Diffusion across particle-laden interfaces in Pickering droplets

Emulsions stabilized by nanoparticles, known as Pickering emulsions, exhibit remarkable stability which enables many applications, from encapsulation, to advanced materials, to chemical conversion. The layer of nanoparticles at the interface of Pickering droplets forms a semi-permeable barrier between the two liquid phases, which can affect the rate of release of encapsulates, and the interfacial transfer of reactants and products in biphasic chemical conversion. The current lack of understanding of diffusion in multi-phase systems with particle-laden interfaces limits the optimal development of these applications. To address this gap, we developed an experimental approach for in-situ, real-time quantification of concentration fields in Pickering droplets in a Hele-Shaw geometry and investigated the effect of the layer of nanoparticles on diffusion of solute across a liquid-liquid interface. The experiments did not reveal a significant hindrance on the diffusion of solute across an interface densely covered by nanoparticles. We interpret this result using an unsteady diffusion model to predict the spatio-temporal effect of particles on diffusion across a particle-laden interface. We find that the concentration field of solute is only affected in the immediate vicinity of the layer of particles, where the area available for diffusion is affected by the particles. This defines a characteristic time scale for the problem, which is the time for diffusion across the layer of particles. The far-field concentration profile evolves towards that of a bare interface. This localized effect of the particle hindrance is not measurable in our experiments, which take place over a much longer time scale. Our model also predicts that the hindrance by particles can be more pronounced depending on the particle size and physicochemical properties of the liquids and can ultimately affect performance in applications.


■ Introduction
Pickering emulsions and foams, which are liquid-fluid dispersions stabilized by solid particles instead of molecular surfactants 1 , find a wide range of applications in advanced materials 2, 3 , food science 4,5 , and chemical conversion 6,7 .Solid particles offer superior stability to emulsions compared to molecular surfactants 1 .Depending on the particle radius, a, and the three-phase contact angle, , the energy cost of removing a particle from a liquid-fluid interface can be 10 2 -10 6 times larger than for a molecular surfactant, resulting in long-term stability of foams and emulsions 8 .
The layer of solid particles at the liquid-fluid interface in a Pickering system is partially permeable, a feature that enables controlled release 9 , selective filtration 3 and exchange of reactants and products in chemical conversion 6,7 .Several studies have explored the macroscopic effect of particles on diffusion across interfaces in Pickering systems, for instance during drop dissolution 10,11 , bubble dissolution [12][13][14] , evaporation of liquid marbles [15][16][17] , and compositional ripening 18,19 .Most of these works report that the addition of particles hinders diffusion across the interface, but the effect has not been quantified because the experiments involved a change in interfacial area, which causes morphological changes of the monolayer (buckling or particle expulsion 12,17 ).It is therefore difficult to isolate the effect of the particles in controlled conditions.
Another challenge in quantifying the effect of interfacial particles on diffusive transport across a liquid-fluid interface lies in performing controlled measurements of concentration fields in multi-component, multi-phase systems.Multicomponent systems exhibit rich https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0 phenomenology upon evolution of their composition 20 that can be difficult to control, for instance nucleation and phase separation 21 , compositional Marangoni effects 22 and density changes 23 .Experiments with pH indicators enable to monitor qualitatively the time evolution of composition in reactive surface nanodroplets 24 , but space-and time-resolved measurements of the concentration field and its evolution in multicomponent droplets remains challenging.
Quantitative measurements of concentration fields have been achieved for a gas-liquid system, using calibrated fluorescence intensity measurements of diffusion of CO2 from a single, dissolving bubble in a Hele-Shaw geometry 25 .The limitation of dissolving bubbles or drops is that the motion of the interface imparts a bulk flow, which complicates the analysis of diffusive transport 22,25 and that, due to the gradual decrease in interfacial area, a layer of particles on the interface would change morphology over time.
In this paper, we design an experimental system to characterize the diffusive transport of a trace amount of fluorescent solute across the interface of a Pickering droplet, which results in a constant droplet volume and constant interfacial area throughout the experiment.We use a Hele-Shaw geometry for quantitative mapping of the quasi-2D concentration field of solute as a function of time.The results can be understood in terms of an unsteady diffusion model where the area through which diffusion occurs is determined by the arrangement of particles at the interface.By comparing the experimental results and model predictions with the case of a bare droplet, we reveal the spatio-temporal effect of particles on the diffusion of solute across the interface.These insights will ultimately pave the way towards rational design of complex, multiphase systems for a variety of applications.

Materials
1-Heptanol (synthesis quality), hexadecane (≥99%) and Rhodamine B (RhB) were purchased from Merck Ltd.RhB, a fluorophore with excitation wavelength λex = 540 nm and emission wavelength λem = 625 nm, was selected after screening many fluorophores, because it is soluble both in water and heptanol and emits a strong fluorescent signal in both solvents.Potassium hydrogen phosphate (≥98%) and potassium dihydrogen phosphate (≥99%) were purchased from Brunschwig Chemie B.V. for the preparation of 20 mmol/L potassium phosphate buffer solution (pH = 7), which was used to prepare 0.1 mmol/L Rhodamine B solutions.Partly hydrophobized fumed silica nanoparticles (HDK H15 ® , Wacker Chemie) were kindly provided by IMCD group (the Netherlands).The primary spherical particles of radius a = 5-15 nm form non-spherical, porous aggregates of approximately 100 nm 26 .Their hydrophobic character originates from substituting 50% of surface hydroxyl groups by dimethylsiloxy groups.Milli-Q water was used, if applicable, for the preparation of aqueous solutions.All chemicals were used as received and the corresponding physical properties are summarized in Table 1.Partition coefficients, , were measured by equilibrating the organic and aqueous phase for 2 days and measuring the equilibrium concentrations with UV-VIS spectroscopy (Hach Lange DR5000).The diffusivity of RhB in water was taken from literature 27 , while the values in hexadecane and heptanol were computed via Einstein-Stokes's law using literature values for the viscosities 28 .All experiments were conducted at room temperature and pressure.and sealing the top with a glass coverslip (VMR).The Hele-Shaw geometry was selected to obtain a quasi-2D concentration profile, i.e., independent of the vertical direction.The concentration of dye RhB in the Hele-Shaw cell could then be calibrated against fluorescent intensity as the cell height was the same in all experiments.The fluorescent intensity calibration was performed for water and heptanol by a series of known RhB concentrations.The details are given in Supporting Information (Figure S1).

Preparation of Pickering emulsion and isolation of single droplet
Figure 1 shows the experimental setup and method to quantify the diffusive transport of a solute across the interface of a Pickering droplet with spatial and temporal resolution.First, a 1 wt% suspension of the silica nanoparticles is formed by dispersing the particles in hexadecane via ultrasonication (20 min).A Pickering emulsion was then prepared by mixing (Grant-bio https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0 vortex mixer) an aqueous solution of 1 mmol/L RhB with the suspension of nanoparticles in hexadecane, forming a water-in-oil emulsion (water/oil = 1:1 v/v), as shown in the schematic of Figure 1(a).The droplet sizes obtained were in the range 100-1000 µm, and the interface was densely covered by nanoparticles as confirmed by optical microscopy.
The dye RhB possesses a hydrophobic part and a positively charged group; it is found to adsorb on the partially hydrophobic silica nanoparticles, either because of hydrophobic-hydrophobic interactions, or because of electrostatic interaction with negatively charged silica.Negatively charged dyes (fluorescein and its derivatives) were tested to prevent adsorption via electrostatic repulsion, but were found to either be insoluble, or to not emit a detectable fluorescent signal, in one of the solvents (see Supporting Information).
The emulsion was left to rest for 1 day, to allow the equilibration of the fluorescent dye between the two liquid phases and on the particles.RhB is soluble in hexadecane ( = 1/1) and adsorbs on the H15 silica nanoparticles.The extent of adsorption on the nanoparticles was quantified in a control experiment by breaking the emulsion by centrifugation, recovering the aqueous phase and measuring the RhB concentration by fluorescent intensity.The concentration of RhB in water after equilibration of the Pickering emulsion is found to be approximately 0.1 mmol/L.
Direct measurement of RhB concentration inside a Pickering drop is not possible due to the layer of nanoparticles at the interface.
To perform the experiment, a small volume of emulsion was transferred onto a glass slide and diluted with fresh hexadecane to facilitate isolation of a single Pickering droplet.In this dilution https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0 step, the addition of pure hexadecane causes the concentration of RhB in the droplets to slightly decrease due to partitioning.The concentration of RhB in the droplet after the dilution step can be calculated after the diffusion experiment has taken place, by measuring the total mass of RhB diffused from the droplets as described below.

Experimental measurement of quasi-2D concentration field
The single Pickering droplet was transferred to a Hele-Shaw cell filled with heptanol, in which RhB is 88 times more soluble than in hexadecane, to initiate diffusion from the droplet through Pickering droplet, where the concentration outside the droplet can be quantified by the fluorescent intensity.We expect that RhB also desorbs from the nanoparticles when the droplet is placed in heptanol due to the higher solubility of RhB in heptane compared to hexadecane.
In control experiments we quantified the mass of RhB desorbing from the nanoparticles and confirmed that it is about 2 orders of magnitude lower than the mass diffusing from the bulk of

Diffusion of solute across a bare interface
To model the diffusion of a solute in the case of a bare interface, we assume that the drop in the Hele-Shaw geometry is a cylinder of radius R. The solute is initially present only in the droplet, with uniform concentration C0 and diffuses to the outer phase through the curved interface of the cylinder.Because the volume fraction of solute is less than 0.01%, the volume of the drop is essentially unchanged even when all the solute is transferred to the outer phase.
The interface of the drop therefore remains static and no bulk convective flow is generated, such that we can describe the evolution of the concentration field, C(r, t), by solving the diffusion equation, where we have assumed that the diffusivity D is constant.
For a cylindrical drop, the concentration only depends on the radial coordinate r.In a cylindrical coordinate system with the origin at the center of the drop, Equation (1) reads: By symmetry, the concentration gradient at r = 0 must satisfy In keeping with the experiments, the droplet phase (r < R) is water, where the diffusivity of solute is  ) , and the outer phase (r > R) is oil, where the diffusivity is  * .Equilibrium partitioning of solute, governed by the partition coefficient , is instantaneously established at the water-oil interface, r = R: https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0 Lastly, we assume that the concentration of solute far from the droplet is unaffected, Concentration profiles C(r) at different times t are obtained from the numerical solution of Equation ( 2) with boundary conditions (3-5) and using the values of parameters in Table 1.It is worth noting that penetration theory is not applicable to this situation, because the concentration in the solute-rich phase is not constant in time.

Three-dimensional model of diffusion of solute across a particle-laden interface
To model the diffusion of solute across a particle-laden interface, we consider for simplicity a planar interface between two semi-infinite domains.The assumption of a planar interface is justified by the large drop to particle size ratio in the experiments, R/a ~ 10 3 -10 4 , where R is the drop radius and a is the particle radius.The assumption is also motivated by the fact that here we focus on the region close to the interface to identify the effect of the particles.
We assume that the particles form a monolayer at the interface, separating two semiinfinite, immiscible phases.a.We checked that this domain size is sufficient for convergence of the concentration field.
The concentration at the interface, x = 0, obeys equilibrium partitioning with partition coefficient  [see Eq. ( 4)].The boundary conditions in the y-and z-direction are periodic.The distance from the interface is normalized with the particle radius, a.The time t is made dimensionless by the characteristic time scale for diffusion of the solute over a distance equal to the particle radius (based on the diffusion coefficient in the water phase,  ) ): https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0 The evolution of the 3D concentration field, C(x, y, z, t), was computed numerically using the Finite Element Method implemented in COMSOL Multiphysics ® .The case of a bare interface was obtained by running a simulation without particles on the interface.

Quasi-1D model of diffusion of solute across a particle-laden interface
We propose a quasi-1D model to calculate average concentration profiles as a function of distance from the interface, (, ).In this model, we account for the effect of the particles as a decrease in the area available for diffusion.The diffusion equation for (, ) becomes: where the effective area available for diffusion (i.e., not occupied by particles), (), is calculated as a function of distance from the interface.Based on the simple geometric argument that the cross-sectional area of a sphere centered at  = 0 varies as  1 $ = ( $ −  $ ) [see With this definition, the effective area available for diffusion as a function of distance from the interface is () = N1 − ()O ( , where  ( = ( = 0).Combining Equations ( 7) and ( 8): Equation ( 9) was discretized and solved using a grid size ∆ =  10 ⁄ and a time step ∆ that satisfies ∆ •  (∆) $ ⁄ to ensure convergence.The numerical solution scheme was implemented in MATLAB (The MathWorks Inc.).We confirmed that this quasi-1D model https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0 correctly reproduces the concentration profile, C(x, t), obtained from the full 3D model, as shown in Figure S3 of the Supporting Information.

■ Results and Discussion
Experimental concentration profiles for bare and Pickering droplets

Model predictions for diffusion across a particle-laden interface
We examine the predictions obtained from the 3D model of diffusion across a particle-laden interface, which give access to a wider range of time scales, and to particle-scale details of the concentration field.First, we consider the spatio-temporal effect of particles on the diffusion of solute across a close-packed hexagonal monolayer of particles ( ( ≈ 0.91) at the planar interface between two immiscible phases.To isolate the effect of the particles alone, in this simulation we set  ) =  * and  = 1, so that the liquid-liquid interface is fictitious.
Figure 4 shows the spatial effect of the particles on the isoconcentration contours for dimensionless times  = 2, 8, and 13 .The iso-concentration contours in the (x, z) plane [Figure 4(a-c)] are affected by the particles only in the immediate vicinity of the interface (up to x/a ≈ 1) and become planar as diffusion progresses further (x/a > 1).Iso-concentration contours in the (y, z) plane are shown within the layer of particles (x/a = 0.5) and just outside the layer of particles (x/a = 1) in panels (d-f) and (g-i) of Figure 4, respectively.As can be seen from both (x, z) and (y, z) cross-sections of the concentration field, the diffusion front advances faster in the center of the gaps between particles.This effect disappears at x/a ≈ 1 for  = 13 [see Figure 4(c, i)].
https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0  (d-e) Evolution of iso-concentration profiles in the (y, z) plane at a distance from the interface x/a=0.5 (within the particle monolayer).(g-i) Evolution of iso-concentration profiles in the (y, z) plane at a distance from the interface x/a=1 (at the outer edge of the particle monolayer).Dashed white circles in (d-i) show the outline of the particle at the interface, x = 0.
To examine the evolution of concentration profiles C(x, t) for comparison with our experiments, we apply the quasi-1D model of diffusion across a particle-laden interface.In Figure 5(c), we turn to dimensional times to compare the evolution of concentration profiles with our experimental observations.The dimensional time t is calculated for the particle size in our experiments, a = 15 nm (primary particle size of fumed silica).As can be seen, after t = 0.01 s, the concentration profiles for bare and Pickering interface start to overlap with each other.If we take as particle size a ≈ 100 nm (typical size of fumed silica aggregates), the dimensional time is approximately 100 times larger, with the concentration profile for a Pickering droplet converging to that of a bare droplet after t ≈ 1 s.After this time, the effect of the particles can no longer be observed.Although this comparison is only qualitative -given the assumption in the model of a monolayer of particles, while in experiment the particles may form multilayers or aggregated microstructures -it does explain why our experiments did not reveal a significant hindrance of the particles on diffusion across a Pickering interface (Figure 3).
To examine the effects of surface coverage and of the physicochemical properties of the fluid phases, we compute the cumulative mass of solute transferred to the solute-poor phase, (), from which we extract the diffusive flux at the interface, |

■ Conclusions
We investigated the effect of a layer of nanoparticles on diffusive transport of a solute across a liquid-liquid interface.The diffusion of fluorescent dye Rhodamine B across the particle-laden interface of a single Pickering droplet was quantified experimentally in a Hele-Shaw geometry using calibrated fluorescence intensity.The experiments revealed a limited hindrance of the particle layer on diffusion, which we explained with the help of a simple diffusion model.A 3D model for the diffusion of solute across a particle-laden interface revealed that the spatiotemporal effect of particles is limited to a distance comparable to the particle radius, and a timescale for diffusion over that distance.This finding also helps to rationalize previous observations reported in the literature: for instance, the effect of particles was limited in https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0 the particle-laden interface.Images were captured by a fluorescence microscope (Olympus BXFM upright microscope, 4× magnification) mounted with a digital camera (DCC1645C, Thorlabs, frame rate 0.1 fps), as shown in Figure 1(b).Due to the confinement in the Hele-Shaw cell (h = 175 µm >> a), the Pickering droplet was squeezed to a cylindrical 'pancake'.All experiments presented in the following for bare and Pickering droplets were performed on such 'pancake' droplets.Figure 1(c) shows a typical image of RhB diffusing outwards from a Figure1(c-d), the concentration was not always radially symmetric, because a small

Figure 1 .
Figure 1.Experimental method for quantification of diffusion of solute across the interface of a Pickering

Figure 2 .
Figure 2. Schematics of the diffusion models for particle-laden interfaces.(a) A planar interface separates Dx https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0 Figure 2(a) shows the schematic of the three-dimensional domain containing a liquid-liquid interface stabilized by solid particles, across which the solute is transferred from the water phase to the oil phase.The spherical particles, with three-phase contact angle  = 90°, are arranged in a hexagonal configuration, covering a fraction of the total interfacial area  ( = of particles per unit area.The governing equation, Eq. (1), is solved in Cartesian coordinates with the x-axis normal to the interface.At the surface of the particles, a no-penetration boundary condition is imposed.The boundary conditions far from interface are ( = ) = 0 and ( = −) =  ( , with L = 500

Figure 3
Figure3shows the concentration profile C(r) of solute RhB outside respectively a bare droplet

Figure 3 (
Figure 3(a) shows that the experimental results of RhB diffusion from a bare droplet (R

Figure 3 .
Figure 3. Experimental measurement of diffusion of solute from bare and Pickering drops.R and r are

Figure 4 .
Figure 4. Three-dimensional model of diffusion of solute across a particle-laden interface reveals the

Figure 5
shows the average solute concentration profiles C(x, t).As shown in Figure5(a), at the start of diffusion (τ = 0.1), the difference between the concentration across a Pickering interface (CPickering/C0, solid lines) and that across the bare interface (Cbare/C0, dashed lines) is minimal.From τ = 1 to https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0 10, the difference becomes more significant.At later times (τ = 100 to 1000), the two sets of concentrations get closer with each other.At all times, in the solute-poor phase CPickering/C0 is lower than Cbare/C0, which highlights the hindrance to diffusion caused by the particles.To isolate the effect of the particles more clearly, in Figure5(b), CPickering/C0 is compared with Cbare/C0 by plotting CPickering/Cbare.The spatial gradient of CPickering/Cbare is only significant in the region x/a < 1 and becomes substantially smaller outside this region, approaching zero in the far field, consistent with the qualitative behavior of the iso-concentration profiles in

Figure 5 .
Figure 5. Evolution of concentration profile for Pickering and bare interfaces.The concentration ( is the total area of the interface, comprising both the particle-free area, () = (1 − ()) ( , and the area occupied by particles.The quantities () and | 1'( contain no information on the spatial concentration profile, rather they are a global measure of the hindrance caused by particles on the diffusion process.In Figure6(a) the mass of solute () is compared for different surface coverages by particles,  ( = 0.58, 0.72, 0.91 for a fictitious interface (Dw = Do and  = 1); n is normalized https://doi.org/10.26434/chemrxiv-2023-9w3w7ORCID: https://orcid.org/0000-0002-0887-500XContent not peer-reviewed by ChemRxiv.License: CC BY-NC 4.0 by n0, the total mass initially present in the solute-rich phase.At early times ( < 0,1) the magnitude of n decreases linearly with the particle-free area, () = (1 −  ( ) ( .For later times (for  > 100) the magnitude of n and the diffusion rate 34 35 of a Pickering interface are the same as for a bare interface for all surface coverages considered.

Figure 6 .
Figure 6.Effect of surface coverage and diffusion coefficients on solute transport across a Pickering

Figure 6 (
c) shows the time evolution of JPickering/Jbare for a Pickering interface with  ( = 0.91 and different values of Do/Dw.The solid line in Figure 6(c)represents the reference case of a fictitious interface (Do/Dw = 1).For Do/Dw = 10, the curve is shifted to earlier times because the solute has higher diffusivity in the solute-poor phase than in the solute-rich phase, and for Do/Dw = 0.1 the curve is shifted to later times.The partition coefficient has negligible effect on the time evolution of the flux (result not shown), because we assumed that the concentrations at the interface are instantaneously in equilibrium.For the water-RhB-heptanol system in our experiments, it is Do/Dw ~ 0.1, in which case the time for the flux to reach the value of a bare interface is longer than for a fictitious interface.The results of Figure6(c) suggest that specific combinations of solvents and solute may accentuate the effect of the particle layer on diffusion.

Table 1 .
Physical properties of solute and solvents.