3.1. Experimental characterisations of static wettability and theoretical analysis

Wettability characterisations were first performed under static conditions. The plate-based measurements aim to quantify the intrinsic solid–fluid affinity, providing general insights into the intrinsic wettability dependence of nanoparticle effects. As shown in figure 1(a), for NAPL droplets, nanoparticles consistently shifted the apparent wettability toward more hydrophilic states under all conditions. Under moderately and weakly water-wet conditions, the aqueous phase contact angle $\theta _{\textit{eff}}$ of nanoparticle suspensions both decreased to $\sim 15^\circ$ , corresponding to a strongly water-wet state.

Several microscopic phase states can exist between the non-aqueous droplet and the nanoparticle-modified substrate: the wetting phase surrounding adsorbed particles is either depleted (state I), retained between particles and the substrate (state II) or stabilised among neighbouring particles (state III). Constrained by intrinsic wettability at the surfaces, the corresponding contact angles can be derived by extending classical wetting models (Quéré Reference Quéré2008), given by $\cos \theta _{{I}} = \cos \theta _w +\, 4 \alpha \cos \theta _p$ , $\cos \theta _{{II}} = \cos \theta _{{I}} + 2 \alpha \cos \theta _p (1 - \sin \theta _w / \sin \theta _p )$ and $\cos \theta _{{III}} = -\alpha (\cos \theta _p - 1)^2 + 1$ , where $\alpha$ is the projected surface coverage, and $\theta _p$ and $\theta _w$ are intrinsic contact angles of the particle and wall surfaces, respectively. The thermodynamically stable state corresponds to the largest contact angle for the surrounding phase, giving $\theta _{\textit{eff}} = \max (\theta _{{I}}, \theta _{{II}}, \theta _{{III}})$ . Detailed derivations are provided in the supporting information. To clarify the role of solid surface adsorption, treated substrates were characterised using SEM. Similar monolayer structures and adsorption capacities were observed across the substrates after 1 h of suspension treatment (insets of figure 1 b), with only minor variations in surface coverage and an average coverage of $\alpha _m = 0.31 \pm 0.05$ (standard deviation over all conditions). Substituting $\alpha _m$ and $\theta _p$ (unmodified silica nanoparticles, $\theta _p = 52^\circ$ ) into the expression for $\theta _{\textit{eff}}$ , the relationship between $\theta _{\textit{eff}}$ and $\theta _w$ can be predicted theoretically. The good agreement between theoretical and experimental results (figure 1 b) quantitatively confirm that nanoparticle adsorption on solid surfaces drives the wettability changes.

In a three-phase system, static wettability can be characterised under two configurations, depending on which phase is defined as the droplet phase. In principle, phase configuration should not strongly affect measured wettability, as it intrinsically reflects the interfacial property. However, as shown in figure 1(c), nanoparticles produced negligible changes in $\theta _{\textit{eff}}$ for aqueous droplets. The variations fell mostly within experimental uncertainty, indicating that nanoparticle adsorption became ineffective.

What causes the transition from significant to negligible wettability alteration? Figure 1(d) illustrates microscopic interfacial states under different configurations. In the NAPL droplet system, the surface is initially occupied by the aqueous phase, and the droplet deposition follows a drainage process. Nanoparticle adsorption occurs symmetrically on both sides of the liquid–liquid interface, increasing the energy difference and thereby decreasing $\theta _{\textit{eff}}$ , as detailed in contact angle derivations. In the aqueous droplet system, adsorption is limited inside the droplet. Substituting the asymmetric adsorption states into the energy balance yields $\cos \theta _{\textit{eff}} = \cos \theta _w - 4 \alpha \sigma _{{NP}}/\sigma \lt \cos \theta _w$ , where $\sigma _{{NP}}$ denotes the water–nanoparticle interfacial energy. This predicts $\theta _{\textit{eff}} \gt \theta _w$ , inconsistent with our observations. The contradiction arises because $\theta _{\textit{eff}}$ should depend only on variations in properties at the contact line (Shardt & Elliott Reference Shardt and Elliott2020). For confined adsorption within the droplet, the contact-line region is barely affected, even though the total energy on the aqueous side is substantially modified. Consequently, the thermodynamic state of the interface remains nearly unchanged, yielding $\theta _{\textit{eff}} \approx \theta _w$ . The discrepancies in nanoparticle-induced wettability variations in earlier studies are also rationalised by the analysis documented earlier: significant wettability alteration from oil-wet to water-wet was observed predominantly for NAPL droplets (Al-Anssari et al. Reference Al-Anssari, Barifcani, Wang, Maxim and Iglauer2016; Alzobaidi et al. Reference Alzobaidi, Wu, Da, Zhang, Hackbarth, Angeles, Rabat-Torki, MacAuliffe, Panja and Johnston2021), whereas relatively minor contact angle changes were reported for aqueous droplets (Lim et al. Reference Lim, Horiuchi, Nikolov and Wasan2015).

Inspired by the striking contrasts in static wettability, we revisit nanoparticle effect on wettability under flow conditions. Evolution of the displacement interface closely resembles the configuration for aqueous droplets, with the pore space initially occupied by the non-aqueous phase. Nanoparticle adsorption lags interfacial propagation and can modify only surfaces already in contact with the aqueous phase. Therefore, wettability alteration requires pre-existing wetting films that allow nanoparticles to access NAPL-occupied surfaces (figure 1 d) and break the interfacial equilibrium. For an isolated droplet on a smooth substrate, this is difficult unless complete wetting occurs. For microflow systems, spontaneous corner flow provides a feasible way for pre-spreading of wetting films, with the criterion given by $\theta _w \lt 90^\circ - \beta /2$ , where $\beta$ is the corner angle (Concus & Finn Reference Concus and Finn1969). For etched microchannels, the corners are always right-angled ( $\beta = 90^\circ$ ), thus $\theta _w \lt \theta _{w,{cr}} = 45^\circ$ , serving as the criterion for nanoparticle-induced wettability alteration during displacement.

Figure 2. Validation of the wettability criterion for displacement in a strongly hierarchical disordered medium by microfluidic experiments. (a) Schematic of the porous structure with grey indicating solid and white indicating pore space. (b) Long-tailed pore size distribution. Inset: top-down view of a local structure with hierarchical features; arrows indicate smaller and larger pores. (c) Comparison of final phase distributions. Significant changes occurred only under the moderately water-wet condition ( $\theta _w = 36^{\circ }$ ). (d,e) Variations in the aqueous phase saturation $S_a$ at the (d) breakthrough and (e) final stages versus $\theta _w$ . (f) Variation in the Euler number of the aqueous phase $\chi _a$ versus $\theta _w$ . Dashed lines in (d–f) denote the wettability criterion ( $\theta _w \lt \theta _{w,{cr}}$ ).

3.2. Displacement performance and mechanisms in a disordered medium

To validate the wettability criterion, displacement experiments were performed in a disordered medium with complex pore geometry (figures 2 a and 2 b). As presented in figure 2(c), nanoparticle suspensions produced significant changes in phase distribution relative to brine only when the corner-flow criterion was satisfied (moderately water-wet, $\theta _w = 36^{\circ }$ ). The invading pattern became more compact with increased branching, corresponding to a more water-wet state. Figures 2(d) and 2(e) show substantial increases in aqueous phase saturation $S_a$ when $\theta _w \lt \theta _{w,{cr}}$ , which expanded after breakthrough. In contrast, variations in $S_a$ were negligible under the weakly water-wet and nearly neutral conditions, where corner flow was inhibited. Differences in material properties other than intrinsic wettability cannot explain these observations, as adsorption capacities on the substrates were similar (figure 1 b). As shown in figure 2(f), the pronounced reduction in the total Euler number of the aqueous phase ( $\chi _a = N_a - L_a + O_a$ , where $N_a$ is the number of isolated units, $L_a$ is the number of redundant loops, and $O_a$ is cavities with $O_a = 0$ for 2D topology) reflects more connected invasion networks by nanoparticle-induced wettability variation.

To identify the displacement pattern controlled by nanoparticles, topological features under the moderately water-wet condition were quantified and compared. Based on the Euler number $\chi _n$ (calculated per ganglion), residual NAPL was categorised into isolated ganglia ( $\chi _n = 1$ ) and clusters ( $\chi _n \lt 1$ ). Using the shape factor $c_n = 4\pi A_n / p_n^2$ ( $A_n$ is the area and $p_n$ is the perimeter), isolated ganglia were further classified as droplets ( $c_n \ge 0.5$ ) or columns ( $c_n \lt 0.5$ ). As shown in figure 3, when initial mainstream paths were established at breakthrough, the phase distribution, droplet ratio and ganglion number were close for brine and suspension displacement. After breakthrough, phase topology showed negligible change for brine due to relatively weak film flow. In contrast, suspension injection generated large amounts of droplets at the post-breakthrough stage, indicating continuous detachment of residual NAPL. Droplet formation was driven by enhanced wetting-film development, which was visible even in deep stagnant zones occupied by NAPL (figure 3 a). The monotonically increasing trend of the droplet ratio (figure 3 b) aligns with pore-scale observations, while fluctuations in the ganglion number (figure 3 c) indicate that droplets were not only peeled off but also mobilised. These results suggest that nanoparticle-induced wettability variation occurs after the establishment of mainstream paths at breakthrough, thus free of the negative effects associated with strongly water-wet states (Lei et al. Reference Lei, Lu, Gong and Wang2023).

Figure 3. Wetting-film development and residual NAPL detachment induced by nanoparticle adsorption. (a) Local view of topological characteristics of the residual non-aqueous phase. (b,c) Evolution of (b) volume ratio of droplets and (c) ganglion number during injection of brine and nanoparticle suspensions. The middle stage corresponds to half the duration of the final stage.

As discussed in § 3.1, spontaneous formation of wetting films driven by corner flow is essential for nanoparticle effect on wettability, while the shift toward stronger hydrophilicity induced by particle adsorption (corresponding to a transition from state IV to state III as depicted in figure 1 d) promotes aqueous film growth and enables detachment of NAPL droplets. The difference in film development can be qualitatively assessed using theoretical models for corner flow (Dong & Chatzis Reference Dong and Chatzis1995), which give the film flow rate in a square channel as $Q_f = aC(2K)^{-1/2} (\sigma /\mu \beta )^{1/2} R^{5/2} t^{-1/2}$ , where $a$ and $K$ are semi-analytical coefficients, $C = 4[\cos \theta \cos (\pi /4+\theta )/\sin (\pi /4) - (\pi /4-\theta )]$ is the shape factor, $\beta$ is the dimensionless flow resistance also dependent on $\theta$ (Ransohoff & Radke Reference Ransohoff and Radke1988), $R = D_h [\cos \theta - \sqrt {\pi /4-\theta +\sin \theta \cos \theta }] / [2(\theta -\pi /4)+2\cos ^2\theta -2\sin \theta \cos \theta ]$ is the curvature radius of corner-film meniscus, $D_h$ is the hydraulic diameter, and $t$ is the characteristic time. Accordingly, the wettability dependence can be expressed as $Q_f \propto C(\theta )[\beta (\theta )]^{-1/2}[R(\theta )]^{5/2}$ . Assuming effective wettability alteration as in static NAPL droplet systems (from $\theta _w \approx 36^{\circ }$ to $\theta _{\textit{eff}} \approx 15^{\circ }$ ), we obtain $Q_{f,{NP}} / Q_{f,0} \approx 16$ , where the subscripts $NP$ and 0 denote nanoparticle-modified and unmodified conditions, respectively, indicating that film accumulation efficiency on nanoparticle-modified surfaces can be over an order of magnitude higher than that on unmodified surfaces. Moreover, tortuous pore spaces in disordered media can induce more complex film transport. As discussed in our previous work (Lei et al. Reference Lei, Lu, Yang, Bagheri and Wang2025), nanoscale roughness from adsorbed particles stabilises wetting films against convex geometries, further amplifying the difference. Note that the effective wettability in pore space may differ from the intrinsic value in state characterisations due to hysteresis effects from surface roughness, dynamic effects during displacement, mixed wettability and so on (Lin et al. Reference Lin, Bijeljic, Berg, Pini, Blunt and Krevor2019; Sun et al. Reference Sun, McClure, Mostaghimi, Herring, Berg and Armstrong2020). Nevertheless, the generality of our wettability criterion remains valid, as it depends on the intrinsic property of the substrate.

3.3. Generality of the wettability criterion with structural hierarchy effects

We further conducted displacement experiments in designed disordered media with varying degrees of structural hierarchy. The pore size distribution progressively evolved from a long-tailed form in the strongly hierarchical structure to a nearly Gaussian form in the weakly hierarchical one (figure 4 a). Single-phase network modelling was performed not to exactly reproduce the phase and velocity distributions under two-phase conditions, but to provide statistical information on intrinsic structural properties based on segmented pore–throat units (see the supporting information for details). The structural heterogeneity factor is defined as $CV_s = \sqrt {\langle (D - D_m)^2 \rangle _V}/D_m$ , where $D$ is the local pore diameter, $D_m$ is the mean pore diameter, and $\langle \rangle _V$ denotes volume-weighted averaging. The flow heterogeneity factor is similarly defined as $CV_f = \sqrt {\langle (U - U_m)^2 \rangle _q}/U_m$ , where $U$ is the local throat velocity, $U_m$ is the injection velocity (identical for all structures), and $\langle \rangle _q$ denotes weighting by local flow rate. As presented in figure 4(b), $CV_f$ exhibits a super-linear dependence on $CV_s$ , highlighting the sensitivity of flow heterogeneity to structural hierarchy.

Figure 4(c) shows the relative displacement performance of nanoparticle suspensions compared with brine. The wettability criterion ( $\theta _w \lt \theta _{w,{cr}}$ ) holds across all tested structures, exhibiting a consistent transition from negligible to significant increase in $S_a$ at the final stage with increasing substrate hydrophilicity. Nevertheless, nanoparticle-induced improvement in efficiency under the effective condition weakened as the structure became more uniform. Variations in droplet ratio (figure 4 d) followed the same trend as displacement efficiency, indicating similar mechanisms as presented in figure 3.

Figure 4. Coupled effects of wettability and structural properties on nanoparticle behaviour during multiphase displacement. (a) Variation in the pore size distribution from strongly to weakly hierarchical disordered media. (b) Relationship between the structural heterogeneity factor $CV_s$ and flow heterogeneity factor $CV_f$ . Insets show schematics of the structures. (c,d) Variations in (c) relative displacement performance $\Delta S_a$ (difference between $S_a$ of suspension and brine) and (d) droplet ratio increase at the final stage for different structures in microfluidic experiments. (e) Local views of phase distributions from experiments and flow fields from lattice Boltzmann simulations. (f) Distribution of the local capillary number $Ca_i = \mu U_i / \sigma$ defined on throat unit $i$ , with $U_i$ from network simulations. The leftmost columns include $Ca_i$ down to zero, and $Ca_m$ denotes the global characteristic capillary number. (g) Illustrative phase diagram showing coupled effects of structural and wettability properties on nanoparticle-enhanced displacement, where $\Delta S_a^* = \Delta S_a / S_{a,0}$ and $S_{a,0}$ denotes the final aqueous phase saturation in brine displacement. Each circle represents the average of three sets of independent experiments.

To probe the link between structural features and flow patterns, local phase distributions in experiments are compared with flow fields from single-phase lattice Boltzmann simulations (figure 4 e; see the supporting information for detailed discussion on two-phase simulations). Fingering pathways during brine displacement largely coincided with the high-speed channels. Instead, detachment of residual NAPL by nanoparticles occurred primarily in low-speed stagnant zones, where a low local capillary number suppresses the main meniscus flow and allows wetting films to develop fully (Lei et al. Reference Lei, Lu, Gong and Wang2023). The higher local capillary number in preferential paths enables ganglion mobilisation under stronger viscous forces (Datta, Ramakrishnan & Weitz Reference Datta, Ramakrishnan and Weitz2014), producing a synergistic effect. By defining $Ca_i = \mu U_i / \sigma$ for each throat unit $i$ , statistics in figure 4(f) show that with increasing structural hierarchy, the volume fractions of regions with extremely low $Ca_i$ ( $\lt 10^{-10}$ ) that benefit film flow and high $Ca_i$ ( $\gt 10^{-4}$ ) that mobilise ganglia both increase, accounting for the enhanced displacement.

Figure 4(g) summarises these findings in an illustrative phase diagram, where the displacement behaviour of nanoparticle suspensions is governed by the interplay between wettability and structural conditions. Constrained by substrate wettability (satisfying the corner-flow criterion) and modulated by structural hierarchy, nanoparticle-induced displacement enhancement is more efficient in hydrophilic media with stronger flow heterogeneity. This is consistent with qualitative trends in the literature: on the one hand, the reported displacement enhancement was generally stronger under more hydrophilic conditions (Zhang et al. Reference Zhang, Ramakrishnan, Nikolov and Wasan2018; Mohammadalinejad et al. Reference Mohammadalinejad, Hosseinpour, Rahmati and Rasaei2019). On the other hand, variations in the extent of improvement among different types of porous media (Zhang et al. Reference Zhang, Ramakrishnan, Nikolov and Wasan2016; Li et al. Reference Li, Jiang, Gao, Huang, Xu and Chen2017; Lu & Wang Reference Lu and Wang2023) may be explained by structural hierarchy effects, as a more random structure typically exhibits a stronger hierarchy and flow heterogeneity, thereby amplifying nanoparticle effects. Although the proposed mechanisms are theoretically general, 3D effects arising from increased topological connectivity and spatial complexity in natural porous media should be carefully considered for engineering applications, as they may strengthen film flow and increase the sensitivity to wettability and structural properties.