Wetting of phase-separated droplets on plant vacuole membranes leads to a competition between tonoplast budding and nanotube formation.

Seeds of dicotyledonous plants store proteins in dedicated membrane-bounded organelles called protein storage vacuoles (PSVs). Formed during seed development through morphological and functional reconfiguration of lytic vacuoles in embryos [M. Feeney et al., Plant Physiol. 177, 241-254 (2018)], PSVs undergo division during the later stages of seed maturation. Here, we study the biophysical mechanism of PSV morphogenesis in vivo, discovering that micrometer-sized liquid droplets containing storage proteins form within the vacuolar lumen through phase separation and wet the tonoplast (vacuolar membrane). We identify distinct tonoplast shapes that arise in response to membrane wetting by droplets and derive a simple theoretical model that conceptualizes these geometries. Conditions of low membrane spontaneous curvature and moderate contact angle (i.e., wettability) favor droplet-induced membrane budding, thereby likely serving to generate multiple, physically separated PSVs in seeds. In contrast, high membrane spontaneous curvature and strong wettability promote an intricate and previously unreported membrane nanotube network that forms at the droplet interface, allowing molecule exchange between droplets and the vacuolar interior. Furthermore, our model predicts that with decreasing wettability, this nanotube structure transitions to a regime with bud and nanotube coexistence, which we confirmed in vitro. As such, we identify intracellular wetting [J. Agudo-Canalejo et al., Nature 591, 142-146 (2021)] as the mechanism underlying PSV morphogenesis and provide evidence suggesting that interconvertible membrane wetting morphologies play a role in the organization of liquid phases in cells.

Droplet Self-Propulsion on Slippery Liquid-Infused Surfaces with Dual-Lubricant Wedge-Shaped Wettability Patterns.

Young's equation is fundamental to the concept of the wettability of a solid surface. It defines the contact angle for a droplet on a solid surface through a local equilibrium at the three-phase contact line. Recently, the concept of a liquid Young's law contact angle has been developed to describe the wettability of slippery liquid-infused porous surfaces (SLIPS) by droplets of an immiscible liquid. In this work, we present a new method to fabricate biphilic SLIP surfaces and show how the wettability of the composite SLIPS can be exploited with a macroscopic wedge-shaped pattern of two distinct lubricant liquids. In particular, we report the development of composite liquid surfaces on silicon substrates based on lithographically patterning a Teflon AF1600 coating and a superhydrophobic coating (Glaco Mirror Coat Zero), where the latter selectively dewets from the former. This creates a patterned base surface with preferential wetting to matched liquids: the fluoropolymer PTFE with a perfluorinated oil Krytox and the hydrophobic silica-based GLACO with olive oil (or other mineral oils or silicone oil). This allows us to successively imbibe our patterned solid substrates with two distinct oils and produce a composite liquid lubricant surface with the oils segregated as thin films into separate domains defined by the patterning. We illustrate that macroscopic wedge-shaped patterned SLIP surfaces enable low-friction droplet self-propulsion. Finally, we formulate an analytical model that captures the dependence of the droplet motion as a function of the wettability of the two liquid lubricant domains and the opening angle of the wedge. This allows us to derive scaling relationships between various physical and geometrical parameters. This work introduces a new approach to creating patterned liquid lubricant surfaces, demonstrates long-distance droplet self-propulsion on such surfaces, and sheds light on the interactions between liquid droplets and liquid surfaces.

Phase field simulation of liquid filling on grooved surfaces for complete, partial, and pseudo-partial wetting cases.

We develop and harness a phase field simulation method to study liquid filling on grooved surfaces. We consider both short-range and long-range liquid-solid interactions, with the latter including purely attractive and repulsive interactions as well as those with short-range attraction and long-range repulsion. This allows us to capture complete, partial, and pseudo-partial wetting states, demonstrating complex disjoining pressure profiles over the full range of possible contact angles as previously proposed in the literature. Applying the simulation method to study liquid filling on grooved surfaces, we compare the filling transition for the three different classes of wetting states as we vary the pressure difference between the liquid and gas phases. The filling and emptying transitions are reversible for the complete wetting case, while significant hysteresis is observed for the partial and pseudo-partial cases. In agreement with previous studies, we also show that the critical pressure for the filling transition follows the Kelvin equation for the complete and partial wetting scenarios. Finally, we find the filling transition can display a number of distinct morphological pathways for the pseudo-partial wetting cases, as we demonstrate here for varying groove dimensions.

Spontaneous phase separation of ternary fluid mixtures.

We computationally study the spontaneous phase separation of ternary fluid mixtures using the lattice Boltzmann method both when all the surface tensions are equal and when they have different values. To rationalise the phase diagram of possible phase separation mechanisms, previous theoretical works typically rely on analysing the sign of the eigenvalues resulting from a simple linear stability analysis, but we find this does not explain the observed simulation results. Here, we classify the possible separation pathways into four basic mechanisms, and develop a phenomenological model that captures the composition regimes where each mechanism is prevalent. We further highlight that the dominant mechanism in ternary phase separation involves enrichment and instability of the minor component at the fluid-fluid interface, which is absent in the case of binary fluid mixtures.

A robust and memory-efficient transition state search method for complex energy landscapes.

Locating transition states is crucial for investigating transition mechanisms in wide-ranging phenomena, from atomistic to macroscale systems. Existing methods, however, can struggle in problems with a large number of degrees of freedom, on-the-fly adaptive remeshing and coarse-graining, and energy landscapes that are locally flat or discontinuous. To resolve these challenges, we introduce a new double-ended method, the Binary-Image Transition State Search (BITSS). It uses just two states that converge to the transition state, resulting in a fast, flexible, and memory-efficient method. We also show that it is more robust compared to existing bracketing methods that use only two states. We demonstrate its versatility by applying BITSS to three very different classes of problems: Lennard-Jones clusters, shell buckling, and multiphase phase-field models.

Control of Superselectivity by Crowding in Three-Dimensional Hosts.

Motivated by the fine compositional control observed in membraneless droplet organelles in cells, we investigate how a sharp binding-unbinding transition can occur between multivalent client molecules and receptors embedded in a porous three-dimensional structure. In contrast to similar superselective binding previously observed at surfaces, we have identified that a key effect in a three-dimensional environment is that the presence of inert crowding agents can significantly enhance or even introduce superselectivity. In essence, molecular crowding initially suppresses binding via an entropic penalty, but the clients can then more easily form many bonds simultaneously. We demonstrate the robustness of the superselective behavior with respect to client valency, linker length, and binding interactions in Monte Carlo simulations of an archetypal lattice polymer model.

Capillary Bridges on Liquid-Infused Surfaces.

We numerically study two-component capillary bridges formed when a liquid droplet is placed in between two liquid-infused surfaces (LIS). In contrast to commonly studied one-component capillary bridges on noninfused solid surfaces, two-component liquid bridges can exhibit a range of different morphologies where the liquid droplet is directly in contact with two, one, or none of the LIS substrates. In addition, the capillary bridges may lose stability when compressed due to the envelopment of the droplet by the lubricant. We also characterize the capillary force, maximum separation, and effective spring force and find that they are influenced by the shape and size of the lubricant ridge. Importantly, these can be tuned to increase the effective capillary adhesion strength by manipulating the lubricant pressure, Neumann angle, and wetting contact angles. As such, LIS are not only "slippery" parallel to the surface, but they are also "sticky" perpendicular to the surface.

Predicting Hemiwicking Dynamics on Textured Substrates.

The ability to predict liquid transport rates on textured surfaces is key to the design and optimization of devices and processes such as oil recovery, coatings, reaction-separation, high-throughput screening, and thermal management. In this work we develop a fully analytical model to predict the propagation coefficients for liquids hemiwicking through micropillar arrays. This is carried out by balancing the capillary driving force and a viscous resistive force and solving the Navier-Stokes equation for representative channels. The model is validated against a large data set of experimental hemiwicking coefficients harvested from the literature and measured in-house using high-speed imaging. The theoretical predictions show excellent agreement with the measured values and improved accuracy compared to previously proposed models. Furthermore, using lattice Boltzmann (LB) simulations, we demonstrate that the present model is applicable over a broad range of geometries. The scaling of velocity with texture geometry, implicit in our model, is compared against experimental data, where good agreement is observed for most practical systems. The analytical expression presented here offers a tool for developing design guidelines for surface chemistry and microstructure selection for liquid propagation on textured surfaces.

Apparent contact angle of drops on liquid infused surfaces: geometric interpretation.

We theoretically investigate the apparent contact angle of drops on liquid infused surfaces as a function of the relative size of the wetting ridge and the deposited drop. We provide an intuitive geometrical interpretation whereby the variation in the apparent contact angle is due to the rotation of the Neumann triangle at the lubricant-drop-gas contact line. We also derive linear and quadratic corrections to the apparent contact angle as power series expansion in terms of pressure differences between the lubricant, drop and gas phases. These expressions are much simpler and more compact compared to those previously derived by Semprebon et al. [Soft Matter, 2017, 13, 101-110].

Critical Pressure Asymmetry in the Enclosed Fluid Diode.

Joint physically and chemically pattered surfaces can provide efficient and passive manipulation of fluid flow. The ability of many of these surfaces to allow only unidirectional flow means they are often termed fluid diodes. Synthetic analogues of these are enabling technologies from sustainable water collection via fog harvesting to improved wound dressings. One key fluid diode geometry features a pore sandwiched between two absorbent substrates-an important design for applications that require liquid capture while preventing back-flow. However, the enclosed pore is particularly challenging to design as an effective fluid diode due to the need for both a low Laplace pressure for liquid entering the pore and a high Laplace pressure to liquid leaving. Here, we calculate the Laplace pressure for fluid traveling in both directions on a range of conical pore designs with a chemical gradient. We show that this chemical gradient is in general required to achieve the largest critical pressure differences between incoming and outgoing liquids. Finally, we discuss the optimization strategy to maximize this critical pressure asymmetry.

Effect of Ageing on the Structure and Properties of Model Liquid-Infused Surfaces.

Liquid-infused surfaces (LISs) exhibit unique properties that make them ideal candidates for a wide range of applications, from antifouling and anti-icing coatings to self-healing surfaces and controlled wetting. However, when exposed to realistic environmental conditions, LISs tend to age and progressively lose their desirable properties, potentially compromising their application. The associated ageing mechanisms are still poorly understood, and results reflecting real-life applications are scarce. Here, we track the ageing of a model LIS composed of glass surfaces functionalized with hydrophobic nanoparticles and infused with silicone oil. The LISs are fully submerged in aqueous solutions and exposed to acoustic pressure waves for set time intervals. The ageing is monitored by periodic measurements of the LIS's wetting properties. We also track the changes to the LIS's nanoscale structure. We find that the LISs rapidly lose their slippery properties because of a combination of oil loss, smoothing of the nanoporous functional layer, and substrate degradation when directly exposed to the solution. The oil loss is consistent with water microdroplets entering the oil layer and displacing oil away from the surface. These mechanisms are general and could play a role in the ageing of most LISs.

Factors controlling the pinning force of liquid droplets on liquid infused surfaces.

Liquid infused surfaces with partially wetting lubricants have recently been exploited for numerous intriguing applications, such as for droplet manipulation, droplet collection and spontaneous motion. When partially wetting lubricants are used, the pinning force is a key factor that can strongly affect droplet mobility. Here, we derive an analytical prediction for contact angle hysteresis in the limit where the meniscus size is much smaller than the droplet, and numerically study how it is controlled by the solid fraction, the lubricant wetting angles, and the various fluid surface tensions. We further relate the contact angle hysteresis and the pinning force experienced by a droplet on a liquid infused surface, and our predictions for the critical sliding angles are consistent with existing experimental observations. Finally, we discuss why a droplet on a liquid infused surface with partially wetting lubricants typically experiences stronger pinning compared to a droplet on a classical superhydrophobic surface.

Phase transitions on non-uniformly curved surfaces: coupling between phase and location.

For particles confined to two dimensions, any curvature of the surface affects the structural, kinetic and thermodynamic properties of the system. If the curvature is non-uniform, an even richer range of behaviours can emerge. Using a combination of bespoke Monte Carlo, molecular dynamics and basin-hopping methods, we show that the stable states of attractive colloids confined to non-uniformly curved surfaces are distinguished not only by the phase of matter but also by their location on the surface. Consequently, the transitions between these states involve cooperative migration of the entire colloidal assembly. We demonstrate these phenomena on toroidal and sinusoidal surfaces for model colloids with different ranges of interactions as described by the Morse potential. In all cases, the behaviour can be rationalised in terms of three universal considerations: cluster perimeter, stress, and the packing of next-nearest neighbours.

Dynamic Morphologies and Stability of Droplet Interface Bilayers.

We develop a theoretical framework for understanding dynamic morphologies and stability of droplet interface bilayers (DIBs), accounting for lipid kinetics in the monolayers and bilayer, and droplet evaporation due to imbalance between osmotic and Laplace pressures. Our theory quantitatively describes distinct pathways observed in experiments when DIBs become unstable. We find that when the timescale for lipid desorption is slow compared to droplet evaporation, the lipid bilayer will grow and the droplets approach a hemispherical shape. In contrast, when lipid desorption is fast, the bilayer area will shrink and the droplets eventually detach. Our model also suggests there is a critical size below which DIBs can become unstable, which may explain experimental difficulties in miniaturizing the DIB platform.

Drop Dynamics on Liquid-Infused Surfaces: The Role of the Lubricant Ridge.

We employ a free-energy lattice-Boltzmann method to study the dynamics of a ternary fluid system consisting of a liquid drop driven by a body force across a regularly textured substrate, infused by a lubricating liquid. We focus on the case of partial wetting lubricants and observe a rich interplay between contact line pinning and viscous dissipation at the lubricant ridge, which become dominant at large and small apparent angles, respectively. Our numerical investigations further demonstrate that the relative importance of viscous dissipation at the lubricant ridge depends on the drop to lubricant viscosity ratio, as well as on the shape of the wetting ridge.

Edge Fracture in Complex Fluids.

We study theoretically the edge fracture instability in sheared complex fluids, by means of linear stability analysis and direct nonlinear simulations. We derive an exact analytical expression for the onset of edge fracture in terms of the shear-rate derivative of the fluid's second normal stress difference, the shear-rate derivative of the shear stress, the jump in shear stress across the interface between the fluid and the outside medium (usually air), the surface tension of that interface, and the rheometer gap size. We provide a full mechanistic understanding of the edge fracture instability, carefully validated against our simulations. These findings, which are robust with respect to choice of rheological constitutive model, also suggest a possible route to mitigating edge fracture, potentially allowing experimentalists to achieve and accurately measure flows stronger than hitherto possible.

On the critical Casimir interaction between anisotropic inclusions on a membrane.

Using a lattice model and a versatile thermodynamic integration scheme, we study the critical Casimir interactions between inclusions embedded in a two-dimensional critical binary mixtures. For single-domain inclusions we demonstrate that the interactions are very long range, and their magnitudes strongly depend on the affinity of the inclusions with the species in the binary mixtures, ranging from repulsive when two inclusions have opposing affinities to attractive when they have the same affinities. When one of the inclusions has no preference for either of the species, we find negligible critical Casimir interactions. For multiple-domain inclusions, mimicking the observations that membrane proteins often have several domains with varying affinities to the surrounding lipid species, the presence of domains with opposing affinities does not cancel the interactions altogether. Instead we can observe both attractive and repulsive interactions depending on their relative orientations. With increasing number of domains per inclusion, the range and magnitude of the effective interactions decrease in a similar fashion to those of electrostatic multipoles. Finally, clusters formed by multiple-domain inclusions can result in an effective affinity patterning due to the anisotropic character of the Casimir interactions between the building blocks.

Phase Separation on Bicontinuous Cubic Membranes: Symmetry Breaking, Reentrant, and Domain Faceting.

We study the phase separation of binary lipid mixtures that form bicontinuous cubic phases. The competition between the nonuniform Gaussian membrane curvature and line tension leads to a very rich phase diagram, where we observe symmetry breaking of the membrane morphologies and reentrant phenomena due to the formation of bridges between segregated domains. Upon increasing the line tension contribution, we also find faceting of lipid domains that we explain using a simple argument based on the symmetry of the underlying surface and topology.

Kinetic Transition Networks for the Thomson Problem and Smale's Seventh Problem.

The Thomson problem, arrangement of identical charges on the surface of a sphere, has found many applications in physics, chemistry and biology. Here, we show that the energy landscape of the Thomson problem for N particles with N=132, 135, 138, 141, 144, 147, and 150 is single funneled, characteristic of a structure-seeking organization where the global minimum is easily accessible. Algorithmically, constructing starting points close to the global minimum of such a potential with spherical constraints is one of Smale's 18 unsolved problems in mathematics for the 21st century because it is important in the solution of univariate and bivariate random polynomial equations. By analyzing the kinetic transition networks, we show that a randomly chosen minimum is, in fact, always "close" to the global minimum in terms of the number of transition states that separate them, a characteristic of small world networks.

Apparent contact angle and contact angle hysteresis on liquid infused surfaces.

We theoretically investigate the apparent contact angle and contact angle hysteresis of a droplet placed on a liquid infused surface. We show that the apparent contact angle is not uniquely defined by material parameters, but also has a dependence on the relative size between the droplet and its surrounding wetting ridge formed by the infusing liquid. We derive a closed form expression for the contact angle in the limit of vanishing wetting ridge, and compute the correction for small but finite ridge, which corresponds to an effective line tension term. We also predict contact angle hysteresis on liquid infused surfaces generated by the pinning of the contact lines by the surface corrugations. Our analytical expressions for both the apparent contact angle and contact angle hysteresis can be interpreted as 'weighted sums' between the contact angles of the infusing liquid relative to the droplet and surrounding gas phases, where the weighting coefficients are given by ratios of the fluid surface tensions.