Granular Character of Particle Rafts

We consider a monolayer of particles floating at a horizontal liquid-gas interface—a particle raft. Upon compressing the monolayer in a Langmuir trough, the particles at first pack but ultimately the monolayer buckles out of the plane. We measure the stress profile within the raft at the onset of buckling and show for the first time that such systems exhibit a Janssen effect: the stress decays exponentially away from the compressing barriers over a length scale that depends on the width of the trough. We find quantitative agreement between the rate of decay and the simple theory presented by Janssen and others. This demonstrates that particle rafts have a granular, as well as elastic, character, which is neglected by current models. Finally, we suggest that our experimental setup may be suitable for exploring granular effects in two dimensions without the complications of gravity and basal friction.

Figure 1

The experimental setup. (a) We use a Langmuir trough with barriers attached to a computer-controlled stepper motor. The width of the trough, $w$, is varied by moving the strips beneath the barriers. The computer is calibrated to infer the surface pressure $\Pi$ from the force on the Wilhelmy plate (red rectangle). In the configuration shown in the main figure it measures ${\Pi }_{\parallel }$. To measure ${\Pi }_{\perp }$, the plate is rotated by 90° in the ($x$, $y$)-plane, as shown in the inset. Dimensions of the trough itself are for the NIMA 600. (b) Images showing the center of the raft at three stages of compression. From left to right: initial state, onset of nonzero pressure and final, highly buckled, state. The scale bar represents 1 cm.

Figure 2

Examples of experimentally measured surface pressure isotherms showing $\Pi$ as a function of the distance between the barriers and the sensor, $d$. $\Pi$ increases rapidly as $d$ decreases but then exhibits a kink (marked by $\times$ in b). This kink corresponds to the onset of buckling. Isotherms are shown for a range of trough widths, $w$, and for different numbers of particles spread on the surface. (a) With a fixed number of particles but decreasing trough width, the value of ${\Pi }_{\parallel }$ measured at the onset of buckling decreases. (b) For fixed trough width but increasing number of particles, the value of ${\Pi }_{\parallel }$ at buckling decreases.

Figure 3

Snapshots of the meniscus formed around the Wilhelmy plate compared to the theoretically predicted meniscus shape. Predicted shapes (solid curves) are obtained by solving the Laplace-Young equation [21] with the value of the surface tension corresponding to the measured surface pressure: $\gamma ={\gamma }_c-\Pi$.

Figure 4

Experimental data showing the pressure measured at the onset of buckling, ${\Pi }^{\mathrm{buck}}$, as a function of the aspect ratio of the trough at this point, $d_c/w$. Results are shown for (a) ${\Pi }_{\perp }$ and (b) ${\Pi }_{\parallel }$. In each panel there are two data sets, corresponding to two automated measures of ${\Pi }^{\mathrm{buck}}$ from surface pressure isotherms: the inflection point (open symbols) and the point where the derivative falls to small values (closed symbols). Fits to the exponential decay of Eq. (4) are shown with characteristic decay parameters $\lambda =1.8$ and 2.3 for ${\Pi }_{\perp }$ and $\lambda =1.6$ for both data sets of ${\Pi }_{\parallel }$. The prefactor in the exponential decay is sensitive to the definition of the buckling point and so is not considered here. (A power law decay is not appropriate judging from the corresponding logarithmic plot and would not yield finite ${\Pi }^{\mathrm{buck}}$ in the limit $d_c/w\to 0$.)