The onset of heterogeneity in the pinch-off of suspension drops

Deforming a dense suspension leads to the rearrangement of the particles, which causes dissipation. The local volume fraction of particles remains constant if the particles are simply sliding or rolling along each other and fluctuates if they move away or toward each other. We hypothesize that the detachment of a suspension drop accelerates because particles start moving away from each other in the vicinity of the neck and refer to this phenomenon as the dislocation of the suspension ligament. The word dislocation is not to be taken in its meaning of defect in a lattice, for there is no order in the suspension, but is to be taken in its etymological meaning of sudden separation. The dislocation mechanism begins when pulling particles apart becomes energetically favorable compared with rearranging them. In other words, dislocation is the growth of local fluctuations of the particle volume fraction.

To describe the dislocation regime, we consider a liquid film of thickness a between two portions of suspension, which represents the local fluctuation of concentration in the neck. The viscosity of the suspension is denoted η; the viscosity of the interstitial liquid is denoted ${\eta }_{\text{f}}$. Pulling the two portions apart at the velocity $\dot{a}$ dissipates the power ${\eta }_{\text{f}}{h}^4{\dot{a}}^2/a^3$ (Materials and Methods). We assume that during the dislocation regime, a is of the order of magnitude of the particle diameter so that $a\sim d$. By balancing the power associated with the capillary forces $\gamma h^2/(t_{\text{c}}-t)$ and the power associated with the viscous dissipation, we obtain the scaling law for the thinning dynamics in the dislocation regime:

Eq. 1 describes the evolution of the thickness of the neck as the capillary pressure makes particles move away from each other and captures well the dynamics in the regime (II) shown in Fig. 2C. We systematically observed this scaling law in all of our experiments for suspensions with volume fraction $\varphi \ge 20\%$. For dilute-enough suspensions, there is a range of volume fractions in which the thinning of the neck shifts directly from the equivalent regime (I) to the interstitial regime (III). Therefore, the dislocation of the suspension only occurs for concentrated-enough suspensions, and the corresponding minimum volume fraction ranges from less than 2% for 20-µm particles to 20% for 500-µm particles.

Fig. 3A reports the evolution of the critical thickness $h^{\mathrm{\star }}$ at the transition between the equivalent fluid regime (I) and the dislocation regime (II) when varying the volume fraction $\varphi$ for monodisperse suspensions with different particle diameters. The thickness $h^{\mathrm{\star }}$ increases both with $\varphi$ and d. Indeed, the larger the particles, the larger the thickness $h^{\mathrm{\star }}$ is, and the denser the suspensions, the larger $h^{\mathrm{\star }}$ is. Interestingly, the heterogeneities brought by the particles start occurring at a much larger length scale than the particle diameter. The critical thickness of the transition between the dislocation regime (II) and the interstitial regime (III), $h^{\prime }$, depends on the particle diameter d but not significantly on the volume fraction, as illustrated in Fig. 3B. This observation is consistent with the concept of dislocation since $h^{\mathrm{\star }}$ marks the onset of heterogeneity when the suspension begins to dislocate, whereas $h^{\prime }$ marks the end of dislocation when the suspension in the neck is so dilute that it cannot be considered a suspension anymore.

Eq. 1 provides a prediction of the critical thickness $h^{\mathrm{\star }}$ as a function of the properties of the suspension. Assuming that before dislocation, the dynamics results from the balance between gravity and the effective viscosity of the suspension, the relevant timescale in the equivalent liquid regime (I) is given by $t^{\mathrm{\star }}=\eta /(\rho \hspace{0.17em}g\hspace{0.17em}{h}^{\mathrm{\star }}\hspace{0.17em})$. By combining this viscous–gravitational timescale with Eq. 1, we obtain $h^{\mathrm{\star }}\hspace{0.17em}/d\sim {\eta }_{\text{r}}{\hspace{0.17em}}^{1/3}\hspace{0.17em}{\left({\ell }_{\text{c}}/d\right)}^{2/3}$, where ${\ell }_{\text{c}}=\sqrt{\gamma /(\rho g)}$ is the capillary length (1.5 mm for the silicone oil used here). We estimate the relative viscosity of the suspension ${\eta }_{\text{r}}\hspace{0.17em}(\varphi )=\eta (\varphi )/{\eta }_{\text{f}}$ using the Maron–Pierce correlation (32), ${\eta }_{\text{r}}\hspace{0.17em}(\varphi )={\left(1-\varphi /{\varphi }_{\text{c}}\right)}^{-2}$, with ${\varphi }_{\text{c}}\simeq 57.8\%$ for the suspensions used here (17). Finally, we obtain the evolution of the critical thickness at the dislocation regime:

Fig. 3C compares the rescaled dislocation threshold $h^{\mathrm{\star }}\hspace{0.17em}/d$ with the quantity ${({\ell }_{\text{c}}/d)}^{2/3}{(1-\varphi /{\varphi }_{\text{c}})}^{-2/3}$, which includes the two input parameters of our experiments: the particle diameter d and the volume fraction $\varphi$. We report in Fig. 3C the experimental results for all monodisperse suspensions considered in this study. The experimental results collapse onto a master line, which matches Eq. 2 with the prefactor 0.6. The prediction only fails for very dilute suspensions of large particles, shown by the purple points in the lower left of Fig. 3C. In this case, the concept of dislocation becomes irrelevant since there are not enough particles in the neck to consider it as homogenous even during the early stage of the pinch-off. The observation that this approach fails only in this most extreme situation emphasizes the robustness of the model, despite the rough assumptions made previously.

The excellent agreement between our experiments and the model demonstrates that the acceleration of the thinning for dense-enough suspensions is indeed induced by the dislocation at the neck. Changing the solvent viscosity and wetting properties from 0.12 Pa×s (silicone oil) to 2.6 Pa×s [poly(ethylene glycol-ran-propylene glycol) monobutyl ether (PEG)] (diamonds in Fig. 3C) does not affect the collapse of the data and the agreement between the model and the experiments. We were also able to extract the value $h^{\mathrm{\star }}$ from recent experiments performed by Moon et al. (13), which provided the thinning dynamics for suspensions of 10-µm particles. The model presented in this paper also captures their experimental data (SI Appendix).

Similarly to Eq. 2, we can derive the threshold $h^{\prime }$ between the dislocation regime (II) and the interstitial regime (III). The typical timescale at this transition is given by $t^{\prime }\sim {\eta }_{\text{f}}/(\rho g{h}^{\prime })$. Combining this timescale with Eq. 1 yields

The corresponding experimental measurements of $h^{\prime }$ plotted with respect to ${({\ell }_{\text{c}}/d)}^{2/3}$ for different monodisperse suspensions are reported in Fig. 3D. Once again, we obtain an excellent agreement between the model of Eq. 3 and the experimental results. The collapse of the data onto the proposed law provides a second proof that the acceleration of the thinning is caused by the dislocation. Similar to what was observed for $h^{\mathrm{\star }}$, the prediction for $h^{\prime }$ fails for very dilute suspensions of large particles. In these cases, it is notable that $h^{\mathrm{\star }}$ and $h^{\prime }$ are very close and smaller than the particle size d. This means that the thinning switches directly from the equivalent regime to the interstitial regime. Indeed, since not enough particles are present in the neck, there is no dislocation mechanism.

Monodisperse suspensions characterized by a single length scale, the diameter d of the particles, are valuable to study due to their simplicity. Nevertheless, actual manufacturing applications or capillary processes primarily involve polydisperse suspensions. In a polydisperse suspension, each possible particle size is theoretically a relevant length scale of the flow. As a first step, we consider the dislocation of bidisperse suspensions. Bidisperse suspensions contain a volume fraction of particles $\varphi$, split among ${\varphi }_{\text{S}}$ of small particles of diameter $d_{\text{S}}$ and ${\varphi }_{\text{L}}$ of large particles of diameter $d_{\text{L}}$. We introduce the volume ratio of small particles $\xi ={\varphi }_{\text{S}}/\varphi$ and the size ratio of particles $\delta =d_{\text{L}}/d_{\text{S}}$ (33). For all of these suspensions and a volume fraction $\varphi$ ranging from 0 to 50%, we observe a thinning dynamic similar to that of the monodisperse case, with the three successive regimes. The main difference is that the thresholds of the dislocation regime, $h^{\mathrm{\star }}$ and $h^{\prime }$, now depend on the composition parameters, ξ and δ.

Polydisperse suspensions are defined by the size distribution of their particles, which we can be described through a volume-averaged particle diameter $\overline{d}$. For a bidisperse suspension, this quantity is $\overline{d}=\xi d_{\text{S}}+(1-\xi )d_{\text{L}}$. The choice of the volume-averaged diameter over the number-averaged diameter will be justified in the last section. In addition to the particle size distribution, polydisperse suspensions also differ by their lower viscosity for a given particle volume fraction than a monodisperse suspension. Indeed, the polydispersity of the particles enables a more efficient filling of the space, described by a larger value of the critical volume fraction ${\varphi }_{\text{c}}(\xi ,\delta )$. The viscosity of a polydisperse suspension can be computed from the jamming fraction of a polydisperse sphere packing using the Maron–Pierce correlation (17, 32), ${\eta }_{\text{r}}\hspace{0.17em}\sim {\left(1-\varphi /{\varphi }_{\text{c}}(\xi ,\delta )\right)}^{-2}$, where the critical volume fraction ${\varphi }_{\text{c}}$ is estimated through the model of Ouchiyama and Tanaka (34). From there, we compute $h^{\mathrm{\star }}$ and $h^{\prime }$ using $\overline{d}$ and ${\varphi }_{\text{c}}(\xi ,\delta )$ in Eqs. 2 and 3.

Fig. 4A reports the evolution of $h^{\mathrm{\star }}\hspace{0.17em}/\overline{d}$ and Fig. 4B reports the evolution of $h^{\prime }/\overline{d}$ compared with the predictions of Eqs. 2 and 3. The colored symbols represent the bidisperse suspensions, and their color indicates the volume ratio of small particles ξ. The red crosses represent experiments performed with tridisperse suspensions composed of one-third 20-µm particles, one-third 140-µm particles, and one-third 500-µm particles and a total particle volume fraction $\varphi$ ranging from 2 to 50%. The black open symbols represent the monodisperse suspensions, already shown in Fig. 3 C and D. We show all experiments on the same graph to highlight the universality of our model beyond the simplest monodisperse case usually considered in the literature.

For polydisperse suspensions, the critical thicknesses defining the thresholds of the dislocation regime, $h^{\mathrm{\star }}$ and $h^{\prime }$, collapse onto the same master line as the monodisperse suspensions. The limitations are the same as in the monodisperse case. More specifically, dilute suspensions of large particles do not dislocate because there are too few particles in the neck. The collapse of $h^{\prime }$ vs. $\overline{d}$ shown in Fig. 4B is even more apparent. Indeed, using a polydisperse suspension allows for tuning the volume-average diameter, and these experiments clarify the dependence of $h^{\prime }$ on the particle diameter. Therefore, it appears that the dislocation regime only depends on the size distribution through the volume-averaged diameter $\overline{d}$ and the critical volume fraction ${\varphi }_{\text{c}}(\xi ,\delta )$.

A notable feature in the pinch-off of suspension drops is that the threshold to the heterogeneous regime $h^{\mathrm{\star }}$ can be much larger than the diameter of the particles d and also depends on the volume fraction $\varphi$. Indeed, a classical interpretation of the transition between a homogeneous and heterogeneous regime is that the equivalent fluid regime stops when the scale of the flow (here, the neck width h) becomes comparable with the particle size (11, 15). Such an interpretation would lead to $h^{\mathrm{\star }}\hspace{0.17em}\sim d$, with a proportionality factor of order one. However, our experiments demonstrate that although for dilute suspensions, $h^{\mathrm{\star }}$ is indeed of the order of the particle diameter, it becomes much larger than d at larger volume fractions. For instance, for $\varphi =50\%$, we measure $h^{\mathrm{\star }}\hspace{0.17em}=1.6$ mm for 140-µm particles, which is more than 11 times the diameter of the particles d. With 20-µm particles, we measure $h^{\mathrm{\star }}\hspace{0.17em}=1.24$ mm, 62 times larger than the particle diameter. More generally, Fig. 3C shows that the ratio $h^{\mathrm{\star }}\hspace{0.17em}/d$ varies over two orders of magnitude and strongly depends on the volume fraction. The polydisperse suspensions exhibit the same trend, as reported in Fig. 4A. Therefore, the heterogeneous structure of dense suspensions plays a role at a scale much larger than the particle diameter. A similar feature has been previously reported for the free surface flow of dense suspensions on an inclined plane. Indeed, Bonnoit et al. (22) have defined a mesoscopic length scale that increases with the volume fraction and diverges at the critical volume transition ${\varphi }_c$. The critical thickness $h^{\mathrm{\star }}$ reported in this study is similar and corresponds to the limit of the homogeneous regime, at the onset of heterogeneity. In the present case, this length would also diverge when $\varphi \to {\varphi }_{\text{c}}$.

Physically, the length $h^{\mathrm{\star }}$ can be seen as the coherence length of the interactions through which momentum diffuses. As a particle undergoes a fluctuation of its position, it exerts a force upon its neighbors through the lubrication film between them. If there are contacts between particles, they transmit the momentum directly. Other forces can also act on the particles. For instance, for dilute suspension, the viscous drag on the particle should also be taken into account. If the particles are small enough, they will be subject to Brownian motion and van der Waals forces. The coherence length can then be seen as the typical length over which a fluctuation will be damped. Therefore, a stress applied on the suspension will be distributed across a volume defined by the coherence length. In the present configuration, during the detachment of the drop, the curvature of the neck generates a capillary pressure gradient between the neck and the rest of the suspension. The resulting stress will be distributed across the suspension if the neck is thicker than the coherence length. However, if the neck is thinner than the coherence length, a fraction of the suspension may experience larger local stress, which pulls the particles apart from each other and leads to the breakup through dislocation. Therefore, the coherence length here controls the stability of the suspension against a concentration fluctuation.

Here, the words fluctuation and coherence length are meant to suggest an analogy with statistical mechanics, even if it is only a rough sketch. As the suspension flows, each particle creates a local disturbance, and one can imagine an ideal state of local equilibrium where the distance between particles is maximum so the global dissipation is minimum. Deviations from this ground state would be caused by the preparation of the suspension and by the constraints applied on the suspension (shape of the container, stress, body force). There are similarities with this approach and the Edwards ensemble (35, 36), in which the volume fraction of granular media plays the role of energy. For suspensions, volume fraction is directly linked to the rate of dissipation (by the viscosity), which is the quantity that should be minimized.

The correlation length also applies to polydisperse suspensions, although the range of sizes of the particles makes the interactions between particles more complex. Assuming that the volume of the lubrication film around a given particle is directly proportional to the volume of that particle, the lubrication pressure between small particles and between large particles is of different magnitude. This explains why $h^{\mathrm{\star }}$ is relatively much larger for 20-µm particles than for larger particles (Fig. 2C). Indeed, for smaller particles, the relative lubrication pressure between them is stronger, and therefore, a perturbation may spread further. Therefore, the coherence length should depend not only on the number of interactions but also, on the relative force of these interactions. As a result, the relevant length scale of polydisperse suspensions is the volume-averaged diameter rather than the number-average diameter. Similarly, the Ouchiyama–Tanaka model (34) with which we compute ${\varphi }_{\text{c}}$ for the polydisperse suspensions is based on a volume average of the local volume fraction. Therefore, the pinch-off dynamics, particularly the transition between the different regimes, of a polydisperse suspension can be inferred from the volume-averaged particle diameter and volume-averaged volume fraction.

Our pinch-off experiments with bidisperse suspensions reveal that they follow dynamics similar to monodisperse suspensions, meaning that pinch-off does not filter particles by size. The dislocation model enables us to understand why; despite the size difference, the movement of small particle remains strongly correlated to that of large particles, and they are not easily separated. This good agreement of our model with the bidisperse suspensions enable us to neglect self-filtration (37, 38) as a mechanism for the accelerated pinch-off.

Contrarily to $h^{\mathrm{\star }}$, the length $h^{\prime }$ does not depend much on the volume fraction $\varphi$ (Fig. 3C). From a neck thickness $h(t)=h^{\prime }$ and smaller, the neck reduces to a filament of interstitial liquid, without any particle in it. Just before this transition, the neck contains two particles, and the film between them experiences all of the stress acting on the neck. Thus, $h^{\prime }$ can be interpreted as the thickness that balances the viscous interaction between the last two particles of the neck and the driving force of the thinning (capillarity and the weight of the drop in the present case).

Various studies on the pinch-off of suspension have reported qualitatively similar results. In particular, the pinch-off of suspension drops from other studies should also be explained by the mechanisms and scaling law presented in this work (18, 19). In both studies, the thinning of the suspension was empirically interpreted in terms of non-Newtonian rheology. We demonstrate in the following that the concept of dislocation can explain these results without involving a non-Newtonian rheology.

Roché et al. (18) studied the thinning of cornstarch suspensions in water, with an average particle diameter of 14-µm. For $\varphi =37\%$, they observed a long filament that would eventually destabilize into “jammed” regions where the strain rate was zero and “flowing” regions where the thinning continues. They measured the wavelength between two flowing regions as 700-µm, although the theory of the Rayleigh–Plateau instability predicts 7 mm. Applying Eq. 2 to their configuration ($d=14\mathrm{\mu }\text{m},\hspace{0.17em}{\ell }_{\text{c}}=$ 2.7 mm for water), we obtain 628-$\mathrm{\mu }\text{m}<h^{\mathrm{\star }}\hspace{0.17em}<875\mathrm{\mu }\text{m}$ in their range of volume fraction ($23\%<\varphi <39\%$). Thus, if the length of the neck is much longer than $h^{\mathrm{\star }}$ when $h(t)=h^{\mathrm{\star }}\hspace{0.17em}$, several dislocations can happen simultaneously, and $h^{\mathrm{\star }}$ is then the wavelength of the destabilization. Although the authors considered the regions not thinning as jammed, it does not necessarily need to be the case, and the thinning simply continues where the viscosity is the lowest (i.e., in the dislocating regions). Pan et al. (19) studied denser suspensions of smaller particles ($1.3\hspace{0.17em}\mathrm{\mu }\text{m}\le d\le 10\hspace{0.17em}\mathrm{\mu }\text{m}$, with $55\%\ge \varphi \ge 59\%$). The thinning dynamics of their aqueous suspensions are similar to those of our viscous liquids; however, they did not observe dislocation. In their configuration, the coherence length of their suspensions, calculated through Eq. 2, is a few millimeters depending on the jamming fraction. The filament is then always shorter than the wavelength of the destabilization and is, therefore, stable.

In a recent study, Château et al. (29) observed the pinching and breakup of capillary bridges of suspensions. Although their configuration is different from a pending drop, they also reported a deviation from the homogeneous liquid regime but did not provide a model to explain the variations of $h^{\mathrm{\star }}$. The scaling law for the thinning regime (Eq. 1) matches their data for $\varphi >20\%$, whereas a linear law fits better for $\varphi \le 20\%$ (SI Appendix). Therefore, it seems that by plotting the last moments of the dynamics, Château et al. (29) were observing the interstitial regime for $\varphi \le 20\%$ and the dislocation regime for $\varphi >20\%$. In the latter case, the interstitial regime may have been too short to be observed. However, their empirical scaling for $h^{\mathrm{\star }}$ differs from ours, probably because the flow is different.

The suspensions are made of polystyrene particles (DynoSeeds from Microbeads) with measured diameters of 21.6 ± 0.9, 80.3 ± 5.0, 144.2 ± 8.3, 249.0 ± 4.2, and 578.1 ± 10.1 µm. These particles are referred to in the article as 80, 140, 250, and 500-µm, respectively. The roughness of the particles is of order 100 nm (39), and their density is in the range $\rho =1050-1060$ kg/m³. The particles are dispersed in a density-matched interstitial liquid to prevent buoyancy effects. We primarily used AP100 silicone oil (from Sigma Aldrich) of shear viscosity ${\eta }_{\text{f}}=120$ mPa×s and surface tension $\gamma =24\pm 2$ mN/m at 20 ${}^{°}$ C, which perfectly wets the particle. We also conducted experiments with particles dispersed in PEG (Sigma Aldrich), for which ${\eta }_{\text{f}}=2.5$ Pa×s and γ  =  45 mN/m. The molar weight being as low as 3,900 g/mol, this solvent can be considered Newtonian. For both interstitial liquids used in this study, the settling time of the particles is much longer than the timescale of the experiments so that the suspensions can be considered as neutrally buoyant.

The bidisperse suspensions are composed of the same interstitial fluid (silicone oil AP100) and made using a couple of particle sizes ($d_{\text{S}},\hspace{0.17em}{d}_{\text{L}}$) chosen among (20, 80-µm), (20, 140-µm), (20, 250-µm), (80, 140-µm), and (80, 250-µm). In these experiments, we also vary the volume fraction of small particles $\xi ={\varphi }_{\text{S}}/({\varphi }_{\text{S}}+{\varphi }_{\text{L}})$. The tridisperse suspensions contains one-third 20-µm particles, one-third 140-µm particles, and one-third 500-µm particles.

The suspensions are transferred to a syringe and then manually extruded through a nozzle of outer diameter 2.75 mm (for wetting liquids, the outer diameter is the relevant length scale). The extrusion is conducted slowly to avoid any inertial effects. The experiments are recorded using a high-speed camera (Phantom VEO 710) equipped with a macro lens (Nikon Micro-Nikkor 200-mm AI-S). To resolve perfectly the contour of the drop and the filament, we place a panel of light emitting diodes (LEDs, from Phlox) behind the experimental setup. The time evolution of the contour of the drop and the minimum diameter of the filament h(t) are extracted through custom-made routines using ImageJ and Python.

To measure $h^{\mathrm{\star }}$, we rescale the time as ${\alpha }_{\eta }(t_{\text{c}}-t)+\mathrm{\Delta }t$ and find the critical thickness at which the rescaled dynamics of the suspension deviates from the dynamics of the interstitial liquid. The stretching parameter ${\alpha }_{\eta }$ accounts for the viscosity difference between the suspension and the Newtonian liquid used for comparison. The time shift $\mathrm{\Delta }t$ accounts for the acceleration of the thinning due to the presence of the particles. Previous work at constant volume fraction $\varphi$ suggested that ${\alpha }_{\eta }$ can be seen as the viscosity ratio between the suspension and the comparative Newtonian liquid ${\eta }_{\text{r}}\hspace{0.17em}=\eta /{\eta }_{\text{f}}$ (17). However, the present study shows that this result does not hold if the difference in viscosity between the suspensions and the viscous liquid used for comparison is too large. Although ${\alpha }_{\eta }$ follows the same divergence as the viscosity, the evolution of ${\alpha }_{\eta }$ over a broad range of volume fraction $\varphi$ does not quantitatively match the viscosity of the suspensions measured with a rheometer. The length $h^{\mathrm{\star }}$ is the thickness at which the rescaled dynamics of the suspension deviates from that of the interstitial fluid. We defined it systematically as the point where the thinning rate $\text{d}h/\text{d}t$ of the suspension differs by more than 5% from that of the interstitial liquid. The length $h^{\prime }$ (smaller than $h^{\mathrm{\star }}$) is defined as the thickness at which the thinning dynamics of the filament becomes linear (i.e., follows a capillary-viscous regime).

To describe the dislocation regime, we consider the stretching of a liquid film of thickness a between two portions of suspension pulled apart at the velocity $\dot{a}$. We estimate the power dissipated in the flow within. Assuming that the suspension on each side of the film is much more viscous than the interstitial liquid (${\eta }_{\text{r}}\hspace{0.17em}\gg 1$), we approximate the flow as that of two plates pulled apart. The liquid film is then set between two cylindrical plates of diameter h at positions z = 0 and z = a. By solving the Stokes equation in the lubrication approximation, we obtain the axial velocity gradient:

We integrate the velocity gradient over the volume of the film and obtain the order of magnitude for the power of viscous dissipation in the neck: