may freelyadsorb to and desorb from the interface19.

For our simulations, we initialise a system volume of size 1283 in lattice units whichis half filled with liquid 1 and half filled with liquid 2 (ρ(1) = ρ(2) = 0.7), such that aninterface forms at x = 64. The top and bottom are closed with solid walls. The particledensity is chosen as ρp = 2 which is an arbitrary choice. The particle is placed at theinterface and is not under the influence of any external forces. We first equilibrate thesystem until the interface diffuses and the particle establishes its equilibrium position, andhence its contact angle, on the interface.33 Then, we apply a constant external force to theparticle. As stated above, the wettability of the particles can be tuned and we determine thecontact angle by subtracting the height of the particle centre of mass above the interface(we linearly interpolate the interface position) and dividing by the particle radius, cos θ =hR .33, 26

To obtain the minimum detachment force, we employ a binary search algorithm: westart the algorithm by using the fact that the particle remains attached at the interface for azero-force, Fatt = 0, and guessing a force which detaches the particle, Fdet. We then runa new simulation with a force, Fnew = 1

2 (Fatt + Fdet) and repeat this procedure until wedetermine Fdet to the desired accuracy. As a next step, we run a single simulation with theminimum detachment force, saving the simulation state frequently. We then run severalsimulations from the saved simulation snapshots (Fig. 3) but now with the particles fixedso that drag, buoyancy and gravity forces can be neglected. We let the systems from eachsnapshot equilibrate, and we measure the resulting force on the particle which is exactlythe resistive force of the interface. This allows us to build a force-distance curve F (x). Wefit F (x) with a fourth-order polynomial, which allows us to capture the linear regime andthe detachment break-off regime accurately, and integrate the fitted function numerically toobtain the detachment energy. The detachment distance is the minimum distance at whichthe resistive force exerted by the interface on the particle is zero. As discussed shortly, theresistive force decreases discontinuously to zero at the point of detachment.

The restoring force provided by the interface to the particle as a function of displace-ment from equilibrium is shown in Fig. 4a. The corresponding fourth-order polynomial

518

(a) (b)

Figure 4: a) The normalized resistive force for a spherical particle of radius R = 10 is approximately linear forsmall displacements as predicted by de Gennes et al.77, 78 and O’Brien.79 The symbols are simulation data andthe dashed lines represent fourth-order polynomial fits to that data. The fourth-order fits are integrated in order toobtain the detachment energy, Eb) Dependence of the detachment energy on the aspect ratio, α, for several different contact angles, θ. Each set ofcoloured data points represents different wettabilities of the particles. Stars are theoretical calculations from ourthermodynamic model in eq. (24) and eq. (25) and symbols are numerical data (Figures reprinted from Davies etal.18).

fits are also shown.Davies et al.18 introduce a simplified thermodynamic model describing the detachment

energies of prolate and oblate spheroids in their equilibrium positions from interfaces.

∆E⊥ =h2

2

(1− α

2G(α)

)− h

2+

α

4G(α), (24)

∆E‖ =h2

2

(1− α

2G(α)

)− h

2+

1

4G(α). (25)

These approximate expressions are simple quadratic functions of the dimensionless heighth and the aspect ratio α. G(α) is a geometry factor. In Fig. 4b we compare the analyticalresults from Eq. (24) and Eq. (25) with our simulation data.

We find good agreement between our thermodynamic model and the numerical simula-tions. For neutrally wetting particles, the measured detachment energies from simulationsare larger than those predicted by the thermodynamic model; this is expected since theparticle has to deform the interface to overcome its resistive force. The differences areof the order of 10%, suggesting that the thermodynamic model, which does not take intoaccount interface deformations, is fairly accurate for the particle aspect ratios we investi-gated. Similarly, we find good agreement for θ = 68 where the numerical data show ahigher detachment energy than predicted by the thermodynamic model, as expected. Forcontact angles θ = 52 and θ = 32, we still find good qualitative agreement betweenthermodynamic theory and numerical simulations for both prolate and oblate spheroids.However, for oblate spheroids the numerical detachment energy is less than the analyticalpredictions, though within errors of the order of the symbol size.

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a)

b)

c)

Caeff = 0.04 Caeff = 0.08 Caeff = 0.12

Figure 5: Side-view of deformed droplets ( a) χ = 0.00, b) χ = 0.27 and c) χ = 0.55). The particles prefer tostay in the center of the channel where the shear flow is weakest. The particles further exhibit tank-treading-likebehaviour: they move around the interface following the shear flow (Figure reprinted from Frijters et al.26).

3.2 Shear induced deformation of particle-armored droplets

Droplets covered by colloidal particles appear as a building block of emulsions, or mightbe produced as individual containers for example for drug delivery purposes. In this sec-tion we review some simulation results on how such droplets behave under an externallyimposed shear flow. The section is based on an article by Frijters et al.26 and the reader isreferred to this article for more details.

A fraction χ of the interface of a droplet immersed in a second fluid is covered withparticles and a shear flow is imposed by moving boundaries at the top and bottom of thesimulation domain, which causes the droplet to deform. The resulting deformation can bequantified using Taylor’s dimensionless deformation parameter80, 81

D ≡ L−BL+B

. (26)

Here L and B are the length and the breadth of the droplet. For small deformations, Taylorpredicts a linear dependence of the deformation of a droplet on the capillary number (seeEq. 27), which becomes D = 35/32Ca for equiviscous fluids. Due to the finite systemsize, we define the capillary and Reynolds numbers in terms of an effective shear rate γeff ,which is measured at the interface of the droplet during the simulation:

Caeff ≡ µmγeffRdσ

, Reeff ≡ ρmγeffR2

d

µm, (27)

These are defined in terms os the dynamic viscosity µm of the medium, the radius of theinitial undeformed droplet Rd and the surface tension σ. The effective Reynolds numberis varied between approximately 0.6 < Reeff < 25.

520

0.04 0.08 0.12Caeff

0.0

0.1

0.2

0.3

0.4

Dd

0.00 0.12Caeff08

1624

Reef

f

(a)

0 40 80 120 160ρp

0.2

0.3

0.4

Dd

(b)

Figure 6: a) The deformation parameter D as a function of the effective capillary number Caeff for χ = 0 (redsquares), χ = 0.27 (blue circles), χ = 0.41 (green diamonds), and χ = 0.55 (purple triangles). The effectof adsorped particles is very weak for low χ, becomes noticeable at χ > 0.4. Taylor’s law is reproduced forsmall Ca (dashed line). The inset depicts that the Reynolds number scales linearly with the capillary number.b) The deformation parameter D as a function of the rescaled mass of the particles m∗p, with χ = 0.55 andCaeff = 0.1. The inertia of the heavier particles causes additional deformation as they drag the droplet interfacein the direction of the shear flow (Figures reprinted from Frijters et al.26).

Let us first understand the effect of nanoparticles on the deformation properties of thedroplet, by discussing how they position themselves at and move over the droplet interface.The fluid-fluid interaction strength is held fixed at gbr = 0.10 and the particles with radiusrp = 5.0 are neutrally wetting (θp = 90) and have a mass density which is identical tothe fluid mass density. The simulation volume is chosen to be nx = ny = 256, nz = 512,with an initial droplet radius of Rinit

d = 0.3 · nx = 76.8, while the number of particlesis varied as np = 0, 128, 256, 320, 384, 446 and 512. This results in surface coveragefractions between χ = 0 and χ = 0.55. The capillary number is changed by changingthe shear rate. Some examples of the deformations thus realised are shown in Fig. 5, forCaeff = 0.04, 0.08, 0.12 and χ = 0.0 (a), χ = 0.27 (b) and χ = 0.55 (c).

The interface of a sheared and thus deformed droplet is increased as compared to itsoriginal spherical shape and more space is available for the particles to move freely overthe interface (cf. Fig. 5). The particles are swept over the interface with increasing velocityas they move away from the center plane of the system and up the shear gradient. If theparticles would not be affected by the shear flow, they would prefer to occupy interface withhigh local curvature as can be explained by a geometrical argument: the interface removedby a spherical particle at a curved interface is larger than the circular area removed from aflat interface. This explains why in this dynamic equilibrium, most particles can be foundat the tips of the droplet (see Fig. 5 b)). Even though the overall structure of the particles onthe droplet interface remains stable over time, individual particles move over the interface,performing a quasi-periodic motion.

The linear dependence of the deformation parameter D on the effective capillary num-ber introduced above is recovered for low capillary number and low particle coverage(cf. Fig. 6a). When the coverage fraction grows beyond χ > 0.40 the deformations inthis regime increase with increasing χ and constant capillary number. Combining Eqns. 27

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results in a relation between the capillary and Reynolds number:

Reeff = σ

(ρmRdµ2m

)Caeff (28)

Since we change Caeff explicitly by changing the shear rate, Reeff is proportional to thecapillary number for a fixed value of the surface tension and inertial effects increase thedeformation. Furthermore, since the particles do not affect the surface tension, all curves ofthe Reynolds number versus capillary number have the same slope (cf. inset of Fig. 6a andEq. (28)). This implies that the increased deformation in the case of added particles is notcaused by changes in inertia of the fluids, but instead the inertia of the particles themselvesplays a decisive role here: As shown in Fig. 6b, we have varied the mass of the particlesover two orders of magnitude. We kept χ = 0.55 and Caeff = 0.1 constant and rescaledthe mass scale with the reference mass: m∗p = mp/524. The particles are acceleratedas long as they are on the part of the droplet interface that experiences a shear flow atleast partially parallel to the particle movement. Eventually, particles have to “round thecorner”. The increased inertia of heavier particles makes it more difficult to change theirmovement, leading to a situation where the droplet interface is in fact initially draggedfarther away in the direction of the shear flow instead. This then explains the increase ofdeformation with increasing particle mass. As our deformation is increased substantially,the system size limits the deformation we can induce. Therefore, the values presented hereare underpredictions of the actual effect of increased mass at high deformations.

3.3 Timescales of emulsion formation caused by anisotropic particles

In this section different types of particle stabilized emulsions and the effect of the particleshape on some of their properties are discussed. The results presented here are a summaryof an article recently published by Gunther et al.35.

We find two different types of emulsions in our simulations, namely the Pickeringemulsion (Fig. 7, left) and the bijel (Fig. 7, right). The choice of parameters (such asparticle contact angle, particle concentration, fluid-fluid ratio, particle aspect ratio) deter-mines the type of emulsions. Parameter studies for emulsions have been discussed in thepast.33, 35, 25 for spherical and ellipsoidal particles, respectively. Here, we limit ourselves toanisotropy effects on the time dependence of the emulsion formation. We use prolate ellip-soids (m = 2; Fig. 7, top), spheres (m = 1; Fig. 7, center) and oblate ellipsoids (m = 1/2;Fig. 7, bottom). The interaction parameter between the two fluids (see Eq. (8)) is chosenas gbr = 0.08 which corresponds to a fluid-fluid interfacial tension of 0.0138. The parti-cles are neutrally wetting (contact angle θp = 90) and the particle volume concentrationis chosen as Ξ = 0.24. The simulated systems have periodic boundary conditions in allthree directions and a side length of LS = 256∆x. Initially, the particles are distributedrandomly. At each lattice node a random value for each fluid component is chosen sothat the designed fluid-fluid ratio is kept (1:1 for the bijels and 5:2 for the Pickering emul-sions). When the simulation evolves in time, the fluids separate and droplets/domains witha majority of one of the fluids form.

The average size of droplets/domains L(t) can be determined by measuring

L(t) =1

3

∑

i=x,y,z

L(t)i, with L(t)i =2π√〈k2i (t)〉

. (29)

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Figure 7: Snapshots of typical simulated Pickering emulsions (left) and bijels (right) after 105 timesteps. Theemulsions are stabilized by prolate ellipsoids (m = 2, top), spheres (m = 1, center) and oblate ellipsoids(m = 1/2, bottom). The parameter determining if one obtains a bijel or a Pickering emulsion is the fluid ratiowhich is chosen as 1:1 for the bijels and 5:2 for Pickering emulsions (Figure reprinted from Gunther et al.15).

〈k2i (t)〉 =

∑k k

2i (t)ς(k, t)/

∑k k

2i (t) is the second-order moment of the three-

dimensional structure function ς(k, t) = (1/ςn)|ϕ′k(t)|. ϕ′ = ϕ− 〈ϕ〉 is the fluctuation ofϕ which is the Fourier transform of the order parameter field ϕ = ρr − ρb.

The time development of L(t) for the three different particle types (prolate, sphericaland oblate (m = 2, 1 and 1/2) and for Pickering emulsions and bijels is shown in Fig. 8a.We can identify three regimes: in the first few hundred timesteps the initial formationof the droplets/domains starts. Then, the growth of droplets/domains is being driven byOstwald ripening. At even later times, droplets/domains grow due to coalescence. Whentwo droplets unify, the area coverage fraction of the particles at the interface is increasedbecause the surface area of the new droplet is smaller than that of the two smaller dropletsbefore. At some point the area coverage fraction of the particles is sufficiently high toprevent further coalescence. The state which is reached at that time is (at least kinetically)stabilized and one obtains a stable emulsion. The values for L(t) are larger for bijels thanfor Pickering emulsions. This can be explained by the way we calculate L(t) (see Eq. (29)and related text) using a Fourier transformation of the order parameter field.

It can clearly be seen that anisotropic particles are more efficient in interface sta-bilization than spheres since they can cover larger interfacial areas leading to smaller

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0.5

1.0

1.5

2.0

2.5

3.0

3.5

0 2 4 6

L [

un

its o

f p

ara

llel p

art

icle

ra

diu

s]

t [105 timesteps]

bijel, m=2 bijel, m=1 bijel, m=1/2 Pickering, m=2 Pickering, m=1 Pickering, m=1/2

(a)

0 5 10 15 20 25 30 35 40

L [

un

its o

f p

ara

llel p

art

icle

ra

diu

s]

t [105 timesteps]

2.85

2.90

2.95

3.00

m=2m=1/2

(b)

Figure 8: a) Pickering emulsion and bijel: Time development of the average domain size L(t) (see Eq. (29)) form = 1 and m = 2. At first view, a steady state is reached after about 105 timesteps. L(t) is larger for bijelsthan for Pickering emulsions, which is due to the measurement being based on the Fourier transform of the orderparameter. Ellipsoids are able to stabilize larger interface areas than spheres leading to smaller L(t).b) Bijel: Zoom form = 2 andm = 1/2: Time development of the average domain sizes depicting the impact ofthe additional timescales. The range of the variation of L(t) is larger as compared to the Pickering emulsion dueto the impact of a small deformation on the larger effective interface of the bijel (Figures reprinted from Guntheret al.15).

fluid domains (note that the simulation volume is kept constant). However, the differ-ence in L(t) for m = 2 and m = 1/2 is small. This can be understood as follows:if a neutrally wetting prolate ellipsoid is adsorbed at a flat interface, it occupies an areaAP,F (m > 1) = m1/3Ap,s, where Ap,s is the occupied interface area for a sphere withthe same volume. This corresponds in the case of m = 2 to the occupied interface beinglarger by a factor of 1.26 as compared to spheres. For an oblate ellipsoid the occupiedinterface area is AP,F (m < 1) = m−2/3Ap,s which for m = 1/2 is by a factor of 1.59larger than the area occupied by spheres. Since in emulsions the interfaces are generallynot flat, these formulae can only provide a qualitative explanation of the behavior of L(t):If the interface curvature is not neglectable anymore, we lose some of the efficiency ofinterface stabilization, which is more pronounced for m < 1. This explains why the valueof L(t) for m = 1/2 is only slightly smaller than for m = 2.

It seems that L(t) reaches a steady state after some 105 timesteps for both types ofemulsions and for all three values of m. However, if one zooms in one can observe thatL(t) develops for a longer time period if the particles have a non-spherical shape. As willbe demonstrated below, the reason for this phenomenon is the additional rotational degreesof freedom due to the particle anisotropy. Furthermore, the time development of L(t) foremulsions stabilized by prolate particles requires more time than that for the oblate ones.If a particle changes its orientation as compared to the interface or a neighboring particlethis generally changes the interface shape. In this way the domain sizes are influenced,leading to changes of L(t) – an effect which is not observed for m = 1.

Fig. 8b depicts a zoom-in of the time development of L(t) for bijels with m = 2 andm = 1/2, respectively. One observes that L(t) decays in both cases. The range of thedecay is larger for m = 2 than for m = 1/2. The time of reordering is much shorterfor m = 1/2 as compared to m = 2. These effects can be explained by the presence of

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additional rotational degrees of freedom for the anisotropic particles. While oblate particleshave only a single additional rotational degree of freedom as compared to spheres, prolateparticles show an even more complex behavior due to their second additional rotationaldegree of freedom.

Th first additional timescale we find can be related to the adsorption of single particlesand their further rotation towards the interface during the initial stages of emulsion forma-tions. However, while the decrease of L(t) in Fig. 8b is certainly of the order of 106 LBtimesteps, we find that the single particle dynamics can only be responsible for effects onscales of the order of 103 timesteps.

By means of time dependent orientational order parameters and pair correlation func-tions we investigate the impact of collective effects of many particles being adsorbed ata flat fluid-fluid interface. For prolate particles, here also the mutual orientation of theparticles is important and one has an additional degree of freedom leading to particle ori-entational ordering. The presence of many particles at an interface leads to two additionaltimescales in the reordering. The first one is the rotation of the particle towards the in-terface. The particle rotates towards its final orientation parallel to the interface. Forhigher particle concentrations the time needed for a particle to come to its final orienta-tion increases. Hydrodynamic as well as excluded volume effects become more important.Above a critical concentration not every particle reaches its “final” orientation. In sum-mary, time scales involved in many-particle effects are found to be of the order of 104 toeven several 105 timesteps.

In emulsions the interfaces are generally not flat. Pickering emulsions usually have(approximately) spherical droplets and a bijel has an even more complicated structure ofcurved interface. The simplest realization of a curved interface is a single droplet. Bysimulating the ordering of many prolate ellipsoids at a droplet surface, it was found thatthe ordering is somewhat faster than in the case of flat interfaces. This is not surprisingbecause capillary interactions between the particles play a substantial role in the case ofcurved interface.

We can understand one of the additional timescales with the behavior of the ellipsoidalparticles at a single droplet. The particles reorder and it can be shown that this leadsto a small change of the shape of the droplet which is (almost) exactly spherical in thebeginning32. A change of the interface shape caused by reordering of anisotropic particlesleads to a change of L(t). The reordering of particle ensembles at flat as well as sphericalinterfaces takes of the order of 105 timesteps. This reordering takes place in idealizedsystems with constant interfaces which do not change their shape considerably. In realemulsions, however, the interface geometry changes substantially during their formation.For example, two droplets of a Pickering emulsion can coalesce. After this unification theparticle ordering starts anew. This explains the fact that the additional timescale we find inour emulsions is of the order of several 106 timesteps.

This reordering is pronounced in the case of curved interfaces, where the movementof the particles leads to interface deformations and capillary interactions. During the for-mation of an emulsion, droplets might coalesce (Pickering emulsions) or domains mightmerge (bijels). After such an event the particles at the interface have to rearrange in orderto adhere to the new interface structure. Due to this, the local reordering is practicallybeing “restarted” leading to an overall increase of the interfacial area on a timescale of atleast several 106 timesteps.

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Our findings provide relevant insight in the dynamics of emulsion formation whichis generally difficult to investigate experimentally due to the required high temporal res-olution of the measurement method and limited optical transparency of the experimentalsystem. It is well known that in general particle-stabilized emulsions are not thermody-namically stable and therefore the involved fluids will always phase separate – even if thismight take several months. Anisotropic particles, however, provide properties which mightallow the generation of emulsions that are stable on substantially longer timescales. Thisis due to the continuous reordering of the particles at liquid interfaces which leads to anincrease in interfacial area and as such counteracts the thermodynamically driven reductionof interface area.

4 Conclusion

We introduced our simulation method for fluid interfaces stabilized by colloidal particles,gave some details of the implementation of a highly efficient and massively parallel simu-lation code and reviewed a number of recent applications.

When an interface is stabilized by colloidal particles, it is of fundamental interest tounderstand how reversible the adsorption of the particles is. For typical particle diametersof the orders of micrometres, the energy required to detach a particle can be of the orderof several thousand kBT . For nanoparticles, however, thermal energies might be of thesame order as the gain in free energy due to the particles being adsorped. In Sec. 3.1and the recent publication by Davies et al.18, we studied this problem in detail and inparticular presented a simplified thermodynamic model to estimate detachment energies ofspheroidal particles. The model shows good agreement with our simulations and allowsthe experimentalist to do ad-hoc estimates in order to for example better understand howstable an emulsion stabilized by colloidal particles of different shapes will be.

Particles and surfactants show some important differences when used for the stabi-lization of fluid interfaces. As shown in Sec. 3.2 and a recent paper by Frijters et al.26,particles do not directly change the interfacial tension, but reduce the interfacial free en-ergy by removing interface area. Furthermore, particles are massive objects and underextreme circumstances such as high mass density and strong shear forces, the inertia of theparticles plays a role. This can, for example, lead to an enhanced deformation of particlearmored droplets in shear flow.

The geometrical shape of the particles used to stabilize an emulsion plays an importantrole for the dynamics and long-term stability of the systems. This was demonstrated inSec. 3.3 and the recent article by Gunther at al.15, where we studied the appearance ofadditional timescales in the dynamics of the formation of emulsions stabilized by oblateor prolate ellipsoids. The dynamics during the adsorption process of a single particle aswell as interfacial jamming or capillary interaction between particles play a crucial role.These effects cause a continuous, long-term reordering of already adsorped particles whichmight be utilized to counteract the inherent thermodynamic instability of these systems byimproved kinetic stabilization.

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Acknowledgments

We thank all the students and collaborators who have contributed to the results reviewedin this lecture and the corresponding original journal articles: Fernando Bresme, Peter V.Coveney, Gary B. Davies, Stefan Frijters, Florian Gunther, Florian Janoscheck, FabianJansen, Badr Kaoui, Timm Kruger, and Qingguang Xie.

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