Sub-Newtonian coalescence dynamics in shear-thickening non-Brownian colloidal droplets

Abstract

Recent investigations into coalescence dynamics of complex fluid droplets revealed the existence of sub-Newtonian behaviour in polymeric fluids (elastic and shear thinning). We hypothesize that such delayed coalescence or sub-Newtonian coalescence dynamics may be extended to the general class of shear thickening fluids. To investigate this, droplets of aqueous corn-starch suspensions were chosen and their coalescence in the sessile–pendant configuration was probed by real-time high-speed imaging. Temporal evolution of the neck (growth) during coalescence was quantified as a function of suspended particle weight fraction, ϕ_w. The necking behavior was found to evolve as the power-law relation, R = at^b, where R is the neck radius, with exponent b ≤ 0.5, implying that it is a subset of the generic sub-Newtonian coalescence. Furthermore, the coalescence dynamics could be demarcated into two distinct regimes, b ∼ 0.5 and b < 0.5, where the emergence of visco-elastic pinch-off response was observed in the latter regime. The particle fraction demarcating these regimes, designated as the critical particle weight fraction, ϕ_w ∼ ϕ_c > 0.35, also coincides with the existence of ‘jamming’ and ‘flowing’ regions within the neck during viscoelastic pinch-off of cornstarch suspensions (Roché et al., Phys. Rev. Lett., 2011, 107, 134503). We also propose a simplistic theoretical model that captures the observed delay in coalescence dynamics implicitly through altered suspension viscosity stemming from increased particle content.

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1. Introduction

Coalescence is the spontaneous merging of individual fluid droplets to form a single entity.^1,2 This process involves the emergence and growth of a liquid bridge between the coalescing droplets, called the neck. Temporal evolution of the neck can be uniquely captured by a single parameter, neck radius R, known to evolve as R = at^b, where a and b are the pre-factor and power-law exponent respectively.^3,4 The magnitude of this power law exponent b is determined by the balance between the driving capillary force and the opposing inertia and viscous forces. There are three possible geometric configurations of coalescence: sessile–sessile, sessile–pendant and pendant–pendant. The present study is restricted to the sessile–pendant configuration, where the exponent b is reported to be universal for coalescing droplets of Newtonian fluids.^5,6 The numerical values of the exponent are 0.5 and 1 in the inertia-dominated and viscous-dominated regimes, respectively.^5,6 The parameter b singularly captures the alterations in the coalescence dynamics arising from changes in fluid properties, miscibility and ambient conditions. Recently, polymeric fluid droplet coalescence has been shown to be slower (b < 0.5) than the corresponding Newtonian coalescence.^7–10 This sluggish response has been attributed to restoring elastic force inherent to macromolecular relaxation processes in polymers. A gradual decrease in b was also observed with an increase in the concentration of polymeric solute in the solution.^8,10 Thus, the existing theoretical framework saw the introduction of a new regime i.e., the elasticity-dominated regime^8,10 in the case of polymeric fluids. It is therefore evident that the signature of the coalescence phenomenon, b, is specific to the class of fluids in question. A new term ‘sub-Newtonian coalescence’⁷ has been coined for the class of fluids that exhibit coalescence dynamics with b < 0.5. So, the question that arises here is, is sub-Newtonian coalescence a characteristic of the restoring elastic force in the fluid? A recent theoretical investigation presented alternative results, where coalescing droplets of shear thinning inelastic fluids in the sessile–sessile configuration¹¹ exhibited sub-Newtonian coalescence. A similar effect has also been observed in coarse grained molecular scale simulations of coalescing droplets laden with surfactants.¹² Introduction of surfactants slowed bridge growth and this deceleration has been attributed to formation of aggregates in droplet bulk that hinder fluid flow into the bridge region coupled with reduction in surface tension that delays initiation of bridge growth between the droplets.¹² Thus, it is evident that sub-Newtonian coalescence or delayed coalescence has been predicted to occur for different classes of complex fluids although the mechanism of occurrence is still unclear and fluid class-specific. In this study, therefore we choose a well-studied and characterised sub-class of complex fluids, i.e., colloidal suspensions. They are also microstructurally different from polymeric fluids and surfactant laden droplets and this difference in their microstructure¹³ arises from the presence of short-range inter-particle forces. These provide the freedom of recovering from applied strain via re-orientation rather than stretching of constituent entities observed in polymers. So, although both polymers and colloidal suspensions deform under imposed stress or flow, the latter have no strain-recovery mechanism. Instead in colloidal suspensions, the imposed stress may give rise to the formation of local aggregates,¹⁴ slippage, etc. as routes of viscous dissipation.

These inter-particle interactions may be modulated via particle content of the colloidal suspension to display a range of complex rheological behaviours as a function of applied shear rates.^15–18 For example, dilute suspensions exhibit Newtonian flow behaviour but with an increase in the particle weight fraction (ϕ_w), the suspension viscosity (μ) increases by several orders of magnitude. A continuous increase in suspension viscosity beyond a critical shear rate ([small gamma, Greek, dot above]_c) marks the onset of continuous shear thickening (CST).^19–21 On the other hand, if the viscosity increase is abrupt, then the behaviour is termed as discontinuous shear thickening (DST). Such transitions from CST to DST with the shear rate and particle content have been the subject of recent investigations.^20,22–26

In the present investigation, we experimentally probe the coalescence of one such class of model colloidal suspensions, i.e., aqueous corn-starch suspensions.^20,27–30 These suspensions are also known to display an array of rheological responses and in particular concentrated corn-starch exhibits a transition from continuous shear thickening (CST) to discontinuous shear thickening (DST)^20,27–35 with increase in the shear rate. The question we probe is, does this rheological behaviour influence the necking dynamics of coalescing colloidal droplets? Our results establish the connection between the rheological response of colloidal suspensions and the coalescence dynamics. Furthermore, we attempt to employ the Cross model as the constitutive equation to account for the rheological behaviour of the suspensions and show that it captures the essential features of coalescence dynamics observed in the experiments.

2. Materials and methods

2.1. Substrate preparation

Glass slides were procured from A-One, Haryana, India and cleaned by ultrasonication in two solvents (acetone followed by deionized water) for 20 minutes each. These cleaned glass slides were dried in an oven at 90 °C for 30 minutes to remove moisture. To have a sessile–pendant configuration, it was essential to have sessile droplets that exhibit a partial wetting state with the substrate. Therefore, these substrates were coated with a hydrophobic layer of polydimethylsiloxane (PDMS). This hydrophobic layer was prepared by mixing the base with the curing agent (Sylgard 184, Silicone Elastomer Kit, Dow Corning) in the ratio of 10 : 1 and vacuum desiccated to remove the air bubbles. The mixture was then spin coated at 5000 RPM for 1 minute on the cleaned slides and cured in the hot air oven at 90 °C for 90 minutes.

2.2. Preparation of corn-starch suspensions

Corn-starch powder having an average particle diameter (d_p, μm) of ∼15.29 ± 4.37 was procured from Sigma-Aldrich. Details of the particle size distribution are provided in the ESI† (Note S1). Corn-starch powder was added to aqueous solutions of cesium chloride (CsCl : water ∼ 51.5 : 48.5 (w/w))^32,33 to mitigate the settling of corn-starch particles. These mixtures were stirred using a magnetic stirrer at 650 RPM for 20 minutes to prepare the cornstarch suspensions of required weight fractions, ϕ_w (w/w). Aqueous solutions of cesium chloride (ϕ_w = 0) are referred to as the control or reference suspensions in the text hereafter. The particle weight fractions were increased incrementally from typically dilute suspensions, ϕ_w = 0.26, to concentrated suspensions, ϕ_w = 0.43. Surface tension (σ, mN m⁻¹) of these suspensions was measured by the pendant drop method using a goniometer (Biolin Scientific, Sweden, Theta lite) and found to lie in the range of 54–62 mN m⁻¹ for the particle weight fractions employed in this study.

2.3. Rheological characterization of cornstarch suspensions

Corn-starch suspensions were characterized by measuring the variation of apparent viscosity (μ) with the imposed shear rate, [small gamma, Greek, dot above] (10⁻²–10³ s⁻¹), using a rheometer (Anton Paar, model: MCR 302) with the cone and plate geometry (plate diameter – 40 mm and cone angle – 1°). The rheological characterization has also been performed with a parallel plate geometry (Anton Paar, model: MCR 302e, 25 mm plate diameter) to check if the observed shear-thickening of the suspensions is dependent on the measurement geometry for the imposed shear rate, [small gamma, Greek, dot above] (10⁰–10³ s⁻¹). Details of the discussion can be found in the ESI,† Note S2.

With the introduction of colloidal particles (ϕ_w ∼ 0.26), the apparent viscosity showed a shear thinning regime at low shear rates followed by a Newtonian plateau at a higher shear rate, [small gamma, Greek, dot above] > 1 s⁻¹ (Fig. 1a). A further increase in particle content showed the occurrence of a shear thickening behaviour at higher shear rates. To distinguish between the continuous and discontinuous shear thickening behaviours, the shear stress responses (Fig. 1b) were fitted to τ ∝ [small gamma, Greek, dot above]^(ψ), where the exponent ψ = 1 corresponds to the Newtonian regime. An increase in particle loading (ϕ_w), i.e., ϕ_w > 0.26, marked the onset of continuous shear thickening (CST) as evidenced by ψ > 1. With a further increase in ϕ_w, i.e., at ϕ_w ∼ 0.41, an abrupt increase in apparent viscosity³⁰ indicated discontinuous shear thickening (DST) behaviour with ψ ∼ 2. Such discontinuous shear thickening behaviour is typical of corn-starch suspensions and its causes are a still a topic of debate in the soft matter community.^{13,23,25,29,30,32,33,36,37} This transition from CST to DST occurs at a particular shear rate, called the critical shear rate, [small gamma, Greek, dot above]_c. Shear-thickening behaviour was found to be consistent even with parallel plate geometry and the reader is referred to the ESI† (Note S2) for further details.

Fig. 1 Rheological behaviour of corn-starch suspensions as a function of particle loading (ϕ_w: w/w). Variation of (a) apparent viscosity (μ) with the applied shear rate ([small gamma, Greek, dot above]). Arrow indicates the direction of increase in ϕ_w; (b) shear stress (τ) with the shear rate ([small gamma, Greek, dot above]) depicting the transition³⁰ of the rheological response from continuous shear thickening (CST) to discontinuous shear thickening (DST). The transition is determined by fitting the data to τ ∝ [small gamma, Greek, dot above]^(ψ), where 0 < ψ < 2 and ψ > 2 indicates CST and DST, resp., with increasing ϕ_w. Fitted lines are shown for completeness.

The extensional behaviours of the colloidal suspensions are characterised by the capillary breakup extensional rheometry dripping on a substrate (CABER-DoS) experiments following the experimental protocol adapted from Dinic et al.³⁸ This involves creation of a pendant drop by pumping the suspension through a nozzle (tip radius, l_c = 0.6 mm) using a syringe pump operated at a constant flow rate of 0.02 mL min⁻¹ (New Era Pump Systems Inc, USA). The pendant drop is then slowly deposited onto a cleaned glass substrate and the capillary bridge formed between the colloidal suspension and substrate is captured by a high-speed camera (20 000 FPS) and compatible light source (Nila Zaila, USA). The gap H_d between the nozzle and the substrate is kept constant and maintained at H_d = 6l_c. Details of the extensional behaviour of the suspension as a function of particle content are discussed in Section 4.2.

2.4. Coalescence experiments

The sessile–pendant configuration is adopted to outline the effect of increasing particle content (ϕ_w) on the coalescence dynamics of colloidal suspensions. Sessile droplets of constant volume (∼7.5 μL) were dispensed onto the substrate. Pendant droplets of identical volume were brought into contact with the sessile drop to initiate coalescence at an approach velocity of 10⁻¹ m s⁻¹ (ref. 9 and 39) by a syringe pump (Fig. 2a). The temporal evolution of the neck formed between the coalescing droplets was captured (Fig. 2b, blow-up) at 170 000 FPS using a high-speed camera (Photron Fastcam Mini, AX – 100) and a suitable light source (Nila Zaila, USA).

Fig. 2 Schematic of the (a) experimental setup (b) coalescing sessile–pendant droplet. Blow-up of the region of interest (neck) shows experimental quantities of interest, neck radius R and semi-neck width H for colloidal suspensions with particle weight fraction ϕ_w. Velocity field employed later in the theoretical framework is shown in the blow-up.

Images were extracted from the captured real-time videos (Supplementary video, S1, ESI†) of coalescence and the neck evolution was characterised by two parameters – radius R and semi-width H using an in-house MATLAB code with sub-pixel accuracy. Details of the analysis can be found elsewhere.^7–10,39,40 A blow-up of the coalescing droplets (Fig. 2b) schematically shows velocity field in the neck region used for developing the theoretical formulation. Experiments were performed for each representative particle weight fraction and the average of at least five trials has been reported. The error range is found to lie within ±5% for the reported neck radii.

3. Theoretical formulation

The flow in the neck region of coalescing droplets in the cylindrical co-ordinate system has been assumed to be incompressible, quasi-steady and quasi-radial, i.e., V = (V_r,0,V_z) := (u,0,v) as shown in Fig. 2(b). Considering the symmetry of the sessile–pendant geometry, the flow field near the neck region can be defined as z = 0, u ≠ 0 and v = 0. In close proximity to the necking region, the flow velocity is given as u⁺ = u⁻ and v⁺ = −v⁻.

The mass and momentum conservation equations at z = 0 can be expressed as eqn (1) and (2), respectively,

where τ is the stress tensor, τ = 2η([small gamma, Greek, dot above])Ḋ with Ḋ as the strain rate tensor. These are further modified for droplets of a shear thickening fluid (STF) by introducing the Cross model.^28,41–47 The transition from CST to DST as a function of particle weight fraction is incorporated through four parameters, η_∞, η₀, Γ and n,where η_∞ and η₀ are the infinite and zero shear viscosity, resp., Γ is the time constant, n is the power law index and is the second invariant of the strain rate tensor. The numerical values of these parameters were extracted by fitting the experimental data with eqn (3) (Table S1, ESI†). This model assumes the fluid viscosity to be a function of only the applied shear rate, [small gamma, Greek, dot above] in tune with rheological behaviour (Fig. 1b) of the suspensions. Modulation of the particle weight fraction is captured exclusively through the altered viscosity behaviour of the fluid and is incorporated through the individual stress tensor components along with the Cross constitutive relation, given by eqn (4),

τ_rz ∼ 0 (4b)

It must be noted that the term 1 + (Γ[small gamma, Greek, dot above])ⁿ has been approximated to ∼(Γ[small gamma, Greek, dot above])ⁿ since O([small gamma, Greek, dot above]) ≫ 1. The r-direction momentum (eqn (2)) is further simplified by introducing the relevant scaling parameters, u ∼ U, r ∼ R, and , while the pressure gradient scaled as , where σ is the measured surface tension of each suspension. The corresponding stress tensor components in terms of scaled variables can be written as and , where R₀ is the initial droplet radius. The momentum conservation with these arguments is reduced to a polynomial equation with two scaling constants, C₁ and C₂

U² + F(Γ,ρ,R,n,η₀,η_∞)U¹⁻ⁿ + G(ρ,R,η_∞)U + H(σ,ρ,R,R₀) = 0 (5)

where the scaling constants have been defined as,

Eqn (5) is solved using an iterative Newton Raphson method with the initial guess corresponding to the Newtonian fluid, i.e., roots of eqn (5) for n = 1. Furthermore, the evolution of the necking radius R is obtained by rewriting the velocity term U as in eqn (5),

Solution of eqn (6) was obtained by the finite difference scheme with a sufficiently small-time step (10⁻⁶ s) to ensure the numerical stability of the scheme. The neck radius R and time t were non-dimensionalized by the length-scale corresponding to the initial droplet radius R₀ and the inertial time scale τ_i = (ρR₀³/σ)^0.5, respectively. These theoretically predicted temporal profiles of the necking radius are shown in the later sections.

4. Results and discussion

The initial contact between the sessile and pendant droplets is a point that gives way to the development of a liquid bridge between the two droplets. In the present study, the lower limit of the region of interest (ROI) is the onset of liquid bridge formation where it can be captured (Fig. 3(a)) by the high-speed camera. Details of the methodology adopted for initiation point detection for coalescing colloidal droplets can be found in the ESI† (Note S3). The reader is also referred to a recent article⁷ for details on discerning the ROI. Evolution of the liquid bridge is quantified by the neck radius R tracked temporally for the colloidal suspensions with incremental increase in the particle weight fraction ϕ_w, i.e., from dilute to concentrated suspensions. It must also be noted that the shear rates involved in the neck region can be estimated as [small gamma, Greek, dot above](coalescence) ∼ U/R ∼ 1/R dR/dt ∼ O(10³), where the typical orders of magnitudes of the neck radius and time are R (m) ∼ O(10⁻⁴) and time t (s) ∼ O(10⁻³) respectively. These shear rates fall within the range of shear rates (Fig. 1(a)) where shear thickening has been observed, i.e., [small gamma, Greek, dot above](shear thickening) ∼ O(10¹) to O(10³) for the particle weight fraction probed in this study. Therefore, shear thickening behaviour is relevant to the reported coalescence phenomenon.

Fig. 3 Temporal evolution of the neck radius R with increasing (L to R) particle weight fractions ϕ_w for colloidal droplets shown via (a) real time images, where R represents the neck radius during coalescence; (b) the corresponding power law behaviour R = at^b, where a: pre-factor and b: power-law exponent, given as slopes of the linear fits. Here, R* and t* are non-dimensionalised neck radius and time, resp. R₀ is the initial droplet radius and τ_i (= ρR₀³/σ)^0.5 is the inertial timescale; σ (mN m⁻¹): suspension surface tension. Inset shows the typical R = at^0.5 for a Newtonian suspension, ϕ_w = 0 with t* × 10⁻²; (c) variation of b with the particle weight fraction ϕ_w, where ϕ_c denotes the critical particle weight fraction corresponding to b < 0.5.

4.1. Colloidal droplet coalescence

Snapshots of coalescing droplets of solvent, ϕ_w = 0, and colloidal suspension, ϕ_w = 0.26, are shown in parallel in Fig. 3(a). It can be observed that the shape of the bridge for the representative colloidal suspension is similar to its suspending solvent. This implies that while the initial bridge shape is unaltered by introduction of micron-sized colloids in droplets (dilute suspensions), its influence on neck growth dynamics is evident. For concentrated suspensions with ϕ_w = 0.42, the initial neck radius is larger compared to the solvent and dilute suspension with ϕ_w = 0.26, but the neck growth occurs at a slower rate. This increase in the initial radius can be attributed to the inherent yield stress of these suspensions as reported by Goel et al.,⁴⁸ and M Roché et al.⁴⁹ The rate of neck growth within the ROI is delayed with increasing particle content.

This is further quantified in Fig. 3(b) where the neck radius R and time are non- dimensionalised by the initial droplet radius R₀ and inertial time scale τ_i (= ρR₀³/σ)^0.5, resp. with the relevant measured suspension surface tension σ (mN m⁻¹). The non-dimensionalised neck radius R* and time t*, resp., exhibited power-law behaviour, R* = at*^b, with b being the magnitude of the power-law exponent (Fig. 3(c)). This power law exponent is found to be specific to each particle weight fraction (ϕ_w). Specificity of the power-law exponent b to the particle weight fraction is further probed. In particular, does the magnitude of b reflect the varying rheological characteristics (Fig. 1) of the colloidal suspensions observed with specific particle content?

Dilute colloidal suspensions with ϕ_w = 0.26, 0.32, and 0.35 exhibited coalescence dynamics (b = 0.5) similar to the suspending Newtonian solvent. But with an increase in particle loading (ϕ_w) from 0.35 to 0.41, a significant drop in the magnitude of b (0.50 to 0.27) was observed. The steepest decrease in the power law exponent (b = 0.38 to 0.27) occurs with an increase in ϕ_w from 0.39 to 0.40. This coincides with the particle weight fraction ϕ_w ∼ 0.33, where jamming was observed to occur on account of imposed extension (Fig. 2(b)) in similar experimental systems.⁴⁹ Evidence of the same can be seen in the oscillatory shear rheology response (Fig. 4) in the present study with at ϕ_w ∼ 0.41 for frequency ω > 5 (s⁻¹).

Fig. 4 Variation of storage (G′, half-open symbols) modulus and loss modulus (G′′, open symbols) with angular frequency ω at 5% shear strain (γ).

Beyond the critical point ϕ_w > ϕ_c, the neck radius continued to exhibit power law behaviour with b ∼ 0.27. However, the suspensions corresponding to ϕ_w = 0.42 and 0.43 consist of particle cluster regions disparately distributed throughout the solvent. These particle clusters give rise to localised density variations within the suspension.³⁴ It has been shown via rotational and capillary rheometry that the macroscopic response of DST by corn-starch suspensions stems from the separation of the colloidal system into ‘low-density-flowing’ and ‘high-density-jammed regions’.³⁴

An underlying implication is that global homogeneity of the suspension is no longer maintained. Taking the analogy further, it can be stated that during coalescence of concentrated suspensions for particle fractions beyond ϕ_c, the flow within the neck is restructured. This restructuring however does not allow the suspension to behave like its Newtonian solvent for then the power law exponent would be its universal value of 0.5. Instead, it lies in the vicinity of b ∼ 0.27, suggesting that the delayed neck growth dynamics comes from local flow restructuring and flow inhomogeneity. Similar evidence of phase separation has been reported in non-Brownian concentrated PVC suspensions.²⁵ Thus, the journey of the neck evolution can be divided into two regimes based on the critical point, denoted by the critical particle weight fraction, ϕ_c, as shown schematically in Fig. 5. We further probe our initial query of rheological behaviour influencing coalescence dynamics of colloidal droplets in these regimes and if the demarcation into regimes is a manifestation of the rheological behaviour of the colloidal suspensions.

Fig. 5 Schematic representation of coalescing colloidal droplets for particle weight fractions beyond the critical point, ϕ_w > ϕ_c, where regions of particle clusters are formed.

To understand this, the extensional behaviour of the colloidal suspensions was further probed via CABER-DoS experiments due to coalescence in the sessile–pendant droplet configuration and therefore predominantly an extensional flow. Another reason that further prompts these experiments is to check if uniaxial tension evokes different responses from these suspensions as a function of particle content.

4.2. Insights from pinch-off dynamics

Coalescence and break-up or pinch-off of liquid droplets involve the evolution of the same geometric entity, i.e., the neck or liquid bridge albeit in an opposing direction. Both the processes therefore can be quantified by the scaling law, R ∼ t^α, where the geometric characteristic length-scale is the same, i.e., R (usually radius) of the evolving liquid bridge with time t and the scaling constant α. These phenomena (coalescence and pinch-off) are essentially geometrically similar although they are kinematically inverse of each other. In spite of being kinematically inverse, the underlying similarities imply that they are governed by an interplay of the same set of forces: inertial, viscous, and capillary, and therefore they can be studied via the same combination of the same group of dimensionless numbers – Reynolds number (Re), capillary number (Ca) and Weber number (We), resp. Furthermore, these dimensionless numbers can be combined to express the scaling laws in terms of Ohnesorge (Oh) number-based units (details in Section 4.3). These similarities have been utilised to understand the mechanistic details of coalescence via the pinch-off process.

4.2.1 Observations from pinch-off experiments. Temporal evolution of the capillary bridge in these experiments was quantified by the non-dimensionalized capillary bridge radius r̄ (r̄ = r/l_c, l_c, r are the nozzle and capillary bridge radius resp.) as a function of the particle weight fraction (Fig. 6). Introduction of colloidal particles into the aqueous cesium chloride solution (Fig. 6a, ϕ_w ∼ 0) altered the characteristic shape of the capillary bridge. A further increase in particle content, ϕ_w ∼ 0.32, resulted in the appearance of a typical beads-on-a-string structure (BOAS) (Fig. 6a) in contrast to the abrupt breakup seen in Newtonian solvent (first row, Fig. 6a). The capillary bridge further stretches into a thin filament (filament thinning), thereby prolonging the pinch-off at ϕ_w ∼ 0.37. The pinch-off time has been employed as a reference timescale for the pinch-off process and this shifted timescale (t_p − t) has been mapped as a function of particle concentration. It is evident from Fig. 6b that an increase in particle concentration prolongs the pinch-off time; for example, the pinch-off time t_p^{36,37,49–56} increased by an order of magnitude, i.e., ∼10⁻² at ϕ_w ∼ 0.32 to ∼O(10⁻¹) at ϕ_w ∼ 0.4 (Fig. 6b). A further increase in the particle weight fraction translates into extended periods of filament thinning and therefore delays the pinch-off time. It must be noted that this also corresponds to the steepest decrease in the power law exponent, b = 0.38 to 0.27, of coalescing colloidal droplets.

Fig. 6 (a) Real time snapshots show the prolonging of the capillary bridge evolution and eventual pinch-off with an increase in particle content, ϕ_w. Left and right columns denote the capillary radius at t = 0 and before pinch-off resp. The dotted line emphasizes filament thinning; (b) corresponding temporal evolution of the non-dimensionalised capillary bridge radius r̄. The arrow indicates the direction of increase in ϕ_w. The dashed circle marks the filament thinning zone.

Thus, it is evident that apart from the dynamics of pinch-off, the filament morphology during pinch-off is also influenced by an increase in particle content of the colloidal suspensions. The abrupt breakup of the reference solvent (ϕ_w ∼ 0) is typical of Newtonian fluids, while the beads-on-a-string structure (BOAS) at ϕ_w ∼ 0.32 is characteristic of viscoelastic fluids. In accordance with coalescence dynamics, the pinch-off dynamics of the suspensions are also influenced by the particle content, where ϕ_w < 0.36 indicates Newtonian characteristics, whereas ϕ_w > 0.36 is reminiscent of pseudo-elastic behaviour as evidenced by the occurrence of the filament thinning stage (Fig. 5b). This implies that the extensional responses are in accordance with the regimes demarcated previously as per the exponent observed in the coalescence dynamics, i.e., for ϕ_w ∼ ϕ_c > 0.36 a delayed coalescence exists, which corresponds to the emergence of visco-elastic characteristics in the pinch-off phenomenon.

To provide a context to the extensional responses observed in the present study, it is compared with a state-of-the-art available literature.

4.2.2 State of the art. The material response of colloidal suspensions^57–59 to extensional flow is fundamentally distinct from that seen in shear flow. However, there are only a handful of studies^{32,36,37,49,60} on the extensional behaviour of corn-starch suspensions and Table 1 shows the responses under different extension modes. The mode of extension determines the tensile loading and therefore the response of the fluid filament under induced stress or strain. Tensile loading applied in the present study is created by placing a sessile drop onto the substrate using a nozzle. This droplet is stretched between the substrate and the nozzle to form a liquid bridge that undergoes self-thinning driven by capillary instabilities and culminates into pinch-off (CABER DoS set-up). This implies that the mode of tensile loading is constant as maintained by the constant pulling velocity ∼0.1 mm s⁻¹ ∼ O(10⁻⁴) in the present study. This is indicated as the first mode of extension, i.e., (i) tensile loading in a standard CABER DoS set up, while other modes have been designated as (ii) sandwiched between a geometry in the rheometer (cone-plate/parallel-plate) indicated as PP/CP and (iii) a pendant droplet that undergoes pinch-off due to gravity indicated as G.

Table 1 State-of-the-art literature on extensional flow of dense colloidal suspensions with corn-starch as the solute

Reference	Solvent	Extension mode	Response	Strain rate or extension rate s^−1
Bischoff White et al. (2010)^36	Water		Filament extension and brittle fracture	0.2–0.6
	Suspensions not density matched using cesium chloride	CP/PP
Fall et al. (2008)^32	De-mineralized water and cesium chloride	CP/PP	Unreported for extensional flow	0.04–2.35
Roché et al. (2011)^49	De-mineralized water and cesium chloride	CABER-DoS	0.29 < ϕw < 0.32
			Filament thinning and cylindrical bridge formation	Unspecified explicitly
			Visco-elastic pinch-off
Majumdar et al. (2017)^60	Water, cesium chloride, and glycerol	CP/PP	v < 3 mm s^−1: viscous liquid	This is not a case of uniaxial tension
			v > 4 mm s^−1: shear jammed region
Wang et al. (2022)^37	Ultrapure water	G	Change in surface texture from glossy to granulated texture	—
Sudheer et al., (2024)	Water and cesium chloride	CABER-DoS	Filament thinning to a beads-on-a-string structure followed by pinch-off	Constant extension rate
Present study			No fracture observed

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The takeaways from Table 1 can be listed as:

(i) Similar BOAS has also been reported³⁷ for breakup of aqueous corn-starch suspensions (corn-starch blended with ultrapure water) corresponding to the particle volume fraction ϕ_v ∼ 0.41, although the suspensions were not density-matched in contrast to the density-matched suspensions in the present study.

(ii) Amongst the reported systems, only that reported by Roche's et al.⁴⁹ matches with the present experimental system, where the extension mode and solvent media (cesium chloride and water) are identical. The difference lies in the absence of filament fracture and retraction with the present study.

(iii) These have been attributed to the formation of local particle clusters and such particle confinement within the filament at higher particle weight fractions (ϕ_w ≥ 0.4) and are also responsible for spike in the measured first normal stress difference.^{36,37,49–56} As per the former study, the suspensions can be divided into less concentrated suspensions that exhibited a landscape of particle clusters giving rise to concentration in-homogeneities in the direction of pull (axial). On the other hand, the relatively concentrated suspensions showed jamming.^{20,27,29–33,35,61}

(iv) Several other signatures occur in response to extensional flow (Table 1), which are indicative of viscoelastic behavior and include direct visual observations such as (a) elastic recoil of the fluid post pinch-off, (b) change in the appearance of the fluid from glassy to matte due to dilatancy, (c) jammed structures and (d) beads on a string structure. Amongst these signatures, only (d) was observed in the present study for non-Brownian colloidal suspensions subjected to extensional flow.

It must be noted here that the assumption of a truly uni-axially extensional flow is questionable and suspension homogeneity often does not hold. Also, the CABER-DoS setup employed herein is essentially a self-extensional flow implying that the fluid drainage by gravity will require infinite time for coalescence to be arrested. As an analogy with the pinch-off process in polymeric solutions, a similar delay caused by filament thinning has also been reported, although the physical origins of filament thinning lie in the macromolecular re-orientations of polymer chains. Thus, filament thinning is a manifestation of polymer viscoelasticity. Similarly, the particle clusters generate the same global response of the colloidal suspensions under extensional flow i.e., signs of filament thinning or pseudo-elasticity. This hypothesis has been recently substantiated by experimental reports on capillary breakup dynamics of colloidal suspensions^37,49,56 with particle weight fractions in the equivalent range as the present study.

4.2.3 Two phase flow or flow through porous particle aggregates?. The question might arise if the extended filament thinning is a manifestation of the solvent flow through porous particle assemblies formed due to jamming at the critical particle weight fraction. To address this, the approach by Smith et al.⁵⁹ is adopted where solvent flow velocity during extension is compared with the estimated fluid velocity arising due to Darcy's flow in such porous constructs. The Laplace pressure (Δp_L) acting across the fluid filament during extension generates a pressure gradient along the filament diameter D given bywhere σ is the surface tension of the suspension. This pressure gradient drives fluid flow through the jammed structures and assuming Darcy's law to be valid, the pressure drop per unit length can be expressed aswhere u(r) is the velocity of the fluid (solvent), i.e., water, K is the permeability and μ is the suspension viscosity. The permeability can be connected to the particle radius a by the empirical correlation K = ka², where k is the modified Carman–Kozeny coefficient.⁶² Equating the two expressions (eqn (1) and (2)) provides an estimate of the fluid velocity,and substitution of the relevant values (Table S4, ESI†) of the parameters in eqn (3) leads to an order estimate of fluid velocity, u ∼ O(10⁻⁶) ∼ μm s⁻¹. The velocity of fluid flow through the filament during extensional flow can be expressed aswhere r is the capillary bridge radius; . Substituting the relevant experimental timescale and length-scale, O(t) ∼ 10⁻³; O(r) ∼ 10⁻⁴, leads to an estimate of O(u_e) ∼ O(10⁻¹) m s⁻¹. Since u_e ≫ u it can be stated that the delayed pinch-off is due to jamming formed by temporary particle clusters that offer a physical or hydrodynamic obstruction rather than (Darcy's flow) solvent drainage through these particle clusters.

4.3. Colloidal droplet coalescence: a subset of sub-Newtonian coalescence

To elucidate the mechanism responsible for neck growth modulation by the particle weight fraction, a discussion of the forces at play in the system is considered hereafter. The force balance is formulated for the suspension in the region of interest i.e., the neck region of the liquid or the liquid bridge connecting the sessile–pendant droplets. Hence, the characteristic length l is equivalent to the neck radius R and the characteristic velocity . The neck growth is driven by the capillary force F_c and opposed by the inertial force F_i, viscous force F_v and additionally by the Brownian force F_b in the case of colloidal suspensions. Thus, the expression for force balance in the coalescence phenomenon for colloidal droplets is

F_c ∼ F_v + F_i + F_b. (11)

The relative magnitudes of these forces can be estimated by the triad of time-averaged dimensionless numbers – Peclet number, ; Reynolds number, and the Ohnesorge number Oh. These have been estimated as a function of particle weight fraction, ϕ_w (Fig. 7). Here, ρ, μ_c, and d_p (= 2a) denote the suspension properties i.e., density, viscosity at the critical shear rate (μ = μ_c) and the average particle diameter, respectively, and k and T are the Boltzmann constant and absolute temperature (298 K).

Fig. 7 Variation of the time averaged dimensionless numbers for the system, Reynolds number (〈Re〉, ■) and Peclet number (〈Pe〉, ), as a function of particle weight fraction, ϕ_w. Dotted lines are a guide to the reader's eyes only.

Note that the choice of viscosity magnitude μ is debatable since, unlike Newtonian fluids, here viscosity is a function of applied shear rate for a specific particle fraction. The possible values include the zero-shear viscosity (μ₀), viscosity at a critical shear rate (μ_c), and infinite shear viscosity (μ_∞). To capture the transition from CST to DST, the analyses here consider the viscosity at a critical shear rate (μ = μ_c; [small gamma, Greek, dot above] = [small gamma, Greek, dot above]_c) for each suspension up to particle weight fractions of ϕ_w = 0.41. This critical shear rate, [small gamma, Greek, dot above]_c marks the onset of shear thickening and is found to lie in the range of [small gamma, Greek, dot above]_c ∼ 5 to 25 s⁻¹, i.e., O(10⁰–10¹) for the investigated particle weight fractions.

The takeaways from the force balance approach can be stated as follows:

(a) Reynolds number – an estimation of time averaged flow Reynolds number reveals that 〈Re〉 > 1 implying that the coalescence phenomenon is inertia-dominated.

(b) Peclet number – the time-averaged Peclet number (Fig. 7) for the range of particle weight fractions investigated in this study is found to be 〈Pe〉 > O(10⁵), implying that the suspensions are non-Brownian. The effects of thermal forces are negligible, which reduces the force balance to

F_c ∼ F_v + F_i. (12)

(c) Ohnesorge number – Ohnesorge number couples the three forces in the system, i.e., inertial, viscous and capillary forces for droplet of diamter, d, as evaluated (Table 2) for the particle weight fraction ϕ_w < 0.36. It is found that Oh < 1 for ϕ_w < 0.36, implying that droplet coalescence for this particle weight fraction range is governed by the balance between the inertial and capillary forces. Furthermore, the surface tension of the colloidal suspensions is almost identical, σ (ϕ_w < 0.36) ∼ 55–58 mN m⁻¹, suggesting that it is the inertia-dominated regime. This is also substantiated by the power law exponent of b ∼ 0.5 obtained for coalescing droplets of colloidal suspensions, which is characteristic of coalescing Newtonian droplets.

Table 2 Variation of the Ohnesorge number (Oh) with the particle weight fraction ϕ_w

Particle weight fraction (ϕw)	0	0.26	0.32	0.35
Ohnesorge number (Oh)	0.04	0.25	0.63	0.80

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4.4. Correlation of the dimensionless number with the onset of discontinuous shear thickening (DST)

The magnitude of the Peclet number estimated at the particle scale, Pe(a) = 6πa³μ[small gamma, Greek, dot above]/kT, is known to govern the onset of discontinuous shear thickening (DST) in colloids.^63–65 Specifically, shear thickening was exhibited by suspensions with dispersed Brownian colloids for Pe(a) ≥ O(10²). A similar estimate, Pe(a), for the present system (Fig. S7, ESI†), where the dispersed colloids are non-Brownian, showed that with an increase in the particle weight fraction, Pe(a) mirrored the trend of suspension viscosity, i.e., a negligible effect at low shear rates (10⁻²–10⁻¹ s⁻¹), while a jump of approximately four orders, from O(10⁵) to O(10⁹) is seen in Pe(a) for [small gamma, Greek, dot above] > 10¹. Overall, the range is O(Pe(a)) ∼ 10⁴ > O(Pe(a)) (Brownian colloids), i.e., even in the case of non-Brownian colloids, the criteria for the inception of shear thickening based on the particle scale Peclet number remains valid and satisfied.³³ Also, it must be noted that the Stokes number Stk ∼ O(10⁻³), Stk ≪ 1, implying that the particles are relaxed in the given flow field irrespective of the particle weight fraction.^63–65

To summarize, coalescing colloidal droplets are a sub-set of the broad umbrella of sub-Newtonian coalescence for ϕ_w ≥ 0.35, whereas it is inertia-dominated Newtonian coalescence for ϕ_w ≤ 0.35. The reduction in the power law exponent b < 0.5 for ϕ_w ≥ 0.35 stems from hydrodynamic and frictional interactions between particle clusters. This is an important and striking contrast since although the deviation from the universal value of Newtonian fluid (b ∼ 0.5) has been reported for polymeric fluid droplets, b ≤ 0.38,^7,9,39 in the inertia-dominated regime. To understand this delayed coalescence of colloidal droplets, consider an analogy with coalescing polymeric droplets. Deviation from the classical exponent, b ∼ 0.5, has been attributed to the reorientation and stretching of macromolecular polymer chains. These re-orientations are the major contributors to the ‘slowed’ merging of polymeric fluid droplets. This stems from the additional degree of freedom available to the macromolecules, i.e., their elasticity. In colloidal droplets this is analogous to the sub-diffusive motion of Brownian particles that is responsible for the slowed merging. This falls under the broad umbrella of coalescence dynamics in complex fluids that has been recently dubbed as sub-Newtonian coalescence but with a distinctive origin for energy dissipation. Through this study, we further delved into the sub-class of colloidal suspensions, i.e., non-Brownian colloidal suspensions, where the particle–particle interactions dominate at higher shear rates, causing the rupture of the solvent film around each particle, and frictional contacts drive the formation of local particle aggregates. We hypothesize that the formation of local particle clusters at ϕ_w ≥ 0.35 and their orientation along the direction of fluid flow in the neck-region act as an additional opposing force. Similar re-arrangements of molecules can occur in a plethora of complex fluids ranging from magneto-rheological fluids (MRFs) to electro-rheological fluids (ERFs) and liquid crystals under the action of pertinent external field.

4.5. Colloidal droplet coalescence: validation and lacunae of the proposed theoretical model

Notably, at the critical concentration, ϕ_w = ϕ_c = 0.35, a sharp decrease in the power law exponent of coalescence b can be observed. This transition is a manifestation of solid-like behaviour of the colloidal suspension due to the presence of dispersed local particle clusters.

Insights – The proposed model explains the behaviour through the scaling constants that are implicit functions of fluid viscosity parameters – μ, μ_∞, μ₀etc. Each term in eqn (5) corresponds to a specific force governing the coalescence phenomenon. The first term (U²) scales with velocity as ∼U² representing the inertial force. The opposing forces can be sub-divided into the (a) frictional force arising from the inter-particle interactions (term 2, F(Γ,ρ,R,n,η₀,η_∞)U¹⁻ⁿ) that is unique to colloidal suspensions such as corn-starch suspensions and the (b) viscous force given by the term 3, G(ρ,R,η_∞)U. These are balanced by the driving capillary force expressed as a function of fluid properties of surface tension, density acting along the characteristic length-scale, i.e., necking radius, H(σ,ρ,R,R₀). For lower particle concentration, i.e., ϕ_w < 0.36 the effect of inter-particle interactions or frictional force (term 2) was found to insignificant compared to the other terms. In such a case, eqn (5) merely reduces to that of coalescing Newtonian droplets, where the characteristic power law exponent b = 0.5 is recovered, as reported by Xia et al.⁶ (please refer to the ESI,† Note S5 and Table S3 for numerical values of the scaling constants).

But for higher particle concentration, ϕ_w ≥ 0.35, the effect of inter-particle interactions comes into picture and the formation of particle clusters is accounted through the changes in suspension viscosity. This is captured by rheometry based characterisation of suspension rheology as well as by extensional rheology. Care has been taken to include the parametric value of infinite shear viscosity, μ_∞, which pertains to the characteristic shear rate encountered during the coalescence process O(10²)–O(10³). A snowball effect of the rheological response can be seen in the phenomenological response, i.e., the coalescence rate. Theoretically, this is considered through the altered viscosity of the suspension and acts as the dominant opposing force in the system. This strong influence of the inter-particle forces is accounted through term 2 and excellent mapping of the experimental data with the proposed theoretical model through this approach is evident (Fig. 8). The present model although primitive captures the distinctive behaviour of the coalescing colloidal droplets based on particle concentration, i.e., Newtonian behaviour for ϕ_w < 0.36 and friction force-dominated for densely populated suspensions, ϕ_w ≥ 0.36. It is evident that the proposed model predicts the experimental necking behaviour successfully.

Fig. 8 Superposition of theoretical temporal profiles (T) of the neck radius over the experimental profiles (E) as a function of the particle weight fraction, ϕ_w. Here, R* and t* are non-dimensionalised neck radius and time, resp. R₀ is the initial droplet radius and τ _i (= ρR₀³/σ)^0.5 is the inertial timescale. σ (mN m⁻¹): suspension surface tension.

Lacunae – It must be stated that a universal model for coalescence of complex fluids may not be feasible since each complex fluid has a distinctive constitution/micro-structure. This constitution influences the response of the fluid during coalescence and is specific to each class of complex fluids. The microstructure of the complex fluid under simple shear showed distinctive power law fluid behavior and to mimic this the Cross constitutive model has been employed in the present study. While this is typically applicable for generalized Newtonian fluids, strictly speaking, it does not capture the viscoelastic nature of the fluid. The emergence of viscoelastic behaviour under simple shear was observed only beyond a critical particle weight fraction, i.e., ϕ_w > 0.36, where oscillatory shear response showed the dominance of the storage modulus over the loss modulus G′ ≫ G′′. Considering extensional response to be a better analogy of coalescence configuration (sessile–pendant), then signatures of viscoelastic behavior in terms of ‘beads-on-a-string structure’ and prolonged filament thinning were also observed for ϕ_w > 0.36. These responses (shear and extensional) (Table 3) coincide with the sudden decrease in the power law exponent b that characterizes the coalescence dynamics; therefore, it is a conjecture that this ‘pseudo’ visco-elastic nature may be responsible for the slow-down in the coalescence dynamics. Furthermore, it must be noted that the same scaling analysis and theoretical model have been employed to capture the neck evolution dynamics during coalescence of sessile–pendant droplets belonging to two different sub-classes of complex fluids like colloidal suspensions (Brownian)⁴⁰ and polymeric solutions,^7–10,39 albeit with appropriate constitutive equations. The only drawback of this model lies in its inability to account changes in coalescence for ϕ = 0.41. To include the higher particle weight fractions, ϕ = 0.41 and 0.42, it would be necessary to quantify the distribution of the particle-cluster-rich region and the solvent-rich region in experiments and incorporate the same in the theoretical framework. This is beyond the scope of the present work and will be addressed in subsequent investigations.

Table 3 Coalescence in complex fluids: colloidal suspensions vis-à-vis polymeric solutions

Sr. no.	Complex fluid system	Shear rheology	Amplitude sweep	Extensional rheology	Exponent, b in R ∝ t^b (coalescence dynamics);
1.	Colloidal suspensions Brownian^40 silica fumed dp ∼ 0.2–0.3 μm	ϕ v ∼ (0.00045–0.044): strongly shear thinning.	G′ > G′′ visco-elastic fluid	• Filament thinning	b < 0.5
		ϕ v ∼ 0.054: approaches sol–gel transition		• Abrupt break-up
				• Negligible extensional elasticity
2.	Colloidal suspensions non-Brownian corn-starch, cesium chloride, and aqueous suspensions dp ∼ 14 μm (current study)	Shear thinning ϕw < 0.32	, response transits from fluid-like to solid-like with signs of pseudo-elasticity	• Filament thinning	b = 0.5
				• Break-up
		CST and DST ϕw > 0.35	, solid like behaviour	• Beads-on-a-string structure	b < 0.5
				• No fracture
3.	Non-colloidal suspensions^40 silica glass spheres dp ∼ 9–13 μm	Weakly shear thinning higher ϕv ∼ 0.054: sol–gel transition	viscous effect is predominant	The same as Brownian colloidal suspensions	b ≤ 0.5
4.	Polymeric solutions^40 polyethylene oxide^9Mw = 5 × 10^6 g mol^−1c = 0.1% g mL^−1	Newtonian^40 for c/c* < 3 and shear thinning^9,40 behaviour for c/c* > 3	G′ > G′′^9	• Filament thinning	b < 0.5
				• Exhibited finite extensional elasticity^66

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The burgeoning interest in the development of microfluidic devices^67–69 for DNA-based applications⁷⁰ and point of care diagnostics involves droplets with suspended micro/nanoparticles, i.e., colloids as the operational entities. This led to the exploration of the four basic droplet operations – creation, splitting, merging (coalescence) and transport, which form the cornerstone of microfluidic devices.⁶⁷ However, the dynamics of colloidal droplet coalescence remains far from being understood. We believe the present study will offer new perspectives on this front and assist in predicting coalescence behaviour as a function of particle content.

5. Conclusions

This work, for the first time, establishes via extensive experiments the role of particle content in modulating coalescence dynamics of colloidal droplets. The model colloidal suspensions chosen in this study were aqueous corn starch suspensions that display shear thinning and discontinuous shear thickening characteristics in dilute and concentrated forms, resp. We report that this typical shear thickening behaviour has a bearing on the coalescence dynamics. In particular for particle weight fractions beyond a critical particle weight fraction, i.e., ϕ_w > ϕ_c (= 0.35), delayed coalescence dynamics were observed. This allows us to conclude that the evolution of the liquid bridge in coalescing colloidal droplets is another example of the sub-Newtonian coalescence. We attribute this delay to local particle cluster formation as a response to applied shear during coalescence. An analysis of the relevant forces prevalent in the system – capillary, inertial, viscous and Brownian, shows that the inter-particle friction can be accounted for through the viscous forces and they are the dominant opposing forces to the driving capillary forces. These viscous forces are accounted for through our proposed theoretical model, showing excellent agreement with the experimentally observed necking behaviour. We hope the present study will serve as a starting point in handling shear thickening non-Brownian colloidal droplets for droplet operations such as coalescence. This will supplement the design of digital microfluidic platforms and diagnostic devices that employ colloidal suspensions or entities heavily for techniques such as DNA assays, proteomics, etc.

Data availability

The data supporting this article have been included as part of the ESI.†

Conflicts of 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.

Acknowledgements

UUG acknowledges generous support received from the SERB POWER grant SPG/2021/003516. AK acknowledges support from the SERB grant CRG/2022/005381. The authors would also like to acknowledge the contribution of Mr Sreyajyoti Mondal, School of Biomedical Engineering, IIT (BHU) for assisting in sketching the schematic of the proposed coalescence mechanism. Further the authors would like to acknowledge Mr. Vivek P. Wagh, Ms. Mansi Sharma and Prof. Ankit Gupta from the Department of Civil Engineering, IIT (BHU) Varanasi and Mr. Rohit Bharti, Department of Chemical Engineering & Technology IIT (BHU), Varanasi for assisting in the rheological characterization required for addressing the reviews.

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Footnotes

† Electronic supplementary information (ESI) available. See DOI: https://doi.org/10.1039/d4sm01389a

‡ Equal contribution.

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