Selective binding of multivalent ligands within a mixture of polyvalent amphiphiles provides, in principle, a simple mechanism for driving domain formation in self-assemblies. Divalent cations are shown here to crossbridge polyanionic amphiphiles, which thereby demix from neutral amphiphiles and form spots or rafts within vesicles as well as stripes within cylindrical micelles. Calcium- and copper-crossbridged domains of synthetic block copolymers or natural lipid (phosphatidylinositol-4,5-bisphosphate) possess tunable sizes, shapes and/or spacings that can last for years. Lateral segregation in these ‘ligand-responsive Janus assemblies’ couples weakly to curvature and proves to be restricted within phase diagrams to narrow regimes of pH and cation concentration that are centred near the characteristic binding constants for polyacid interactions. Remixing at high pH is surprising, but a theory for strong lateral segregation shows that counterion entropy dominates electrostatic crossbridges, thus illustrating the insights gained into ligand-induced pattern formation within self-assemblies.

We study the origin of the scaling behavior in frictionless granular media above the jamming transition by analyzing their linear response. The response to local forcing is non-self-averaging and fluctuates over a length scale that diverges at the jamming transition. The response to global forcing becomes increasingly nonaffine near the jamming transition. This is due to the proximity of floppy modes, the influence of which we characterize by the local linear response. We show that the local response also governs the anomalous scaling of elastic constants and contact number.

We conduct experiments on two-dimensional packings of colloidal thermosensitive hydrogel particles whose packing fraction can be tuned above the jamming transition by varying the temperature. By measuring displacement correlations between particles, we extract the vibrational properties of a corresponding ``shadow'' system with the same configuration and interactions, but for which the dynamics of the particles are undamped. The vibrational properties are very similar to those predicted for zero-temperature sphere packings and found in atomic and molecular glasses; there is a boson peak at low frequency that shifts to higher frequency as the system is compressed above the jamming transition.

We probe the nature of the jamming transition of frictional granular media by studying their vibrational properties as a function of the applied pressure $p$ and friction coefficient $\mu$. The density of vibrational states exhibits a crossover from a plateau at frequencies $\omega \gtrsim {\omega }^{*}(p,\mu )$ to a linear growth for $\omega \lesssim {\omega }^{*}(p,\mu )$. We show that ${\omega }^{*}$ is proportional to $\mathrm{\Delta }z$, the excess number of contacts per grain relative to the minimally allowed, isostatic value. For zero and infinitely large friction, typical packings at the jamming threshold have $\mathrm{\Delta }z\to 0$, and then exhibit critical scaling. We study the nature of the soft modes in these two limits, and find that the ratio of elastic moduli is governed by the distance from isostaticity.

We compare the elastic response of spring networks whose contact geometry is derived from real packings of frictionless discs, to networks obtained by randomly cutting bonds in a highly connected network derived from a well-compressed packing. We find that the shear response of packing-derived networks, and both the shear and compression response of randomly cut networks, are all similar: the elastic moduli vanish linearly near jamming, and distributions characterizing the local geometry of the response scale with distance to jamming. Compression of packing-derived networks is exceptional: the elastic modulus remains constant and the geometrical distributions do not exhibit simple scaling. We conclude that the compression response of jammed packings is anomalous, rather than the shear response.

We investigate correlations between low-frequency vibrational modes and rearrangements in two-dimensional colloidal glasses composed of thermosensitive microgel particles, which readily permit variation of the sample packing fraction. At each packing fraction, the particle displacement covariance matrix is measured and used to extract the vibrational spectrum of the “shadow” colloidal glass (i.e., the particle network with the same geometry and interactions as the sample colloid but absent damping). Rearrangements are induced by successive, small reductions in the packing fraction. The experimental results suggest that low-frequency quasilocalized phonon modes in colloidal glasses, i.e., modes that present low energy barriers for system rearrangements, are spatially correlated with rearrangements in this thermal system.