Interaction between Nearly Hard Colloidal Spheres at an Oil-Water Interface

We show that sterically stabilised ("nearly hard") colloids at a water-oil interface behave to a good approximation as charge monopoles. Interparticle potentials, $U(r)$, are extracted via a reverse Monte Carlo scheme which provides a best fit to the radial distribution functions measured by fluorescence microscopy; the results are then validated by particle tracking in a blinking optical trap. We postulate that the long range repulsion we observe arises mainly through interactions between neutral holes on a charged interface. In agreement with this interpretation, we find that the interaction can be tuned by varying salt concentration. The interaction also depends on the nature of the grafted polymer used to stabilise the colloids.

The interaction between particles adsorbed to a liquid interface, and therefore their microstructure, affects the rheological properties of that interface [1].These properties play a role in the formation and stability of systems with large interfacial area, such as particlestabilised emulsions and foams [1][2][3], which have wellknown and widely used applications in the personal care, mineral, and food sectors [4][5][6][7].Understanding the interparticle interaction is therefore important to understand the properties of Pickering systems.
Previous work has considered the microstructure and interactions of charge stabilised particles at liquid-air and liquid-liquid interfaces.Pieranski [8] showed that, for polystyrene particles at a water-air interface, the interaction can be described by a combination of a screened Coulomb potential and a long range dipole-dipole interaction.More recently, studies on polystyrene particles at oil-water interfaces [9,10] concluded that the colloidal repulsion observed there might be due to residual charges either on the oil [9] or water [10] side of the particle.
In contrast, there has been less work investigating the nature of the interaction between sterically stabilised interfacial particles, which can behave as nearly hard spheres [11].Like charge stablised particles, sterically stabilised colloids can be used to stabilise large interfaces.A common particle choice is poly(methyl methacrylate)(PMMA) with polymer hairs grafted to the surface to prevent aggregation due to Van der Waals forces [11,12].PMMA stabilised with poly(12hydroxystearic acid) (PHSA) is often used in dodecane as a model hard sphere system [13,14], although it has recently been noted that when these particles attach to a dodecane-water interface the particles appear to show a long range repulsion [15] -the origin of this source is unclear as these particles have been shown to behave as hard spheres in dodecane [13,14] and are not stable in water [15].Additionally, it was found that PMMA-PHSA particles display a dipole-dipole repulsion on interfaces between water and a cyclohexyl bromide-alkane mixture which is prone to light-induced dissociation [16,17].This arises because PMMA particles suspended in the bromide component acquire an effective charge -this is consistent with them forming colloidal crystals with large lattice spacings in this solvent [18].
In the present work we investigate the long range interaction of sterically stabilised PMMA particles on a dodecane-water interface and propose a new model for its origin.We use two methods to find the pair potential for these particles: fitting of radial distribution functions by a reverse Monte Carlo method and use of a blinking optical trap.We observe a negligible dipole-dipole contribution but a clear screened Coulomb potential in both cases.We find that this interaction can be tuned in two ways.We can bring the particles closer to a hard sphere by changing the stabilising polymer and we can favour aggregation by introducing salt in the water phase.Due to the negligible dipole component, previous models used for charge stabilised particles [9,10] are inapplicable to our system.We instead attribute the long range interaction to the repulsion between neutral holes on a homogeneously charged interface.
We use two types of colloidal particles in this work, PMMA stabilised with PLMA (poly(lauryl methacrylate)) with diameter 2.4 µm and polydispersity of 2.5% (determined by Static Light Scattering, SLS) (synthesised following [19]), and PMMA stabilised with PHSA with diameter 2.2 µm and polydispersity 2.4% (SLS) (synthesised following [13]).These are referred to as PMMA-PLMA and PMMA-PHSA particles respectively.The PLMA has a radius of gyration of 2.5 nm in good sol-vent (n-dodecane; Acros organics, 99%) from Dynamic Light Scattering (DLS), while the PHSA has a radius of gyration of 2.6 nm from DLS and an end-to-end distance of 19 nm when grafted to the colloid surface [20].PMMA-PLMA has a contact angle of 123°at the water/oil interface (determined by a Light Extinction technique, LE [21]) and PMMA-PHSA has a contact angle of 121°at the water/oil interface (LE).
For measurements at a salt solution/oil interface, PMMA-PHSA was used with a diameter of 3.0 µm and polydispersity of 5% (SLS) as well as PMMA-PLMA with a diameter of 2.4 µm and polydispersity 2.5% (SLS).All particles are kept as dispersions in n-dodecane which had been filtered 3 times through an alumina column to remove polar impurities.Distilled and deionized water (Milli-Q, resistivity 18 MΩcm) was used as the subphase in all interfacial experiments.We use sodium chloride solutions to perform measurements with a salt solution subphase at 0.1 M and 1.0 M.
All interfaces were prepared using the same method.A small polytetrafluoroethylene well was filled with water to a sharp aluminium ledge in order to pin the interface.Above the water layer, 3 ml of low volume fraction dispersion ( 0.005%) of PMMA in dodecane was gently spread over the water layer and the flat part of the pinning ledge, shown in Fig. 1(a).This setup was left for 1-2 hours in order to allow the particles to settle at the interface.
A fluorescence microscope (Nikon Eclipse E800, 10×/0.3NA objective) was used to take at least 600 snapshots of the interface at an interval of 1 s -an example snapshot is shown in Fig. 1(b).The radial distribution function, g(r), was found from these images using Python code written in-house.Enough snapshots were taken such that the noise in g(r) (quantified by the standard deviation) was ≤ 0.03 at large separations (where g(r) itself is ∼ 1).These g(r) were then converted to pair potentials, U (r), via a reverse Monte Carlo scheme.A parameterised pair potential was used to run a Monte Carlo simulation and g(r) was extracted from the results.The parameters were then varied to find an optimum fit, corresponding to a minimum in a χ 2 parameter.The parameterised potential and form of χ 2 are given in equation (1), where we have used a screened Coulomb potential as our parameterised potential and ∆ i is the measured error on measurement point i.This parameterisation was chosen as an Ornstein-Zernicke inversion scheme [22,23] for dilute interfacial concentrations shows there is a negligible dipole contribution to the pair potential.From the radial distribution function shown in Fig. 1(c) we can see long range order in this system, with measurable correlations persisting up to ∼ 30 particle diameters.The source of this repulsion is unclear, with PMMA thought to be a hard sphere in dodecane [13,14] and unstable in water [15].
A few remarks are in order.First, there are multiple values of (A, κ) which provide similar values of χ 2 (Fig. 2(c)) -the order of magnitude is the same though.This is expected as phase behaviour should largely depend on the second virial coefficient (rather than on A and κ separately).Second, the fit for PMMA-PHSA is worse (Fig. 2(d)), suggesting that the interparticle potential there might be more complicated.Third, we note the large value of the Debye screening length, κ −1 ∼ 3 µm, implying that the interaction propagates mainly through the oil phase rather than water, which has a maximum Debye length of ∼ 1 µm at very low ionic strengths [24] [25] Based on these considerations, charges on the oil side of the particle [9] and/or finite-ion size effects [10] cannot completely explain our observations as they would fea- ture non-negligible dipole-dipole interactions.For these reasons, we propose an alternative model based on the idea of neutral holes in a charged plane.It is known that water/alkane interfaces can become charged [26,27].Invoking superposition, we can consider that an array of neutral holes on a charged sheet will behave as an array of charged holes on a neutral sheet as far as in-plane interactions are concerned (we neglect the homogeneous electric field perpendicular to the interface as it does not contribute to the pair interaction).The holes will have an effective charge given by Q eff = aσ, where a is the cross-sectional area of the particle at the surface and σ is the surface charge density of the bare liquid interface.These effectively charged holes will therefore repel with a screened Coulomb interaction.
If the repulsion we observe is caused by neutral holes existing in a charged plane, it is fundamentally an electrostatic one.We therefore argue that it should be possible to alter the interaction by adding salt to the water phase, as this will affect electrostatic screening.Figs.3(a,b) shows the results of salt addition in 0.1 M and 1 M solutions with PMMA-PHSA.We observe aggregation [28] into colloidal clusters of self-limiting size (i.e., microphase separation).This can be explained as follows.Initially, the salt screens the electrostatic repulsion, allowing capillary and Van der Waals interactions to facilitate aggregation (see below for a more quantitative discussion of these).As the aggregates grow, the area of interface blocked by that aggregate increases and therefore so does the effective charge of that neutral hole.We therefore observe aggregates eventually behaving as larger, interfacially adsorbed particles which have their own long range repulsion and order.For the PMMA-PLMA case, we see a less dramatic effect upon addition of salt.Measurements of g(r) at comparable surface coverage (but varying aqueous salt concentration) show a decrease in order upon increased salt concentration, quantified by a lowering of the first peak height as salt concentration is increased, Fig. 3(c).This decrease in order can be explained in one of two ways, a decrease in the effective charge or a decrease in the screening length.
A change in effective charge could be achieved by a changing contact angle as the area blocked by the particle is given by A = πR 2 sin 2 (θ).A light extinction technique to measure contact angle [21] was used to check this and it was found that, for PMMA-PLMA, there is no apparent dependence of contact angle on salt concentration.So, the decrease in order with increasing salt concentration is governed by a decreasing screening length, κ −1 .Previously we noted that the relatively large value of κ −1 implies the interaction propagates through the oil phase.We see here that at least part of the interaction propagates through the water phase as well, as NaCl does not substantially dissolve in dodecane.
So far, we have not estimated possible sources of attraction between the particles.As in the bulk, we expect Van der Waals forces should be counteracted by the steric stabiliser.Capillary forces, however, may be present.The Bond number gives the ratio of gravitational to surface tension effects, Bo = R 2 ∆ρg/γ(1−cos θ) where ∆ρ is the density difference between the particle and the lower phase, g is acceleration due to gravity, γ is the interfacial tension and θ is the contact angle [29].For our particles Bo ∼ 10 −8 ≪ 1, indicating that gravitational effects are negligible and therefore there should be no flotation capillary forces [30].Surface roughness, however, due to the polydispersity in stabiliser length, could induce capillary attractions, which may cause attraction when the electrostatic repulsion is suppressed by a change in contact angle and a shorter screening length [31].
A quantitative analysis of the relative strength of repulsion in PMMA-PLMA and PMMA-PHSA reveals a seemingly puzzling discrepancy.In particular, a comparison between Figs. 2(c,d) shows that PMMA-PLMA has weaker interactions compared to PMMA-PHSA, which is evident from the fewer long range oscillations and faster decay in g(r) at the same surface coverage in Fig. 2(c) (or by looking at best fit values for A).This result cannot be explained by a difference in contact angle as both contact angles were found to be the same.To determine the origin of the difference between PMMA-PLMA and PMMA-PHSA at a water-oil interface, we used a blinking optical trap to determine the interparticle potential both at the interface and in bulk dodecane.
A dilute layer (surface coverage ≪ 1%) of particles was adsorbed onto the oil-water interface, and two particles were trapped using a blinking optical trap (BOT) with a power of 0.46W and a wavelength 1064nm (Diode pumped Nd:YAG Laser, IPG photonics).The particles were brought to a separation where the potential is expected to be small.The optical trap then blinked on and off at a frequency of 20Hz.During the time that the lasers were off the particles' motions were tracked and the diffusion coefficient and speeds were measured from respectively mean squared displacement (MSD) vs time and displacement vs time plots.The force was then calculated using the Stokes-Einstein relation where v is the speed, D is the diffusion coefficient and k B T is the thermal energy.This was repeated at closer and closer separations.Interparticle potentials were then calculated via a numerical integration using the cumulative trapezoidal method.This is shown in Figs.4(a) and (b) for both PMMA-PHSA and PMMA-PLMA.The energy curves were fit to a screened Coulomb interaction [32] given in equation (1).The prefactor, A, is given by where ǫ is taken to be the permittivity of the oil phase and κ is the inverse screening length.We take ǫ as the relative permittivity of the oil phase rather than the water phase as the value of κ −1 > 1µm implies the interaction propagates mainly through the oil phase and the majority of the volume of the particle is in the oil phase given a contact angle > 120°.From this, we can calculate the effective charge, Q, and find the surface charge density of the oil-water interface, σ = Q πR 2 sin 2 (θ) .Doing this we find σ = 1.4 ± 0.3 nC•cm −2 .Using the prefactor value obtained from inverting g(r) we obtain σ = 1.7±0.3nC•cm −2 , which is in fair agreement.Differences between BOT and g(r) inversion results can be attributed to the heterogeneity of the interparticle interaction between different particle pairs, where g(r) inversion involves the entire ensemble, whereas the BOT experiment relies on a specific pair of particles [33].
To determine whether our energy curves are better described by a functional form having a screened Coulomb, equation (1), plus dipole term, taken as B r 3 , or a screened monopole term only, we performed a Bayesian model comparison.This analysis shows that the posterior probability ratio for each curve is 4 to 8 in favour of the model with a screened monopole term only.
One might think that this long range force is not caused by the presence of the interface, but there may be a long range force in bulk dodecane.We can see in Figs.4(b) and (d) that this is in fact true for PMMA-PHSA (also see [34]) but not for PMMA-PLMA.There exists a long range force when dispersed in dodecane for PMMA-PHSA which seems to contradict previous measurements [13,14].PMMA-PLMA however, only has a relatively small force, approaching hard-sphere-like behaviour and therefore the long range interaction at the interface is indeed due to the presence of the interface.Here we also see why PMMA-PHSA exhibits more long range order compared to PMMA-PLMA, seen in Figs.2(c) and (d), as the repulsion between PMMA-PHSA is due to a combination of bulk repulsion and interfa-cial repulsion due to neutral holes.The observation of a screening length of 3 µm, i.e. between that of bulk water (1 µm) and bulk dodecane (10 µm from fitting curve in fig 4(c)), aligns with the claim that the interaction propagates through both phases.
In conclusion, we have shown that sterically stabilised, nearly hard-sphere particles exhibit long range repulsion when attached to an oil-water interface.We attribute this to the particles acting as neutral holes in the charged plane of the oil-water interface.It should be noted that this is a very generic statement, applying to any Pickering system where the fluid-fluid interface has a charge.This interaction can be altered by changing the contact angle of the particles used.We can alter this in more than one way, by changing the salt concentration in the water phase and the steric stabiliser of the colloidal particle.
I. Muntz acknowledges studentship funding from EP-SRC under grant number EP/L015110/1.J. H. J. Thijssen acknowledges The University of Edinburgh for a Chancellor's Fellowship.The authors acknowledge A. Law and D. M. A. Buzza for useful discussions, L. Kobayashi Frisk for preliminary measurements, T. Wallner for preliminary measurements and data analysis, and M. Turley and J. Arlt for help in setting up contact angle measurements.

FIG. 1 .
FIG. 1. (Colour online)(a) Schematic of colloidal particle at a liquid-liquid interface.(b) Experimental micrograph (PMMA-PLMA)showing the structure of these colloidal particles when adsorbed to an interface (zoomed in from original image for clarity).Gravity points into the page.Scale bar is 100 µm.(c) The radial distribution function, g(r), extracted from a series of these micrographs at two different surface coverages, φ, 0.89% and 2.53%.The lower surface fraction is shifted by 0.5 for clarity.Errors in g(r) are of the same order as the symbol size.

FIG. 2 .
FIG. 2. (a) Simulated snapshot of particles at an interface, scale bar is 100 µm.(b) Contour plot of χ 2 as a function of λ = κ −1 and A for PMMA-PLMA.Optimal fits are minima in this plot.(c) Comparison of experimental (red line) and simulated (green line) g(r) for PMMA-PLMA.(d) Comparison of experimental (red line) and simulated (green line) g(r) for PMMA-PHSA.

FIG. 3 .
FIG. 3. (a,b) Snapshots of a collection of PMMA-PHSA colloids at a dodecane-salt solution interface, at 0.1M (a) and 1.0M (b) NaCl, showing aggregation upon addition of salt.Surface coverages were 2.45% (for the 0.1M case) and 3.6% (for the 1M case).(These surface fractions lead to similar number fractions of aggregates plus individual particles.)The scale bars are both 100 µm.(c) Plot of g(r) at various salt concentrations of PMMA-PLMA at an water-oil interface.r is scaled by r, the average interparticle separation based on surface coverage.

FIG. 4 .
FIG. 4. (Colour available online) Energy profiles for PMMA-PHSA when adsorbed to a dodecane-water interface (a) and in bulk (c), measured with a blinking optical trap.(b) and (d) show the same for PMMA-PLMA.Separation is core-to-core separation.