Self-assembled nanoparticle-coated interfaces: Capillary pressure, shell formation and buckling

a r t i c l e i n f o


Introduction
Particle accumulation at a liquid/liquid or liquid/gas interface may confer distinctive properties to the interface and support unique applications such as the stabilization of emulsions and foams i.e., Pickering emulsion [1,2], enhanced oil recovery [3], drug delivery [4], or augmented sealing capacity for geological CO 2 storage [5]. The properties of the particle-coated interface evolve during the contraction of the interface. Overall, the interface exhibits a viscoelastic response at small strains and strain rates [6,7] as observed in interfacial rheology measurements [6,8]. Consequently, some deformation analyses have considered the interface as an elastic sheet [9][10][11]. However, elastic models ignore the granular nature of the particle raft, overlook the complexity of particleinterface interactions [12][13][14][15], and fail to capture evolving phenomena during contraction and expansion, such as buckling [16], particle expulsion [16][17][18], and the formation of multilayers [17,19].
Previous experimental studies characterized the deformation of the particle-coated interface using Langmuir-Blodgett troughs and Wilhelmy plates to determine the evolving surface pressure while decreasing the surface area [20][21][22]. Reported results show that the surface stress is nonuniform, probably due to the granular nature of the particle-coated interface and the large surface area of Langmuir troughs [10,23,24].
Furthermore, the flat interface in trough experiments differs from mixed-fluid conditions in porous media where interfaces are small and inherently curved. The capillary pressure DP across a curved interface depends on the interfacial tension c and the mean curvature j as predicted by the Young-Laplace equation: DP = 2cj. Capillary phenomena dominate fluid displacement patterns in porous media [25,26], determine residual saturations [27,28] and give rise to pore-scale instabilities such as Haines jump [29,30] and snap-offs [31,32]. The contraction and expansion of droplets resemble the evolution of fluid interfaces as they traverse a porous medium where they experience sequentially converging and diverging cross-sections. However, only a few studies have investigated the deformation of particle-coated bubbles or droplets [33][34][35], and changes in capillary pressure caused by particlecoated interfaces remain unclear.
This study explores the mechanical response of nanoparticlecoated droplets during contraction and expansion. We measure the capillary pressure, explore the asymmetric buckling behavior of nanoparticle-coated interfaces, and investigate implications on multi-phase flow in porous media.

Setup for droplet studies
The experimental setup resembles the microtensiometer used for measuring capillary pressure, interfacial tension and interfacial rheology [34,36,37], and consists of a precision syringe pump, a high-resolution pressure transducer, and a glass capillary immersed in a liquid bath (Fig. 1a). Experiments use borosilicate glass capillaries (Drummond Scientific Company, ID = 0.56 mm and OD = 0.79 mm). We covalently graft perfluorodecyltrichlorosilane (FDTS) through molecular vapor deposition for tests that require hydrophobic capillaries (MVD100E, AMST). The advancing h A and receding h R contact angles on the hydrophobic FDTScoated glass surfaces are h A = 105°± 1°and h R = 72°± 1°(2 lL deionized water drops dispensed/retracted at 0.2 lLÁs -1 ).
Tests involve a nanofluid (described below) and mineral oil (Mineral oil light, Sigma-Aldrich; density q = 877 kg/m 3 , dynamic viscosity l = 21 mPaÁs and oil-water interfacial tension c ow = 34.2 mN/m). Experiments start by filling the bath with either fluid; then the syringe pump (Braintree Scientific, Inc., BS-8000) delivers the other fluid to the tip of the glass capillary to create a droplet of the desired volume. The droplet rests for a preset adsorption time t ad to allow nanoparticles to diffuse and selfassemble at the interface. Then, we reduce the droplet volume by withdrawing the fluid at a low flow rate (q = 0.2 ml/hr) to ensure that the nanofilm is in a quasi-static condition during contraction. A microscope (Stemi 2000-CS, ZEISS) records the droplet deformation and the pressure transducer (PX40, OMEGA) measures the capillary pressure throughout the droplet expansion and contraction stages. The pressure difference due to viscous loss between the pressure transducer and the oil-water interface is in the order of 1 Pa for the flow rate used in these experiments, and it is much smaller than the range of the measurements (>100 Pa). We calibrate the pressure transducer using hydrostatic pressure to remove any offset caused by trapped air. There are no dynamic instabilities in our experiments, therefore, the system compliance does not affect pressure measurements [29,38,39].

Nanofluids: concentration optimization
Our experiments involve silica nanoparticles (Sigma-Aldrich; size d = 88 nm from dynamic light scattering -Zetasizer ZS). We modify the nanoparticle surface with the cationic surfactant cetyltrimethylammonium bromide CTAB (Sigma-Aldrich, purity l Dynamic viscosity (mPaÁs) / Polar angle (°) >99%). The preparation of the nanofluids starts by mixing nanoparticles with the CTAB solution at a solids fraction of 5 g/L, followed by sonication for 40 min. We vary the CTAB concentration from 10 -6 M to 2 Â 10 -3 M. Nanoparticles are negatively charged (zeta potential f = À60 mV, measured by Zetasizer Nano ZS, Malvern Instruments), therefore, CTAB adsorption to the silica nanoparticles prevails and the surfactant is almost completely removed from the bulk aqueous phase. The interfacial tension measured for the supernatant water after centrifugation is indistinguishable from the deionized water interfacial tension (see also [40]).
Preliminary droplet contraction experiments show that: (1) the interface behaves as a liquid-liquid interface when the concentration of CTAB is less than 10 -5 M; (2) granular jamming and shelllike interfaces develop when the concentration of CTAB is between 10 -5 M and 2 Â 10 -4 M; and (3) the nanofluid becomes unstable at higher CTAB concentrations (Fig. 1b). Furthermore, there is evidence that NaCl and NaOH promote the adsorption of surfactants and increase the stability of nanofluids [41]. Consequently, the engineered nanofluid used in all tests reported in this article consists of 5 g/L silica nanoparticles, 8 Â 10 -5 M CTAB, 1 mM NaCl, and has a pH = 10 (adjusted by NaOH).

Results
We tested two different scenarios: (a) the contraction of a ''nanofluid droplet in an oil bath" and (b) the contraction of an ''oil droplet in a nanofluid bath". In both cases, we used FDTS-coated capillary tubes and allowed t ad = 10 min for adsorption. The following results highlight salient features of particle-coated interfaces observed in multiple repeatable experiments. Stage B (images 2 and 3): while the droplet remains spherical, the capillary pressure decreases rapidly towards DP ? 0 during this stage. Apparently, repulsive particle-particle interactions reduce the apparent interfacial tension to minimal values c A ? 0.

Contraction of a nanofluid droplet in an oil bath
Stage C (images 3 to 5): a distinct shell-like response emerges and the particle-coated interface begins to wrinkle. The capillary pressure remains constant and very low DP = 10 Pa as the droplet contraction continues.
Stage D (image 5): the capillary pressure decreases below zero and the film folds into the capillary tube.

Contraction of an oil droplet in a nanofluid bath
The response of an ''oil droplet in a nanofluid bath" is strikingly different from the case of the ''nanofluid droplet in an oil bath" shown above. There are three salient stages in this case ( Fig. 2b see Supplementary Material V2 for a complete video).
Stage A (before image 1): the droplet remains spherical as it contracts at a constant pressure.
Stage B (images 1 to 4): the pressure decreases monotonically while the droplet contracts with convex shapes. Eventually a granular shell forms. The stiff shell contracts past the tube edge and a few localized wrinkles initiate around the tube edge (images 3 and 4). The capillary pressure DP = P oil -P nanofluid decreases below zero and the granular shell resists contraction; in fact, the particle-coated interface sustains a significant negative pressure difference DP = À100 Pa.
Stage C (images 4 and 5): finally, the particle-coated interface experiences a sudden collapse and the shell loses its ability to resist the large pressure difference DP (image 5). Changes between images 4 and 5 suggest a pronounced re-structuring of the nanoparticle assembly at the interface as early localized wrinkles vanish after the shell collapses.

Effect of boundary wall wettability on the buckling pattern
Buckling patterns appear to be affected by the boundary conditions against the capillary wall. We tested this observation by comparing contraction experiments for a ''nanofluid droplet in an oil bath" using both FDTS-coated glass capillaries and hydrophilic uncoated glass capillaries (Note: all other experimental parameters are identical). Multiple experimental observations show that hydrophobic FDTS coated glass capillaries always produce crumpling patterns (multiple folds, Fig. 3a -Supplementary Material V1) while hydrophilic glass capillaries trigger buckling patterns (a few folds, Fig. 3b -Supplementary Material V3). Apparently, the hydrophobic interaction between the FDTS-coated glass and the CTAB-modified silica particles generates a ''pinned joint" which constrains the displacement between the particles at the interface and the glass capillary [42]. On the other hand, the untreated glass capillaries lead to a ''roller joint" so that particles at the interface displace along the glass edge and only a few large folds emerge (Fig. 3c).

Particle adsorption at the oil-water interface
A solid particle's affinity for water defines its equilibrium position at the interface. The free energy DE associated with the migration of one spherical particle of radius a from the water phase to a planar oil-water interface is a function of the solid-water contact angle h and the oil-water interfacial tension c ow ( [43], Fig. 4a): Clearly, completely hydrophobic (h = 180°) or hydrophilic (h = 0°) particles prefer to be fully immersed in oil or water, while partially hydrophobic silica nanoparticles take an intermediate position at the interface. Soft particles, such as microgels, experience flattening at the interface because of interfacial tension which could result in higher adsorption energy compared with hard particles [44].
In their initial condition, silica particles are hydrophilic and show no preferential adsorption to the oil-water interface. However, the addition of the surfactant CTAB modifies their wettability and equilibrium position at the interface. Fig. 4b suggests the potential interaction between the negatively charged silica nanoparticles and CTAB as a function of the surfactant concentration: CTAB coverage gradually increases until its molecules saturate all sites on the nanoparticle surface. Hydrophobic interactions between carbon chains lead to the adsorption of additional CTAB molecules on the surface, and the formation of micelles at high CTAB concentration. Thus, the wettability of silica particles changes from hydrophilic to hydrophobic with the increase in CTAB concentration, and reverses back to hydrophilic at high concentrations [40]. The intermediate CTAB concentration selected in this study to promote silica nanoparticle adsorption onto the oil-water interface, in agreement with these observations (Fig. 1b).

Apparent interfacial tension for particle-coated interfaces
The apparent interfacial tension of particle-coated interfaces combines the interfacial tension of the liquid-liquid interface and inter-particle interactions. For simplicity, let us consider an oilwater interface with interfacial tension c ow . Nanoparticles rest on the interface at an inter-particle distance H and experience an inter-particle repulsive force F(H) which is a function of the particle-particle distance. Particles typically arrange in a hexagonal pattern when dispersed at an interface [45,46], yet, other patterns such as square packing have also been predicted and experimentally observed [47][48][49][50]. Consider then the hexagonal and square patterns sketched in Fig. 5. The inter-particle force F is the sum of all particle-particle interactions, which may include electrostatic repulsion, Van der Walls attraction, hydrophobic attraction, and contact force [42,45,51]. The line integral of the tension along the perimeter of the representative area relates the apparent interfacial tension c A to the oil-water interfacial tension c ow and interparticle forces F. The apparent interfacial tension c A is: In general, the apparent interfacial tension c A for nanoparticle coated interfaces is: where b is a packing factor: b ¼ 1 for square pattern and b ¼ ffiffiffi 3 p for hexagonal pattern. Clearly, repulsive particle-particle interactions  reduce the apparent interfacial tension. The long-range electrostatic repulsion is dominant before particles jam [45]; thereafter, contact force repulsion opposes hydrophobic attraction.

Particle assembly and re-arrangement during droplet contraction
The particle assembly at the interface evolves in time as particles diffuse from the nanofluid to the interface. Furthermore, an initially loose particle packing at the interface gradually tightens and particles undergo in-plane rearrangement during droplet contraction. Eventually, the granular interface buckles and shears.
Let us assume spherical droplets of radius R, and irreversible particle adsorption to render a surface coverage of particles c = A particle /A interface . After a certain adsorption time t ad , the droplet is at the initial state (R 0 , c 0 ) shown as Point 1 in Fig. 2a (Fig. 6a). The inter-particle force F(H) is small when the surface coverage is low and the particle-particle distance H is large. As the droplet contracts, particles come closer together and the apparent interfacial tension c A gradually decreases (Eq. (2c)). The packing density increases and reaches the jamming condition (R L , c L ) identified as Point 2 in Fig. 2a (Fig. 6a); repulsive interparticle contact forces reduce the apparent interfacial tension and there is a rapid reduction in capillary pressure.
The granular packing evolves and nanoparticles finally reach a dense packing state (R D , c D ) identified as Point 3 in Fig. 2a. We assume that the limiting packing density at the onset of buckling corresponds to the hexagonal packing, c D = p/(2 p 3) = 0.907 (last stage in Fig. 6a).
The typical duration of a droplet contraction experiments (time scale in seconds - Fig. 2) is much shorter than the adsorption time  t ad (time scale in minutes - Fig. 6). Therefore, the increase in the number of particles at the interface during the test is negligible. Then, let's consider a constant number N 0 of nano-particles of radius a forming a monolayer at the interface. The evolving droplet radius R and the surface concentration c during droplet contraction satisfy the following geometric constraint: Therefore the droplet size ratios are related as (refer to Fig. 6): at jamming : at buckling : and : Fig . 6b shows the experimentally measured ratios R D /R 0 , R L /R 0 , and R D /R L for droplets contraction tests run with different adsorption times t ad = 5, 10, 30 and 120 min. The droplet size ratio R D /R L is around 0.9 and is independent of the adsorption time. The computed surface coverage at jamming is c L % 0:73 (from Eq. (4c)) which is smaller than the coverage for a square packing (c = 0.785); this hints to the role of electrostatic interactions on packing stability.
The size ratios at jamming R L /R 0 and at the verge of buckling R D / R 0 increase with adsorption time (Fig. 6b); in other words, the initial surface coverage c 0 increases with adsorption time t ad . Fig. 6c shows the relationship between the initial surface coverage and the adsorption time calculated from the droplet radius R D at the verge of buckling and the initial droplet size R 0 : While diffusive transport defines the initial concentration c 0 at time t ad before droplet contraction, the adsorption rate decreases with increasing surface coverage because of steric effects at the interface and the time-decreasing concentration of nanoparticles within the droplet volume [52][53][54].

Inter-particle forces
Differences in capillary pressure measured for water-oil droplets (i.e., without interacting particles) and nanofluid-oil droplets (i.e., with interacting particles) reflect the strength of particle interactions (Fig. 2a). The force equilibrium analysis along a droplet equatorial plane relates the interfacial tension c ow to the capillary pressure DP = P oil -P w (Fig. 7a), Therefore, the capillary pressure is always positive for an oil droplet in water because of the interfacial tension.
However, experimental result shows that a particle-coated droplet is able to withstand negative capillary pressure DP = P in -P out (Fig. 2a and b). An analogous Laplacian-type analysis for a particle-coated droplet DPpR 2 = 2pRc A involves the apparent interfacial tension c A (Eq. (2)). For contacting particles (Fig. 7b), Therefore, the inter-particle force F acting on a nanoparticle is: square pattern : The calculated particle-particle contact force ranges from F = 4.9 nN (hexagonal packing) to F = 8.5nN (square packing) for the measured capillary pressure DP = -100 Pa (Fig. 2b), droplet size R = 1 mm, nanoparticle radius a = 50 nm, and interfacial tension c ow = 34.2mN/m (Note: these interparticle contact forces correspond to~1 MPa in-plane stress for a monolayer thick shell). Clearly, self-organized nanoparticles on the surface of small contracting droplets can sustain higher interparticle forces that those obtained from Langmuir trough experiments [55]. In either case, the strong hydrophobic attraction between the adsorption layers of silica particles contributes to shell stability [55,56].

Asymmetric behavior
Droplet contraction tests for an ''oil droplet in a nanofluid bath" and for a ''nanofluid droplet in an oil bath" exhibit distinct shapes and capillary pressure signatures (Fig. 2). This observation indicates an asymmetric nanoparticle-coated interface behavior. Consider a particle at a liquid-liquid interface (see optical microscopy images in [21]; freeze-fracture SEM images in [57]; and cyro-SEM images in [8]). Since the buoyant weight is negligible for submicron particles, the relative position of the particle at the interface depends on the contact angle h. The capillary force F c that acts on the spherical particle of radius a when it is displaced to a polar angle / with respect to the oil-water interface at equilibrium is ( Fig. 8a -see also [58]): Fig. 8b plots the computed capillary force F c as a function of the polar angle / for various values of the contact angle h. The capillary force increases when the particle deviates from its equilibrium position and reaches maxima at / = h/2 and / = À(p-h)/2.
Let's extend the single particle analysis to a hinged particle arch with an inter-particle angle a (Fig. 8c). The free-body analysis of the intermediate particle 2 shows that the resultant of contact forces F 12 (the inter-particle force between particles 1 and 2) and F 23 (between particle 2 and 3) is normal to the arch and is balanced by the capillary force F c .
This analysis allows us to compare a particle chain forced to bend towards either the wetting phase (oil - Fig. 8d) or the nonwetting phase (water - Fig. 8e). The interparticle force required to bend the chain towards the wetting phase decreases as the bending angle a increases, and failure happens as soon as bending starts (Fig. 8f). However, when the chain is forced to deform towards the non-wetting phase, the force required to bend the chain increases with the bending angle a, the particle chain is self-stabilizing, and it can sustain a significant out-of-plane loading. The maximum normal force estimated by the particle chain model (F N = 7.8 nN -Eq. (9)) is consistent with the particleparticle force inferred from measurements (F N = 4.9 nN-to-8.5 nN -Eq. (7)). Admittedly, other effects such as rolling friction and contact angle hysteresis may play a role but are not considered in this first-order analyses [59][60][61]. Results from interfacial rheology measurements show that the shear modulus of particle monolayers also depends on the particle wettability [8,62].
The position of the interface relative to the nanoparticles creates a preferential bending direction towards the liquid interface with respect to the interparticle contacts and biases the direction of instabilities: a ''nanofluid droplet in an oil bath" develops protruding ridges (Fig. 9a), while an ''oil droplet in a nanofluid bath" buckles towards the inside forming invasive crevices (Fig. 9b).

Implications: effects on immiscible displacement in porous media
Fluid saturated porous media often experience the invasion of an immiscible fluid. This common phenomenon has important implications to environmental remediation (LNPL's and DNPLs in fresh water aquifers), resource recovery (enhanced oil production) and waste disposal (geological CO 2 storage). Interconnected pores in soils and rocks consist of sequentially converging and diverging cross-sections, thus, fluid interfaces contract and expand as they traverse the porous medium. Previous studies show that transient changes in surfactant concentration affect the evolution of the capillary pressure across pore constrictions [63]. Similarly, we anticipate that changes in particle concentration can alter the invasion of a particle coated immiscible fluid interface.
A single-constriction experiment illustrates the effect of particle-coated interfaces on the capillary pressure during oil displacement ( Fig. 10-a &b -see Supplementary Material V4 for the complete video; for comparison, the capillary pressure evolution for a similar test without nanoparticles is provided in Supplementary Material S2). The constricted capillary tube connects to the syringe pump (Braintree Scientific, Inc., BS-8000) and a pressure transducer (PX40, OMEGA) mounted next to the inlet monitors the pressure (see details for a similar setup in [63]). The nanofluid is gradually injected into an oil filled capillary tube. The capillary pressure DP = P oil -P nanofluid increases as the interface curvature decreases near the pore constriction and the water-based nanofluid invades the capillary (Phase A -before point 1). However, the capillary pressure starts to decrease with increased curvature as the granular shell forms and arches against the pore wall (Phase B -points 1 to 4). The shell wrinkles and crumples as it squeezes through the pore throat (points 3 and 4). Then, the interface   10. The influence of a particle-coated interface on oil displacement. (a) A particle-coated interface passes through a pore constriction. (b) The evolution of capillary pressure DP = P oil -P nanofluid . Note: numbers of markers refer to the sequence of micro photographs. Uncoated borosilicate hydrophilic capillary tube with internal diameter ID = 0.97 mm; constriction diameter 0.3 mm. expands and the capillary pressure recovers after passing the constriction (Phase C -points 5 and 6).
The interface expands after constrictions, and particles rearrange by self-assembly. As the interfacial tension homogenizes, the droplet recovers the Laplacian spherical shape (point 4).
Asphaltenes, biofilms and fine particles can adsorb at oil-water or contaminant-water interfaces and may exhibit similar behaviors as the particle-coated interfaces explored in this research [64][65][66][67]. Clearly, the formation of particle-coated interfaces during multiphase fluid flow through complex pore networks can have important implications for engineered and natural processes.

Conclusions
We investigated the mechanical properties of silica nanoparticle-coated interfaces by analyzing the shapes of interfaces during droplet contraction and expansion, and the associated capillary pressure signature. Our study and analyses emphasize the particulate nature of the modified interface.
Initially, particles diffuse and reach the interface where they self-organize. The interface undergoes a transition from fluid-like to shell-like as the droplet contracts and the self-assembled particles jam at a loose packing configuration. Then, packing evolves to a dense configuration before the particle-coated interface buckles. Particle-particle interactions result in a reduced apparent interfacial tension for the particle-coated interface. In fact, the apparent interfacial tension may become negative, as well as the capillary pressure.
Previous work observed various buckling patterns when particle-coated interfaces were compressed and suggested that the elastoplastic characteristics of the thin film determined the deformed topology [15,35,68]. Here, we show that the buckling patterns also depend on the interaction between the particlecoated interface and the solid boundary. Constrained displacement (''pinned joints") leads to crumpling patterns, while free displacement (''roller joints") results in a few buckles.
Furthermore, results demonstrate that particle-coated interfaces exhibit asymmetric mechanical behavior. The ''nanofluid droplet in an oil bath" buckles at a negligible pressure difference DP ? 0 in agreement with previous work [34], while an ''oil droplet in a nanofluid bath" withstands a significant pressure difference. A particle-level analysis shows that the liquid-liquid interface contributes a larger restoring force when the particle chain bends towards the non-wetting phase. This asymmetric mechanical behavior creates a preferential bending direction and biases the direction of instabilities.
Our experimental results also show that the contraction and expansion of particle-coated interfaces affect multiphase displacement in converging-diverging pores. This pore-scale phenomenon remained undetected in previous core-flooding experiments and pore-scale observations [3,69]. In particular, we show that the formation of the granular shell-like interface hinders the increase in capillary pressure during the invasion of wetting fluids across pore constrictions. Therefore, the presence of nanoparticles at fluid interfaces alters mixed fluid displacement in porous media. This interfacial phenomenon may have significant implications for environmental remediation, oil recovery and CO 2 injection.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.