Onsager's variational principle in active soft matter.

Onsagers variational principle (OVP) was originally proposed by Lars Onsager in 1931 [L. Onsager, Phys. Rev., 1931, 37, 405]. This fundamental principle provides a very powerful tool for formulating thermodynamically consistent models. It can also be employed to find approximate solutions, especially in the study of soft matter dynamics. In this work, OVP is extended and applied to the dynamic modeling of active soft matter such as suspensions of bacteria and aggregates of animal cells. We first extend the general formulation of OVP to active matter dynamics where active forces are included as external non-conservative forces. We then use OVP to analyze the directional motion of individual active units: a molecular motor walking on a stiff biofilament and a toy two-sphere microswimmer. Next we use OVP to formulate a diffuse-interface model for an active polar droplet on a solid substrate. In addition to the generalized hydrodynamic equations for active polar fluids in the bulk region, we have also derived thermodynamically consistent boundary conditions. Finally, we consider the dynamics of a thin active polar droplet under the lubrication approximation. We use OVP to derive a generalized thin film equation and then employ OVP as an approximation tool to find the spreading laws for the thin active polar droplet. By incorporating the activity of biological systems into OVP, we develop a general approach to construct thermodynamically consistent models for better understanding the emergent behaviors of individual animal cells and cell aggregates or tissues.

Stochastic dynamics of an active particle escaping from a potential well.

Active matter systems are driven out of equilibrium by the energy directly supplied at the level of constituent active particles that are self-propelled. We consider a model for an active particle in a potential well, characterized by an active velocity with a constant magnitude but a random orientation subject to white noises. We are interested in the escape of the active particle from the potential well in multiple-dimensional space. We investigate two distinct optimal paths, namely, the shortest arrival-time path and the most probable path, by using the analytical and numerical techniques from optimal control and rare event modeling. In particular, we elucidate the relationship between these optimal paths and the reachable set using the Hamiltonian dynamics for the shortest arrival-time path and the geometric minimum action method for the most probable path, respectively. Numerical results are presented by applying these techniques to a two-dimensional double-well potential.

Controllable ingestion and release of guest components driven by interfacial molecular orientation of host liquid crystal droplets.

Controllable construction and manipulation of artificial multi-compartmental structures are crucial in understanding and imitating smart molecular elements such as biological cells and on-demand delivery systems. Here, we report a liquid crystal droplet (LCD) based three-dimensional system for controllable and reversible ingestion and release of guest aqueous droplets (GADs). Induced by interfacial thermodynamic fluctuation and internal topological defect, microscale LCDs with perpendicular anchoring condition at the interface would spontaneously ingest external components from the surroundings and transform them as radially assembled tiny GADs inside LCDs. Landau-de Gennes free-energy model is applied to describe and explain the assembly dynamics and morphologies of these tiny GADs, which presents a good agreement with experimental observations. Furthermore, the release of these ingested GADs can be actively triggered by changing the anchoring conditions at the interface of LCDs. Since those ingestion and release processes are controllable and happen very gently at room temperature and neutral pH environment without extra energy input, these microscale LCDs are very prospective to provide a unique and viable route for constructing hierarchical 3D structures with tunable components and compartments.

Superconducting characteristics of 4-A carbon nanotube-zeolite composite.

We have fabricated nanocomposites consisting of 4-A carbon nanotubes embedded in the 0.7-nm pores of aluminophosphate-five (AFI) zeolite that display a superconducting specific heat transition at 15 K. MicroRaman spectra of the samples show strong and spatially uniform radial breathing mode (RBM) signals at 510 cm(-1) and 550 cm(-1), characteristic of the (4, 2) and (5, 0) nanotubes, respectively. The specific heat transition is suppressed at >2 T, with a temperature dependence characteristic of finite-size effects. Comparison with theory shows the behavior to be consistent with that of a type II BCS superconductor, characterized by a coherence length of 14 +/- 2 nm and a magnetic penetration length of 1.5 +/- 0.7 mum. Four probe and differential resistance measurements have also indicated a superconducting transition initiating at 15 K, but the magnetoresistance data indicate the superconducting network to be inhomogeneous, with a component being susceptible to magnetic fields below 3 T and other parts capable of withstanding a magnetic field of 5 T or beyond.

Contact line motion in confined liquid-gas systems: Slip versus phase transition.

In two-phase flows, the interface intervening between the two fluid phases intersects the solid wall at the contact line. A classical problem in continuum fluid mechanics is the incompatibility between the moving contact line and the no-slip boundary condition, as the latter leads to a nonintegrable stress singularity. Recently, various diffuse-interface models have been proposed to explain the contact line motion using mechanisms missing from the sharp-interface treatments in fluid mechanics. In one-component two-phase (liquid-gas) systems, the contact line can move through the mass transport across the interface while in two-component (binary) fluids, the contact line can move through diffusive transport across the interface. While these mechanisms alone suffice to remove the stress singularity, the role of fluid slip at solid surface needs to be taken into account as well. In this paper, we apply the diffuse-interface modeling to the study of contact line motion in one-component liquid-gas systems, with the fluid slip fully taken into account. The dynamic van der Waals theory has been presented for one-component fluids, capable of describing the two-phase hydrodynamics involving the liquid-gas transition [A. Onuki, Phys. Rev. E 75, 036304 (2007)]. This theory assumes the local equilibrium condition at the solid surface for density and also the no-slip boundary condition for velocity. We use its hydrodynamic equations to describe the continuum hydrodynamics in the bulk region and derive the more general boundary conditions by introducing additional dissipative processes at the fluid-solid interface. The positive definiteness of entropy production rate is the guiding principle of our derivation. Numerical simulations based on a finite-difference algorithm have been carried out to investigate the dynamic effects of the newly derived boundary conditions, showing that the contact line can move through both phase transition and slip, with their relative contributions determined by a competition between the two coexisting mechanisms in terms of entropy production. At temperatures very close to the critical temperature, the phase transition is the dominant mechanism, for the liquid-gas interface is wide and the density ratio is close to 1. At low temperatures, the slip effect shows up as the slip length is gradually increased. The observed competition can be interpreted by the Onsager principle of minimum entropy production.

Nucleation of wetting films on cylindrical and spherical substrates: a numerical study by the string method.

Using the mean-field diffuse-interface model for liquid-vapor system and employing the numerical string method, we study the critical nuclei involved in the prewetting transitions on curved substrates. We first introduce three distinct kinds of critical nuclei, namely, the disklike, bandlike, and layerlike ones, which respectively correspond to three possible growth modes of wetting films. We show the disklike growth mode to be the only mode for infinite planar substrates. We then turn to cylindrical and spherical substrates, the two simplest but most important geometries in the real world. We focus on the critical nuclei of finite size, through which the wetting films may be formed with finite thermodynamic probabilities. It is shown that the disklike growth mode is always the most probable for wetting film nucleation and growth as long as a disklike critical nucleus exists. It is also shown that on a cylindrical substrate, the disklike critical nucleus can no longer exist if the substrate radius is smaller than some critical value, comparable to the radius of the disklike critical nucleus on planar substrate. We find that on a cylindrical substrate whose radius is below the critical value, the nucleation and growth of a wetting film can only occur through the bandlike critical nucleus. It is worth emphasizing that the results concerning the bandlike and layerlike growth modes can only be obtained from the diffuse-interface model, beyond the macroscopic description based on the line and surface tensions.

Generalized Lorentz reciprocal theorem in complex fluids and in non-isothermal systems.

The classical Lorentz reciprocal theorem (LRT) was originally derived for slow viscous flows of incompressible Newtonian fluids under the isothermal condition. In the present work, we extend the LRT from simple to complex fluids with open or moving boundaries that maintain non-equilibrium stationary states. In complex fluids, the hydrodynamic flow is coupled with the evolution of internal degrees of freedom such as the solute concentration in two-phase binary fluids and the spin in micropolar fluids. The dynamics of complex fluids can be described by local conservation laws supplemented with local constitutive equations satisfying Onsager's reciprocal relations (ORR). We consider systems in quasi-stationary states close to equilibrium, controlled by the boundary variables whose evolution is much slower than the relaxation in the system. For these quasi-stationary states, we derive the generalized LRT and global Onsager's reciprocal relations (GORR) for the slow variables at boundaries. This establishes the connection between ORR for local constitutive equations and GORR for constitutive equations at boundaries. Finally, we show that the LRT can be further extended to non-isothermal systems by considering as an example the thermal conduction in solids and still fluids.

Application of the string method to the study of critical nuclei in capillary condensation.

We adopt a continuum description for liquid-vapor phase transition in the framework of mean-field theory and use the string method to numerically investigate the critical nuclei for capillary condensation in a slit pore. This numerical approach allows us to determine the critical nuclei corresponding to saddle points of the grand potential function in which the chemical potential is given in the beginning. The string method locates the minimal energy path (MEP), which is the most probable transition pathway connecting two metastable/stable states in configuration space. From the MEP, the saddle point is determined and the corresponding energy barrier also obtained (for grand potential). Moreover, the MEP shows how the new phase (liquid) grows out of the old phase (vapor) along the most probable transition pathway, from the birth of a critical nucleus to its consequent expansion. Our calculations run from partial wetting to complete wetting with a variable strength of attractive wall potential. In the latter case, the string method presents a unified way for computing the critical nuclei, from film formation at solid surface to bulk condensation via liquid bridge. The present application of the string method to the numerical study of capillary condensation shows the great power of this method in evaluating the critical nuclei in various liquid-vapor phase transitions.

Generalized nematohydrodynamic boundary conditions with application to bistable twisted nematic liquid-crystal displays.

Parallel to the highly successful Ericksen-Leslie hydrodynamic theory for the bulk behavior of nematic liquid crystals (NLCs), we derive a set of coupled hydrodynamic boundary conditions to describe the NLC dynamics near NLC-solid interfaces. In our boundary conditions, translational flux (flow slippage) and rotational flux (surface director relaxation) are coupled according to the Onsager variational principle of least energy dissipation. The application of our boundary conditions to the truly bistable pi -twist NLC cell reveals a complete picture of the dynamic switching processes. It is found that the thus far overlooked translation-rotation dissipative coupling at solid surfaces can accelerate surface director relaxation and enhance the flow rate. This can be utilized to improve the performance of electro-optical nematic devices by lowering the required switching voltages and reducing the switching times.

A phenomenological approach to the deposition pattern of evaporating droplets with contact line pinning.

When an evaporating droplet of colloidal suspension dries on a solid surface with the contact line pinned, the solute particles are driven by the solvent flow toward the edge and form a ring-like deposition pattern. In this work, we take into account the contact angle hysteresis and incorporate it into the effective model of Man and Doi (2016 Phys. Rev. Lett. 116 066101) which is based on Onsager's variational principle. We show that single-ring pattern is formed when the contact line pinning and/or friction are sufficiently strong. We demonstrate that there exists an appropriate range for contact line pinning and friction in which two rings can be formed in the deposition pattern, one at the initially pinned contact line and the other a bit closer to the center of droplet.

Thermal singularity and contact line motion in pool boiling: Effects of substrate wettability.

The dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)] is employed to model the growth of a single vapor bubble in a superheated liquid on a flat homogeneous substrate. The bubble spreading dynamics in the pool boiling regime has been numerically investigated for one-component van der Waals fluids close to the critical point, with a focus on the effect of the substrate wettability on bubble growth and contact line motion. The substrate wettability is found to control the apparent contact angle and the rate of bubble growth (the rate of total evaporation), through which the contact line speed is determined. An approximate expression is derived for the contact line speed, showing good agreement with the simulation results. This demonstrates that the contact line speed is primarily governed by (1) the circular shape of interface (for slow bubble growth), (2) the constant apparent contact angle, and (3) the constant bubble growth rate. It follows that the contact line speed has a sensitive dependence on the substrate wettability via the apparent contact angle which also determines the bubble growth rate. Compared to hydrophilic surfaces, hydrophobic surfaces give rise to a thinner shape of bubble and a higher rate of total evaporation, which combine to result in a much faster contact line speed. This can be linked to the earlier formation of a vapor film and hence the onset of boiling crisis.

Hydrodynamic slip boundary condition at chemically patterned surfaces: a continuum deduction from molecular dynamics.

We investigate the slip boundary condition for flows past a chemically patterned surface. Molecular dynamics simulations show that fluid forces and stresses vary laterally along the patterned surface. A subtraction scheme is developed to verify the validity of the Navier slip boundary condition, locally, for the patterned surface. A continuum hydrodynamic model is formulated using the Navier-Stokes equation and the Navier boundary condition, with a slip length varying along the patterned surface. Steady-state velocity fields from continuum calculations are in quantitative agreement with those from molecular simulations.

A Variational approach to thin film hydrodynamics of binary mixtures.

In order to model the dynamics of thin films of mixtures, solutions, and suspensions, a thermodynamically consistent formulation is needed such that various coexisting dissipative processes with cross couplings can be correctly described in the presence of capillarity, wettability, and mixing effects. In the present work, we apply Onsager's variational principle to the formulation of thin film hydrodynamics for binary fluid mixtures. We first derive the dynamic equations in two spatial dimensions, one along the substrate and the other normal to the substrate. Then, using long-wave asymptotics, we derive the thin film equations in one spatial dimension along the substrate. This enables us to establish the connection between the present variational approach and the gradient dynamics formulation for thin films. It is shown that for the mobility matrix in the gradient dynamics description, Onsager's reciprocal symmetry is automatically preserved by the variational derivation. Furthermore, using local hydrodynamic variables, our variational approach is capable of introducing diffusive dissipation beyond the limit of dilute solute. Supplemented with a Flory-Huggins-type mixing free energy, our variational approach leads to a thin film model that treats solvent and solute in a symmetric manner. Our approach can be further generalized to include more complicated free energy and additional dissipative processes.

Anisotropic particle in viscous shear flow: Navier slip, reciprocal symmetry, and Jeffery orbit.

The hydrodynamic reciprocal theorem for Stokes flows is generalized to incorporate the Navier slip boundary condition, which can be derived from Onsager's variational principle of least energy dissipation. The hydrodynamic reciprocal relations and the Jeffery orbit, both of which arise from the motion of a slippery anisotropic particle in a simple viscous shear flow, are investigated theoretically and numerically using the fluid particle dynamics method [Phys. Rev. Lett. 85, 1338 (2000)]. For a slippery elliptical particle in a linear shear flow, the hydrodynamic reciprocal relations between the rotational torque and the shear stress are studied and related to the Jeffery orbit, showing that the boundary slip can effectively enhance the anisotropy of the particle. Physically, by replacing the no-slip boundary condition with the Navier slip condition at the particle surface, the cross coupling between the rotational torque and the shear stress is enhanced, as manifested through a dimensionless parameter in both of the hydrodynamic reciprocal relations and the Jeffery orbit. In addition, simulations for a circular particle patterned with portions of no-slip and Navier slip are carried out, showing that the particle possesses an effective anisotropy and follows the Jeffery orbit as well. This effective anisotropy can be tuned by changing the ratio of no-slip portion to slip potion. The connection of the present work to nematic liquid crystals' constitutive relations is discussed.

Determining hydrodynamic boundary conditions from equilibrium fluctuations.

The lack of a first-principles derivation has made the hydrodynamic boundary condition a classical issue for the past century. The fact that the fluid can have interfacial structures adds additional complications and ambiguities to the problem. Here we report the use of molecular dynamics to identify from equilibrium thermal fluctuations the hydrodynamic modes in a fluid confined by solid walls, thereby extending the application of the fluctuation-dissipation theorem to yield not only the accurate location of the hydrodynamic boundary at the molecular scale, but also the relevant parameter value(s) for the description of the macroscopic boundary condition. We present molecular dynamics results on two examples to illustrate the application of this approach-one on the hydrophilic case and one on the hydrophobic case. It is shown that the use of the orthogonality condition of the modes can uniquely locate the hydrodynamic boundary to be inside the fluid in both cases, separated from the molecular solid-liquid interface by a small distance Δ that is a few molecules in size. The eigenvalue equation of the hydrodynamic modes directly yields the slip length, which is about equal to Δ in the hydrophilic case but is larger than Δ in the hydrophobic case. From the decay time we also obtain the bulk viscosity which is in good agreement with the value obtained from dynamic simulations. To complete the picture, we derive the Green-Kubo relation for a finite fluid system and show that the boundary fluctuations decouple from the bulk only in the infinite-fluid-channel limit; and in that limit we recover the interfacial fluctuation-dissipation theorem first presented by Bocquet and Barrat. The coupling between the bulk and the boundary fluctuations provides both the justification and the reason for the effectiveness of the present approach, which promises broad utility for probing the hydrodynamic boundary conditions relevant to structured or elastic interfaces, as well as two-phase immiscible flows.

Molecular scale contact line hydrodynamics of immiscible flows.

From extensive molecular dynamics simulations on immiscible two-phase flows, we find the relative slipping between the fluids and the solid wall everywhere to follow the generalized Navier boundary condition, in which the amount of slipping is proportional to the sum of tangential viscous stress and the uncompensated Young stress. The latter arises from the deviation of the fluid-fluid interface from its static configuration. We give a continuum formulation of the immiscible flow hydrodynamics, comprising the generalized Navier boundary condition, the Navier-Stokes equation, and the Cahn-Hilliard interfacial free energy. Our hydrodynamic model yields interfacial and velocity profiles matching those from the molecular dynamics simulations at the molecular-scale vicinity of the contact line. In particular, the behavior at high capillary numbers, leading to the breakup of the fluid-fluid interface, is accurately predicted.

Single-bubble dynamics in pool boiling of one-component fluids.

We numerically investigate the pool boiling of one-component fluids with a focus on the effects of surface wettability on the single-bubble dynamics. We employed the dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)], a diffuse-interface model for liquid-vapor flows involving liquid-vapor transition in nonuniform temperature fields. We first perform simulations for bubbles on homogeneous surfaces. We find that an increase in either the contact angle or the surface superheating can enhance the bubble spreading over the heating surface and increase the bubble departure diameter as well and therefore facilitate the transition into film boiling. We then examine the dynamics of bubbles on patterned surfaces, which incorporate the advantages of both hydrophobic and hydrophilic surfaces. The central hydrophobic region increases the thermodynamic probability of bubble nucleation while the surrounding hydrophilic region hinders the continuous bubble spreading by pinning the contact line at the hydrophobic-hydrophilic intersection. This leads to a small bubble departure diameter and therefore prevents the transition from nucleate boiling into film boiling. With the bubble nucleation probability increased and the bubble departure facilitated, the efficiency of heat transfer on such patterned surfaces is highly enhanced, as observed experimentally [Int. J. Heat Mass Transfer 57, 733 (2013)]. In addition, the stick-slip motion of contact line on patterned surfaces is demonstrated in one-component fluids, with the effect weakened by surface superheating.

Pancake bouncing on superhydrophobic surfaces.

Engineering surfaces that promote rapid drop detachment1,2 is of importance to a wide range of applications including anti-icing3-5, dropwise condensation6, and self-cleaning7-9. Here we show how superhydrophobic surfaces patterned with lattices of submillimetre-scale posts decorated with nano-textures can generate a counter-intuitive bouncing regime: drops spread on impact and then leave the surface in a flattened, pancake shape without retracting. This allows for a four-fold reduction in contact time compared to conventional complete rebound1,10-13. We demonstrate that the pancake bouncing results from the rectification of capillary energy stored in the penetrated liquid into upward motion adequate to lift the drop. Moreover, the timescales for lateral drop spreading over the surface and for vertical motion must be comparable. In particular, by designing surfaces with tapered micro/nanotextures which behave as harmonic springs, the timescales become independent of the impact velocity, allowing the occurrence of pancake bouncing and rapid drop detachment over a wide range of impact velocities.

Hydrodynamics of Leidenfrost droplets in one-component fluids.

Using the dynamic van der Waals theory [Phys. Rev. E 75, 036304 (2007)], we numerically investigate the hydrodynamics of Leidenfrost droplets under gravity in two dimensions. Some recent theoretical predictions and experimental observations are confirmed in our simulations. A Leidenfrost droplet larger than a critical size is shown to be unstable and break up into smaller droplets due to the Rayleigh-Taylor instability of the bottom surface of the droplet. Our simulations demonstrate that an evaporating Leidenfrost droplet changes continuously from a puddle to a circular droplet, with the droplet shape controlled by its size in comparison with a few characteristic length scales. The geometry of the vapor layer under the droplet is found to mainly depend on the droplet size and is nearly independent of the substrate temperature, as reported in a recent experimental study [Phys. Rev. Lett. 109, 074301 (2012)]. Finally, our simulations demonstrate that a Leidenfrost droplet smaller than a characteristic size takes off from the hot substrate because the levitating force due to evaporation can no longer be balanced by the weight of the droplet, as observed in a recent experimental study [Phys. Rev. Lett. 109, 034501 (2012)].

Molecular dynamics simulations for the motion of evaporative droplets driven by thermal gradients along nanochannels.

For a one-component fluid on a solid substrate, a thermal singularity may occur at the contact line where the liquid-vapor interface intersects the solid surface. Physically, the liquid-vapor interface is almost isothermal at the liquid-vapor coexistence temperature in one-component fluids while the solid surface is almost isothermal for solids of high thermal conductivity. Therefore, a temperature discontinuity is formed if the two isothermal interfaces are of different temperatures and intersect at the contact line. This leads to the so-called thermal singularity. The localized hydrodynamics involving evaporation/condensation near the contact line leads to a contact angle depending on the underlying substrate temperature. This dependence has been shown to lead to the motion of liquid droplets on solid substrates with thermal gradients (Xu and Qian 2012 Phys. Rev. E 85 061603). In the present work, we carry out molecular dynamics (MD) simulations as numerical experiments to further confirm the predictions made from our previous continuum hydrodynamic modeling and simulations, which are actually semi-quantitatively accurate down to the small length scales in the problem. Using MD simulations, we investigate the motion of evaporative droplets in one-component Lennard-Jones fluids confined in nanochannels with thermal gradients. The droplet is found to migrate in the direction of decreasing temperature of solid walls, with a migration velocity linearly proportional to the temperature gradient. This agrees with the prediction of our continuum model. We then measure the effect of droplet size on the droplet motion. It is found that the droplet mobility is inversely proportional to a dimensionless coefficient associated with the total rate of dissipation due to droplet movement. Our results show that this coefficient is of order unity and increases with the droplet size for the small droplets (~10 nm) simulated in the present work. These findings are in semi-quantitative agreement with the predictions of our continuum model. Finally, we measure the effect of liquid-vapor coexistence temperature on the droplet motion. Through a theoretical analysis on the size of the thermal singularity, it can be shown that the droplet mobility decreases with decreasing coexistence temperature. This is observed in our MD simulations.