Amorphous to amorphous transition in particle rafts

Space-filling assemblies of athermal hydrophobic particles floating at an air-water interface, called particle rafts, are shown to undergo an unusual phase transition between two amorphous states, i.e., a low density “less-rigid” state and a high density “more-rigid” state, as a function of particulate number density ($\Phi$). The former is shown to be a capillary bridged solid and the latter is shown to be a frictionally coupled one. Simultaneous studies involving direct imaging as well as measuring its mechanical response to longitudinal and shear stresses show that the transition is marked by a subtle structural anomaly and a weakening of the shear response. The structural anomaly is identified from the variation of the mean coordination number, mean area of the Voronoi cells, and spatial profile of the displacement field with $\Phi$. The weakened shear response is related to local plastic instabilities caused by the depinning of the contact line of the underlying fluid on the rough surfaces of the particles.

Figure 1

Micrographs of hydrophobized silica particles at the air-water interface in the form of (a) disjoint clusters (patchy state) and (b) a homogeneous “compressed” state of the raft in presence and (c) absence (“relaxed” state) of longitudinal stress, with scale bars of 10 mm. (d) The left (right) panel showing the plot of the radial density pair correlation function, $g\left(r\right)$, of the “relaxed” (“compressed”) state of the system.

Figure 2

Experimental setup: hydrophobized microscope cover-slip attached to a piezo stage oscillating along $x$ producing a sinusoidal shear deformation. The cantilever with an attached microscope cover-slip, placed along $y$ at a distance ($D=60$) mm away from the piezo stage measures the shear stress (${\sigma }_{xy}$). A parallel-plate capacitor, marked as the force sensor, is used to measure the longitudinal stress (${\sigma }_{xx})$.

Figure 3

Variation of longitudinal stress (${\sigma }_{xx}$) and shear stress (${\sigma }_{xy}$) plotted as a function of $\Phi$ in (a) and (b), respectively, for $c1$ (open symbols) and $e1$ (filled symbols) cycles. The arrows in (a) show the direction in which $\Phi$ changes. The region over which ${\sigma }_{xy}$ shows a softening is marked as region I. In region II, ${\sigma }_{xy}$ increases rapidly. The different states of the system, i.e., patchy, “compressed,” and “relaxed,” are marked in the figure. The transition region separating I and II is shaded.

Figure 4

The variation of (a) longitudinal modulus ($K_A$) and (b) shear modulus ($G$) and (c) the ratio $K_A/G$ as a function of $\Phi$ for the first expansion cycle. The longitudinal modulus ($K_A$) is calculated by numerically differentiating smoothed (using a polynomial fit) ${\sigma }_{xx}$ with respect to $\Phi$.

Figure 5

Displacement field of the particles and the probability distribution of angles $P\left(\theta \right)$ associated with the displacement vector ($\vec{s}$) measured during the first expansion cycle, shown in the right and left panel, respectively. Here $\theta ={\text{tan}}^{-1}\left(\frac{s_y}{s_x}\right)$. For values of $\Phi <0.74$ the displacement field is inhomogeneous and exhibits a large number of complex cellular features (right panel). This is reflected in the broad distribution of $P\left(\theta \right)$ (left panel). However, for values of $\Phi >0.74$, the displacement field becomes spatially correlated over large distances and $P\left(\theta \right)$ narrows considerably. The arrow indicates the progressive sequence of $\Phi$; i.e, the system moves from a state of high density, i.e., $\Phi =0.77$, to a state of low density, i.e., $\Phi =0.69$.

Figure 6

(a) Typical displacement curves of a few particles (${\Delta }_i$), measured from its position in the “relaxed” state as a function of ${\sigma }_{xx}$. For visual clarity the curves are shifted vertically. The corresponding scale bar of the displacement is shown in the figure. The data are plotted for a compression cycle. (b) The variation of the cumulative strain ($\lambda =\sum {\Delta }_i/L_x^0$, where the summation is over all the particles) with ${\sigma }_{xx}$ for successive compression (open symbols) and expansion (filled symbols) cycles. The uncertainty in the measurement of ${\Delta }_i$ and $\lambda$ is shown by the accompanying band.

Figure 7

The variation of (a) the mean coordination number $Z$ and (b) the mean Voronoi cell area $A$ normalized with the particle's area ($\pi a^2$) as a function of $\Phi$. The solid lines show their linear variation in region I. The dotted lines which are a linear extrapolation of initial variation highlight the later deviation of the graphs. The data presented here are for $e1$.

Figure 8

Variation of (a) $K_A$ and (b) $G$ as a function of $Z$. The filled symbols correspond to $e1$ while the open ones correspond to $c2$. The shaded regions indicate the spread in the data. The “less-rigid” state [schematically shown as a capillary bridged interaction in (a)] and “more-rigid” state [schematically shown as a friction dominated interaction in (a)] are marked in the figure. The inverted cusp in $G$ vs $Z$ corresponds to the slipping of the contact line [schematically shown in (b) with a greater width of a sliding contact line]. The inset of (b) shows the variation of $G$ of the less-rigid (triangles) and more-rigid (square) states of the raft as a function of particle diameter ($2a$). The solid lines passing through the data show $a^{-1}$ and $a^{-2}$ variations of $G$ for the less-rigid and more-rigid states, respectively.

Figure 9

(a) Frequency dependence of $G$ measured at $\Phi =0.77$. (b) The time dependence (creep effect) of ${\sigma }_{xy}$ shown for different $\Phi$.

Figure 10

Variation of ${\sigma }_{xx}$ and ${\sigma }_{xy}$ for various compression (open symbols) and expansion (filled symbols) cycles as a function of $\Phi$ for particles of diameter 1 mm. The sequences of expansion and compression are as follows: first compression (data are not shown for reasons discussed in the main text), first expansion ($e1$: filled triangles), second compression ($c2$: open triangles), second expansion ($e2$: filled circles), third compression ($c3$: open circles), third expansion ($e3$: filled stars), and fourth compression ($c4$: open stars). The repeated compression-expansion cycles result in annealing of the raft and thus later cycles show a reduced but finite hysteresis.