1. Introduction
Dense turbulent suspensions at fluid interfaces are commonly encountered in a variety of natural and industrial settings, from microplastics on the ocean surface to mineral processing by froth flotation. Yet, understanding their behaviour remains a formidable challenge. These systems involve an intricate interplay of turbulence, interfacial tension and multi-particle interactions across a wide range of scales, defying simple predictive models. Historically, research on particle-laden flows has focused either on dense suspensions dominated by viscous forces and short-range interactions (Wagner & Brady Reference Wagner and Brady2009; Brown & Jaeger Reference Brown and Jaeger2014; Denn & Morris Reference Denn and Morris2014) or dilute suspensions of inertial particles in turbulence (Toschi & Bodenschatz Reference Toschi and Bodenschatz2009; Balachandar & Eaton Reference Balachandar and Eaton2010; Monchaux, Bourgoin & Cartellier Reference Monchaux, Bourgoin and Cartellier2012; Brandt & Coletti Reference Brandt and Coletti2022). Bridging those extremes, however, has proven challenging, as experiments and simulations become increasingly complex when both high concentration and turbulence coexist (Matas, Morris & Guazzelli Reference Matas, Morris and Guazzelli2003; Picano, Breugem & Brandt Reference Picano, Breugem and Brandt2015; Baker & Coletti Reference Baker and Coletti2019; Hogendoorn et al. Reference Hogendoorn, Breugem, Frank, Bruschewski, Grundmann and Poelma2023).
Particles at fluid interfaces introduce additional complexities, which have been studied extensively both for non-Brownian (Singh & Joseph Reference Singh and Joseph2005; Madivala, Fransaer & Vermant Reference Madivala, Fransaer and Vermant2009) and colloidal suspensions (Fuller & Vermant Reference Fuller and Vermant2012; Garbin Reference Garbin2019). In this case, the collective behaviour is driven by fundamentally different mechanisms: while dense suspensions cluster due to hydrodynamic interactions (Guazzelli, Morris & Pic Reference Guazzelli, Morris and Pic2011) and turbulent suspensions do because of particle/fluid inertia (Brandt & Coletti Reference Brandt and Coletti2022), in interfacial suspensions, agglomeration is driven by capillarity (Protière Reference Protière2023). Reaching a predictive understanding of dense turbulent interfacial suspensions is a formidable challenge because of the coexistence of such mechanisms.
In systems comprising a large number of particles, an especially relevant aspect is the tendency to percolate, i.e. to form connected networks spanning the entire domain (Stauffer & Aharony Reference Stauffer and Aharony2018). In dense particle suspensions, percolating networks of contacts (lubricated or direct) among particles typically emerge at a concentration threshold close to but below the jamming limit (Morris Reference Morris2020). Percolating clusters are critical for the rheological and transport properties, occurring for particles of different nature, size, shape and deformability; for recent examples, see Möbius et al. (Reference Möbius, Tesser, Alards, Benzi, Nelson and Toschi2021), Alicke, Stricker & Vermant (Reference Alicke, Stricker and Vermant2023), van der Naald et al. (Reference van der Naald, Singh, Eid, Tang, de Pablo and Jaeger2024), Girotto et al. (Reference Girotto, Scagliarini, Benzi and Toschi2024), Marin & Souzy (Reference Marin and Souzy2024), Kim & Hilgenfeldt (Reference Kim and Hilgenfeldt2024). The tendency of the system to percolate is usually assessed as a function of concentration and applied stress. While the protocols vary across the various settings, the applied stress commonly comes from geometrically simple compressive/extensive strain or pressure-driven flow of the carrier phase. In dense turbulent suspensions, though, the main source of stress may be the fluid fluctuating energy; this is the case in the present study.
We recently reported on the behaviour of dense turbulent suspensions at the interface of turbulent liquids (Shin & Coletti Reference Shin and Coletti2024; Vowinckel Reference Vowinckel2024; Shin, Stricker & Coletti Reference Shin, Stricker and Coletti2025). These were experimentally realised in electromagnetically driven fluids laden with millimetric spherical particles, over broad ranges of concentration, interfacial tension and turbulence intensity. We investigated the competing role of viscous drag, capillary attraction and lubrication, and demonstrated how their balance determines the transition between different regimes in terms of particle energy, diffusivity, clustering properties and tendency to percolate.
Previously (Shin & Coletti Reference Shin and Coletti2024; Shin et al. Reference Shin, Stricker and Coletti2025), we used particles less than 2 mm in diameter and with density close to that of the carrier fluid. For those, buoyancy does not appreciably deform the liquid interface and the capillary inter-particle attraction is dominated by the short-range quadrupolar mode of interfacial distortion (Stamou, Duschl & Johannsmann Reference Stamou, Duschl and Johannsmann2000; Botto et al. Reference Botto, Lewandowski, Cavallaro and Stebe2012; Liu, Sharifi-Mood & Stebe Reference Liu, Sharifi-Mood and Stebe2018). In many practical situations, larger particle size and/or density mismatch induce significant interfacial deformation, and the capillary interaction is dominated by the monopole mode that decays much slower (roughly logarithmically; see Vella & Mahadevan Reference Vella and Mahadevan2005). We ask the following questions. How does such a longer-range interaction change the clustering behaviours and the percolation threshold? Do dense turbulent interfacial suspensions admit a unified description, encompassing both quadrupolar and monopolar modes of surface deformation?
To address these questions, we consider both spherical and disc-shaped particles and examine the distinct clustering and percolation behaviours. By introducing an effective capillary length scale, we reconcile monopolar and quadrupolar regimes into a single predictive framework, enabling quantitative predictions for a wide range of conditions. The remainder of the paper is organised as follows: in § 2, we describe the experimental methodology and the parameter space; in § 3, we present and discuss the results, first in terms of the prevalent mode of capillary attraction and then in terms of clustering and percolation; in § 4, we draw conclusions and discuss the outlook.
2. Experimental methodology
We employ different fluid–particle combinations, as summarised in table 1. Polyethylene spheres with diameters smaller than 2 mm exhibit quadrupole-dominant interactions, as shown by Shin & Coletti (Reference Shin and Coletti2024). Larger polypropylene spheres and 1 mm-thick polypropylene discs, however, interact prevalently by monopolar capillarity, as we will show. Two fluid-layer configurations are studied. In the single-layer (SL) case, the particles float at the air–water interface of a 7-to-8 mm-deep conductive aqueous solution (10 % CuSO
$_4$
by mass, density
$\rho _f = 1.08$
g mL
$^{{-}1}$
, kinematic viscosity
$\nu = 1.0 \times 10^{{-}6}$
m
$^2$
s
$^{{-}1}$
). In the double-layer (DL) case, a 2-to-4 mm layer of mineral oil (
$\rho _f = 0.84$
g mL
$^{{-}1}$
,
$\nu = 1.9 \times 10^{{-}5}$
m
$^2$
s
$^{{-}1}$
) is added on top of the same aqueous phase, and particles primarily reside within the oil layer, partially wetted by the underlying water. The particle/interface configurations are schematically illustrated in figure 1.
Figure 1. Schematic illustration of the particle–fluid configurations investigated: (a) a sphere in the single-layer (SL) configuration; (b) a sphere in the double-layer (DL) configuration; and (c) a disc in the double-layer (dDL) configuration.
Table 1. Summary of the main experimental parameters investigated in this study, including the Reynolds number
$Re$
, the capillary number
$\textit{Ca}$
, the areal fraction
$\phi$
, the Bond number
$Bo$
and the interaction Bond number
$\textit{Bo}_c$
, all defined in the text. PE and PP refer to polyethylene and polypropylene, respectively.
The capillary force is obtained by pair-approach experiments. Pairs of identical particles are placed at distance of approximately 10 mm from each other in a tray containing the fluid layer(s). The pairs approach each other due to capillarity, and their positions and trajectories are measured with sub-pixel accuracy using the same imaging equipment and methodology as of Shin & Coletti (Reference Shin and Coletti2024).
The particle-laden fluid layers are stirred into quasi-two-dimensional (Q2-D) turbulent flows using the apparatus described in detail in earlier studies (Shin, Coletti & Conlin Reference Shin, Coletti and Conlin2023; Shin & Coletti Reference Shin and Coletti2024; Shin et al. Reference Shin, Stricker and Coletti2025). This features a
$320 \times 320$
mm
$^2$
tray above an
$8 \times 8$
array of cylindrical neodymium-iron-boron magnets. These are arranged in a chequerboard pattern with alternating polarities, which combine with the DC current between copper electrodes at opposite sides of the tray to induce Lorentz forces. The centre-to-centre magnet spacing
$L_F = 35$
mm provides the forcing length scale in the turbulent Reynolds number
$Re = u_{rms} L_F / \nu$
, where
$u_{rms}$
is the root-mean-square flow velocity.
The Q2-D flow achieves a fully turbulent character, as extensively characterised by Shin et al. (Reference Shin, Coletti and Conlin2023, Reference Shin, Stricker and Coletti2025). In particular, the scaling of the third-order longitudinal structure function of the velocity fluctuations transitions from cubic to linear for separations
$\sim L_F$
. Correspondingly, the energy spectrum displays the expected scaling
$k^{{-}3}$
in the enstrophy-cascade subrange, i.e. for wavenumbers
$k\gt 2\pi /L_F$
(Boffetta & Ecke Reference Boffetta and Ecke2012).
We vary the areal fraction
$\phi \equiv N_{{p}}(\pi d_{{p}}^{2}/4)/A_{{FOV}}$
between 1 % and 76 %, where
$N_{{p}}$
is the average number of particles in the field of view of area
$A_{{FOV}}$
. The relative strength of the applied stress is quantified by the capillary number
$\textit{Ca}$
: this compares viscous drag imposed by the fluid strain rate (separating particle pairs) and capillary attraction, both evaluated at contact,
$\textit{Ca} \equiv F_{drag,r=d_p}/F_{cap,r=d_p}$
.
We explain how the capillary force is evaluated in the following section, while the scaling of the drag force is briefly sketched here (for details, see Shin & Coletti Reference Shin and Coletti2024). Assuming a Stokesian formulation, the force
$F_\textit{drag}$
pulling two particles apart is linear in the relative velocity between the particles and the surrounding fluid. Considering a pair of adjacent particles aligned with the extensional eigenvector of the local strain rate tensor, the relative velocity of the fluid across the pair is
$\Delta u_f \approx \dot {\varepsilon }_{max} d_p$
, where
$\dot {\varepsilon }_{max}$
is the maximal principal strain rate. Neglecting the incipient motion of the particles away from each other,
$\Delta u_f$
equals the relative velocity between fluid and particles. The strain rate is estimated as
$\dot {\varepsilon }_{max} \sim u_{rms}/L_F$
, with the numerical prefactor determined by approximating the flow as an array of Taylor–Green vortices. Here, we have assumed that ‘strain cells’ at the scale of the particles are mainly responsible for breakup of adjacent particles. This assumption is consistent with measurements of the aggregate breakup rate, as discussed by Qi, Li & Coletti (Reference Qi, Li and Coletti2025b
).
Contact is defined based on a search radius around each centroid of approximately 1.2
$d_p$
, determined from low-
$\textit{Ca}$
snapshots where tightly bound clusters and isolated particles are clearly distinguishable. We define clusters as sets of four or more adjacent particles (the specific minimum number being inconsequential for the final conclusions). We monitor the clustering fraction
$\chi _{cl}$
(the number of clustered particles normalised by the total number of imaged particles) and the percolation fraction
$\chi _p$
. The latter is the fraction of images in which a cluster spans the field of view side-to-side. We stress that, via imaging, we can only assess connectivity percolation which is based on contact/adjacency, and not rigidity percolation which requires evaluating inter-particle forces (Sedes et al. Reference Sedes, Makse, Chakraborty and Morris2022).
Moreover, due to the high particle concentration achieved, the fluid velocity is measured in the single-phase experiments. Using the latter in the evaluation of the drag implies the assumption that the fluid forcing is approximately unaffected by the particles. In the dense regimes, the dispersed phase is in fact expected to influence the fluid. However, the agreement between the observations and the regime transitions theorised by Shin & Coletti (Reference Shin and Coletti2024) suggests that the assumption is, to first order, tenable. Moreover, the pair dispersion measurements by Shin et al. (Reference Shin, Stricker and Coletti2025) indicate that the energy dissipation rate due to lubrication is responsible for the reduction in particle kinetic energy at high concentration, rather than the clusters damping the fluid flow energy.
3. Results and discussion
We begin by assessing the prevalent mode of capillary attraction through pair-approach experiments. As shown by Shin & Coletti (Reference Shin and Coletti2024), for the smaller spheres (
$d_p$
= 1.09 and 1.84 mm), the quadrupolar contribution dominates (Liu et al. Reference Liu, Sharifi-Mood and Stebe2018):
\begin{equation} F_{cap} = {-}\frac {3 \pi \gamma h_{qp}^2 d_p^4}{r^5}, \end{equation}
where
$\gamma$
is the interfacial tension and
$h_{qp}$
is the amplitude of the quadrupolar mode. The latter is related to the particle surface roughness and is estimated by a least-square fit of the centre-to-centre inter-particle separation
$r$
as a function of time
$t$
:
\begin{equation} r_{0}^6 {-}r^6(t) = \frac {12 \gamma h_{qp}^2 d_p^3 t}{\mu }, \end{equation}
where
$r_0$
is the initial separation and
$\mu$
is the dynamic viscosity of the fluid in which the particles are immersed (Shin & Coletti Reference Shin and Coletti2024). Balancing
$F_{cap}$
against Stokes drag and lubrication yields the following expression of the relative velocity:
\begin{equation} v_{rel}(r) = \frac {2 \gamma h_{qp}^2 d_p^3 G(x)}{\mu r^5}, \end{equation}
with dimensionless separation
$x=r/d_p$
, and
$G(x)\approx 1{-}3/(4x)+1/(8x^3){-}15/ (64x^4){-}4.46/1000(2x{-}1.7)^{{-}2.867}$
the hydrodynamic mobility accounting for reduced relative velocities at small particle separations due to lubrication effects (Batchelor Reference Batchelor1976). Here,
$v_{rel}$
initially increases as particles move closer and capillarity grows stronger, and subsequently decreases as lubrication forces dominate at small separations. In contrast, the larger particles (3.9 mm spheres and 10 mm discs) exhibit longer-range capillary attractions associated with the monopolar mode of interfacial deformation:
\begin{equation} F_{cap}(r) = {-}C_s K_1 \left ( \frac {r}{l_c} \right ), \end{equation}
where
$C_s$
is a prefactor determined by vertical force balance between buoyancy and interfacial tension,
$l_c \equiv \sqrt {\gamma /(\Delta \rho _f g)}$
is the capillary length with
$\Delta \rho _f$
the density difference between two fluids,
$g$
is the gravitational acceleration, and
$K_1$
is a modified Bessel function of the second kind (Vella & Mahadevan Reference Vella and Mahadevan2005; Dalbe et al. Reference Dalbe, Cosic, Berhanu and Kudrolli2011). In this case, one obtains a relative velocity:
\begin{equation} v_{rel}(r) = \frac {2C_s G(x)}{3 \pi \mu d_p}K_1\left (\frac {r}{l_c}\right ). \end{equation}
As the drag is approximately linear (Shin & Coletti Reference Shin and Coletti2024), the full capillary-driven interaction force and the associated relative velocity are obtained as the sum of the quadrupolar and monopolar components. As shown in figure 2, the behaviour of the millimetric spheres in case 2DL (as well as in case 1SL, not shown) is dominated by the quadrupolar components at small separations, while the slowly decaying monopolar component takes over at further distances (figure 2 a). However, for the larger spheres (as well as the discs, not shown), the monopolar component prevails at all separations (figure 2 b). As capillary attraction becomes stronger at small separations, for simplicity, we refer to the two behaviours as quadrupole-dominant and monopole-dominant, respectively.
Figure 2. Measured relative velocity
$v_{rel}(r)$
compared with theoretical predictions obtained by superposing monopolar and quadrupolar capillary interactions for the (a) 2DL and (b) 4SL, respectively. Error bars represent the standard deviations obtained from five individual experiments.
Having determined the prevalent capillary interaction modes, we now examine the clustering and percolation behaviour for both classes of particles. Figure 3(a,b) show the clustering fraction
$\chi _{cl}$
across the explored range of
$\textit{Ca}$
and
$\phi$
for the quadrupole-dominant (1SL, 2DL) and monopole-dominant cases (4SL, 4DL, 10dDL), respectively; figure 3(d,e) present the corresponding percolation fraction
$\chi _p$
for these two cases. When
$\textit{Ca} \gt 1$
, both classes exhibit essentially identical behaviour: drag dominates particle dynamics, rendering capillary interactions secondary. When
$\textit{Ca} \lt 1$
, however, the monopole-dominant cases demonstrate notably stronger clustering and a lower percolation threshold. The difference in these observables between the two cases is quantified by the following metrics:
\begin{align} \Delta \chi _{cl} = \chi _{cl}^{M} {-} \chi _{cl}^{Q}, \quad \Delta \chi _{p} = \chi _{p}^{M} {-} \chi _{p}^{Q}, \end{align}
where the superscripts
$M$
and
$Q$
denote the monopole-dominant and quadrupole-dominant cases, respectively. Maps of
$\Delta \chi _{cl}$
and
$\Delta \chi _{p}$
(figure 3
c,f) highlight the enhanced tendency towards clustering and percolation in the monopole-dominant cases at low
$\textit{Ca}$
. This is attributed to the difference in spatial variation between the quadrupolar interaction, decaying as
$r^{{-}5}$
according to (3.1), and the monopolar one, falling off approximately as
$r^{{-}1}$
(since
$K_1(r) \approx 1/r$
for
$r \rightarrow 0$
). Thus, even for the same
$\textit{Ca}$
(defined at particle contact), monopolar interactions maintain significant strength at much greater separations.
Figure 3. Maps of clustering fraction
$\chi _{cl}$
for (a) quadrupole-dominant and (b) monopole-dominant cases in the original
$\textit{Ca}{{-}}\phi$
parameter space, with (c) the difference
$\Delta \chi _{cl}$
. Maps of percolation fraction
$\chi _{p}$
for (d) quadrupole-dominant and (e) monopole-dominant cases,
with (f) the difference
$\Delta \chi _{p}$
. Panels (c) and (f) highlight the enhanced clustering and percolation tendencies of the monopole-dominant cases, particularly at
$\textit{Ca} \lt 1$
.
To rationalise these differences, we introduce an effective interaction length
$r_c$
defined as the separation at which the capillary attraction equals the strain-induced drag pulling particles apart, i.e.
$F_{cap}(r_c) = F_\textit{drag}(r_c)$
. By this definition, particle pairs feel a net attractive force and tend to aggregate for
$r\lt r_c$
and disperse for
$r\gt r_c$
. If
$r_c \lt d_p$
(i.e.
$\textit{Ca} \gt 1$
), drag prevails even at particle contact. For particles experiencing predominantly quadrupolar-mode capillarity,
$F_{cap}(r) = F_{cap}(d_p)(d_p/r)^5$
. Thus, at the separation
$r^Q$
such that
$F_{cap}(r_c^Q) = F_\textit{drag}(r_c^Q)$
, we have
\begin{equation} F_{cap}(r) = F_{cap}(d_p)\left ( \frac {d_p}{r_c^Q} \right )^5, \end{equation}
which yields
\begin{equation} \frac {r_c^Q}{d_p} = Ca^{{-}1/6}. \end{equation}
For monopole-dominant particles, however, the spatial dependency of the capillary attraction involves the modified Bessel function, leading to
\begin{equation} \frac {r_c^M}{d_p} = \frac {1}{\textit{Ca}}\frac {K_1(r_c^M/l_c)}{K_1(d_p/l_c)}. \end{equation}
As this relation involves the modified Bessel function
$K_1$
, it cannot be inverted analytically, and thus
$r_c^M/d_p$
must be computed numerically. Moreover, unlike
$r_c^Q/d_p$
, the monopolar interaction length scale explicitly depends on the fluid–fluid interface through the capillary length
$l_c$
. This is illustrated in figure 4(a), plotting the dimensionless interaction length
$r_c/d_p$
as functions of
$\textit{Ca}$
. The monopole-dominant cases, which are configuration-dependent, exhibit larger interaction lengths especially at low
$\textit{Ca}$
. At
$\textit{Ca} \gt 1$
, all cases yield
$r_c \lt d_p$
by definition: clusters tend to break up regardless of the dominant capillary attraction mode, consistent with the observations made in figure 3.
Figure 4. (a) Dimensionless effective capillary interaction length
$r_c/d_p$
as a function of the capillary number
$\textit{Ca}$
for quadrupolar (3.8) and monopolar (3.9) cases. (b) Clustering fraction
$\chi_{cl}$
and (c) percolation fraction
$\chi _{p}$
for both quadrupole- and monopole-dominant cases presented in the unified
$\textit{Bo}_c$
–
$\phi$
parameter space. The red dashed line indicates the theoretical percolation threshold (3.13).
To unify monopolar- and quadrupolar-dominant behaviours, we define an ‘interaction Bond number’:
\begin{equation} Bo_{c} \equiv \left (\frac {d_p}{r_c} \right )^2. \end{equation}
In analogy with the classical Bond number
$Bo = (d_p/l_c)^2$
(also listed in table 1),
$\textit{Bo}_c$
compares the particle size with the dynamically relevant length scale, though it quantifies the balance between drag and capillary attraction rather than between gravity and surface tension. From (3.8), for the quadrupolar-dominant cases,
\begin{equation} Bo_c^Q = Ca^{1/3}, \end{equation}
whereas the monopolar-dominant cases require numerical evaluation of
$\textit{Bo}_c^M$
. Figure 4(b,c) display
$\chi _{cl}$
and
$\chi _p$
in the
$\textit{Bo}_c {{-}} \phi$
space: the smooth and monotonic behaviour indicates that this pair of parameters captures the unified behaviour of both monopolar and quadrupolar classes.
We remark that, because an explicit analytical relation between
$\textit{Ca}$
and
$\textit{Bo}_c$
is not available for the monopolar case, the
$\textit{Bo}_c {{-}} \phi$
space cannot be obtained by a simple mapping from the
$\textit{Ca} {{-}} \phi$
space. Therefore, the definition of
$\textit{Bo}_c$
is necessary to unify the monopolar and quadrupolar cases. Moreover, introducing the interaction length allows a concise rationalisation of the problem: because the areal fraction is determined by the particle diameter and the interparticle distance,
$\phi = \pi d_p^2 / (4r_{int}^2)$
, the entire parameter space is expressed in terms of the three defining scales of the system:
$d_p$
,
$r_c$
and
$r_{int}$
.
Unlike
$\textit{Bo}_c$
, however,
$\textit{Ca}$
can be directly evaluated from the input parameters, and therefore the
$\textit{Ca}{-}\phi$
representation introduced by Shin & Coletti (Reference Shin and Coletti2024) remains advantageous to describe the main regimes characterising the system.
As shown by Shin et al. (Reference Shin, Stricker and Coletti2025), percolation at
$\textit{Ca} \gt 1$
is governed by geometric constraints: it occurs for areal fractions above a constant value
$\phi _0 \sim 0.55$
, approximately equal to the saturation limit of a synthetic set of particles generated by random sequential adsorption (RSA, see Evans Reference Evans1993).
This approach virtually distributes non-overlapping circles, neglecting any forcing length scale and inter-particle interactions except for the excluded volume. Shin et al. (Reference Shin, Stricker and Coletti2025) indeed showed how, in the high-
$\textit{Ca}$
limit, the cluster statistics match those of RSA-generated configurations. This reinforces the interpretation that sufficiently strong turbulent forcing enforces stochastic configurations, in which case, the onset of percolation is controlled by excluded-volume effects rather than by cohesive interactions. In contrast, at
$\textit{Ca} \lt 1$
, enhanced capillary attraction leads to the formation of fractal-like clusters which trigger percolation at lower areal fractions, with a threshold
$\phi _c$
dependent on
$\textit{Ca}$
. Despite this observation, no explicit predictive framework was provided by Shin et al. (Reference Shin, Stricker and Coletti2025) to account for the reduced percolation threshold in the capillary-dominated regime. However, the concept of interaction length naturally leads to a quantitative unified criterion for percolation: if the mean interparticle separation
$r_{int}$
is shorter than
$r_c$
, the particles experience a net mutual attraction, facilitating percolation at lower particle concentrations. Approximating the mean interparticle separation as
$r_{int} \sim d_p (\phi _0/\phi )^{1/2}$
, the condition for percolation in the range
$\textit{Ca} \lt 1$
(i.e.
$r_c \gt d_p$
or
$\textit{Bo}_c \lt 1$
) becomes
\begin{equation} r_c \geqslant d_p \left (\frac {\phi _{0}}{\phi }\right )^{1/2}. \end{equation}
Overall, we write the percolation criterion as
\begin{equation} \phi \geqslant \phi _{c} = \begin{cases} \phi _{0}Bo_c, \quad & Bo_c\lt 1,\\ \phi _{0}, & Bo_c\geqslant 1. \end{cases} \end{equation}
Figure 5. Percolation fraction
$\chi _p$
versus (a) particle areal fraction
$\phi$
, showing distinct percolation transitions across different experimental conditions. (b) Universal collapse of percolation data obtained by normalising
$\phi$
by the predicted percolation threshold
$\phi _c$
, clearly illustrating a transition at
$\phi /\phi _c \approx 1$
. The dashed vertical line marks the theoretical threshold, validating the unified percolation criterion.
That such a criterion represents well the data is evident in figure 4(c), where (3.13) is indicated by a dashed line.
Some mismatch may be due to aspects such as nonlinearity in the drag force and modifications of the fluid flow at high concentrations. These and other aspects neglected here deserve to be investigated, though their influence on the percolation threshold does not appear to be dominant. We further confirm this threshold by plotting
$\chi _p$
from all experiments as a function of the areal fraction in figure 5. While individual cases exhibit distinct percolation behaviour at different
$\phi$
(figure 5
a), plots of
$\chi _p$
versus
$\phi /\phi _c$
display a consistent step-like transition around unity (figure 5
b).
We note that the sharp transition between small and large values of
$\chi _p$
reflects a distinct process with respect to general clustering. The latter, measured by
$\chi _{cl}$
, is determined by a dynamic equilibrium between a multitude of breakup/aggregation events, as recently described for the present system (though at more dilute concentrations) by Qi et al. (Reference Qi, Li and Coletti2025b
). In contrast, the percolation fraction concerns the existence of a system-spanning cluster: once formed, the gain/loss of particles by such an object marginally affects
$\chi _p$
. This is consistent with the threshold behaviour predicted by classic percolation theory; see, e.g. Stauffer & Aharony (Reference Stauffer and Aharony2018).
4. Conclusion
We have investigated clustering and percolation in dense suspensions of spheres and discs at turbulent fluid interfaces, focusing on the two most common regimes of capillary interaction: quadrupolar-dominant and monopolar-dominant.
Including discs has allowed us to test the robustness and generality of the presented framework, while extending the parameter space: particles with similarly large diameter would not be fully immersed in the thin fluid layers. Despite the fundamentally different scaling of the capillary force, a unified description emerges by defining an effective interaction length
$r_c$
at which capillary attraction and strain-induced drag are in balance. This allows data from monopolar and quadrupolar cases to be described by a common parameter space defined solely by the interaction Bond number
$\textit{Bo}_c$
and the areal fraction
$\phi$
. The percolation transition is quantitatively predicted when the average inter-particle distance
$r_{int} = r_c$
, leading to a simple definition of the threshold concentration. Importantly, the only parameter of such threshold,
$\phi _0$
, is purely geometrical and is obtained a priori by RSA calculations.
The present results unify prior observations in quadrupole-dominant systems (Shin & Coletti Reference Shin and Coletti2024; Shin et al. Reference Shin, Stricker and Coletti2025) with the behaviour of their monopole-dominant counterparts, creating a framework valid for a broad range of interfacial turbulent suspensions. Practical applications span diverse scales and contexts, from microscopic colloids to macroscopic debris or biomaterials floating in natural bodies of water. Further extensions to other capillary interaction modes, such as dipolar forces from asymmetric particles, could further test the universality of the framework.
While capillarity has specific traits which we have attempted to unify here, the phenomenology of clustering and percolation displayed by our system can be interpreted in the broad framework of attractive particle suspensions subject to agitation. Such systems admit a jamming phase diagram in which the balance between cohesive forces and effective temperature (thermal energy or mechanical agitation) governs the transition between fluid-like and solid-like states (Trappe et al. Reference Trappe, Prasad, Cipelletti, Segre and Weitz2001). Analogously, in the present system, the turbulent motion of the fluid may play the role of the effective temperature: at low
$\textit{Bo}_c$
(or equivalently low
$\textit{Ca}$
), particles rapidly form a connected network; at high
$\textit{Bo}_c$
, clustering is suppressed and percolation only occurs near the geometric RSA limit in the
$\textit{Bo}_c{{-}}\phi$
space. However, turbulence differs fundamentally from uncorrelated thermal agitation in that it is correlated over relatively large spatio-temporal scales. Indeed, while turbulent agitation in the present system results in higher diffusivity of the particle transport, the diffusion process transitions from ballistic to diffusive around temporal scales dictated by the eddy-turnover time (Shin et al. Reference Shin, Stricker and Coletti2025). Further investigations of how closely dense turbulent interfacial suspensions follow the canons of gas/liquid/solid phase transitions are underway.
Future research is warranted to address current limitations. First, turbulent free-surface flows with substantial depth exhibit compressible surface velocity fields not captured by the present Q2-D system (Lovecchio, Marchioli & Soldati Reference Lovecchio, Marchioli and Soldati2013; Li et al. Reference Li, Wang, Qi and Coletti2024; Qi, Li & Coletti Reference Qi, Shin and Coletti2025 a; Qi, Xu & Coletti Reference Qi, Xu and Coletti2025 c). Additionally, the temporal resolution of our measurements is not sufficient to discern whether clusters come to span the domain with a well-defined percolation front velocity. Identifying the latter would be important to verify whether the present system falls under the umbrella of directed percolation, which has been successfully used to interpret a wide range of phenomena including Leidenfrost effect (Chantelot & Lohse Reference Chantelot and Lohse2021), liquid–liquid demixing (Goyal, van der Schoot & Toschi Reference Goyal, van der Schoot and Toschi2021) and laminar-to-turbulent transition (Hof Reference Hof2023). Our experiments used spherical and disc-shaped particles, leaving unexplored the impact of particle polydispersity, shape (Madivala et al. Reference Madivala, Fransaer and Vermant2009; Botto et al. Reference Botto, Lewandowski, Cavallaro and Stebe2012) and electrostatic repulsion (van Baalen, Vialetto & Isa Reference van Baalen, Vialetto and Isa2023), which may significantly influence clustering dynamics. Finally, comparative studies with alternative systems, such as active colloids or self-propelled particles at interfaces (Bourgoin et al. Reference Bourgoin, Kervil, Cottin-Bizonne, Raynal, Volk and Ybert2020; Calascibetta et al. Reference Calascibetta, Giraldi, El Khiyati and Bec2024; Yang et al. Reference Yang, Jiang, Picano and Zhu2024), could yield valuable insights into universal aggregation phenomena.
Declaration of interests
The authors report no conflict of interest.
Open access
