Nonclassical Surface Nucleation of 6CB at the Air-Liquid Interface of a 6CB Oil-in-Water Nanoemulsion.

The surface tension of a freshly extruded pendant drop of a nanoemulsion, 4-cyano-4'-hexylbiphenyl or 6CB (a liquid crystal) in water, exhibits an unusual surface nucleation phenomenon. Initially the surface tension is that of pure water; however, after a surface nucleation time, the surface tension decreases suddenly in magnitude. This nucleation time, of hundreds to thousands of seconds, depends strongly upon (i) the 6CB concentration in water, (ii) the 6CB nanodroplet size, and (iii) the temperature. Similar behavior is observed in both the isotropic and nematic phases of 6CB; thus, this surface nucleation phenomenon is unrelated to this system's liquid crystalline properties. The observed surface nucleation behavior can be explained via considerations of the nanoemulsion's bulk entropy together with the number of 6CB nanodroplets in the vicinity of the surface.

Free-energy landscapes of intrusion and extrusion of liquid in truncated and inverted truncated conical pores: Implications for the Cassie-Baxter to Wenzel transition.

As the simplest model of transition between the superhydrophobic Cassie-Baxter (CB) and Wenzel (W) states of a macroscopic droplet sitting on a microscopically rough or corrugated substrate, a substrate whose surface is covered by identical truncated or inverted truncated conical pores is considered. The free-energy landscapes of the intrusion and extrusion processes of a liquid into single pore are analyzed when the liquid is compressed or stretched so that the liquid phase is either stable or metastable relative to the vapor phase. Therefore, this model is also relevant to the stability of the superhydrophobic submerged substrates. In this study, the macroscopic classical capillary theory is adopted. Even within this simplified model, two simple geometries of truncated and inverted truncated cones lead to completely different free-energy landscapes. A simple criterion for the stability of the CB state based on Laplace pressure is shown not to be sufficient to understand the destruction and recovery of the CB state. The free-energy landscapes indicate that a gradual and an abrupt destruction of CB state is possible, which depends on the orientation of the conical pore and whether the liquid is compressed or stretched. The extensions of these theoretical results to more complex geometries are briefly discussed.

Effect of line tension on axisymmetric nanoscale capillary bridges at the liquid-vapor equilibrium.

The effect of line tension on the axisymmetric nanoscale capillary bridge between two identical substrates with convex, concave, and flat geometry at the liquid-vapor equilibrium is theoretically studied. The modified Young's equation for the contact angle, which takes into account the effect of line tension, is derived on a general axisymmetric curved surface using the variational method. Even without the effect of line tension, the parameter space where the bridge can exist is limited simply by the geometry of substrates. The modified Young's equation further restricts the space where the bridge can exist when the line tension is positive because the equilibrium contact angle always remains finite and the wetting state near the zero contact angle cannot be realized. It is shown that the interplay of the geometry and the positive line tension restricts the formation of capillary bridge.

Spreading law on a completely wettable spherical substrate: The energy balance approach.

The spreading of a cap-shaped spherical droplet on a completely wettable spherical substrate is studied. The nonequilibrium thermodynamic formulation is used to derive the thermodynamic driving force of spreading including the line-tension effect. Then the energy balance approach is adopted to derive the evolution equation of the spreading droplet. The time evolution of the contact angle θ of a droplet obeys a power law θ∼t^{-α} with the exponent α, which is different from that derived from Tanner's law on a flat substrate. Furthermore, the line tension must be positive to promote complete wetting on a spherical substrate, while it must be negative on a flat substrate.

Nucleation and growth of a core-shell composite nucleus by diffusion.

The critical radius of a core-shell-type nucleus grown by diffusion in a phase-separated solution is studied. A kinetic critical radius rather than the thermodynamic critical radius of standard classical nucleation theory can be defined from the diffusional growth equations. It is shown that there exist two kinetic critical radii for the core-shell-type nucleus, for which both the inner-core radius and the outer-shell radius will be stationary. Therefore, these two critical radii correspond to a single critical point of the nucleation path with a single energy barrier even though the nucleation looks like a two-step process. The two radii are given by formulas similar to that of classical nucleation theory if the Ostwald-Freundlich boundary condition is imposed at the surface of the inner nucleus and that of the outer shell. The subsequent growth of a core-shell-type postcritical nucleus follows the classical picture of Ostwald's step rule. Our result is consistent with some of the experimental and numerical results which suggest the core-shell-type critical nucleus.

Spreading law of non-Newtonian power-law liquids on a spherical substrate by an energy-balance approach.

The spreading of a cap-shaped spherical droplet of non-Newtonian power-law liquids, both shear-thickening and shear-thinning liquids, that completely wet a spherical substrate is theoretically investigated in the capillary-controlled spreading regime. The crater-shaped droplet model with the wedge-shaped meniscus near the three-phase contact line is used to calculate the viscous dissipation near the contact line. Then the energy balance approach is adopted to derive the equation that governs the evolution of the contact line. The time evolution of the dynamic contact angle θ of a droplet obeys a power law θ∼t^{-α} with the spreading exponent α, which is different from Tanner's law for Newtonian liquids and those for non-Newtonian liquids on a flat substrate. Furthermore, the line-tension dominated spreading, which could be realized on a spherical substrate for late-stage of spreading when the contact angle becomes low and the curvature of the contact line becomes large, is also investigated.

Topography- and topology-driven spreading of non-Newtonian power-law liquids on a flat and a spherical substrate.

The spreading of a cap-shaped spherical droplet of non-Newtonian power-law liquids on a flat and a spherical rough and textured substrate is theoretically studied in the capillary-controlled spreading regime. A droplet whose scale is much larger than that of the roughness of substrate is considered. The equilibrium contact angle on a rough substrate is modeled by the Wenzel and the Cassie-Baxter model. Only the viscous energy dissipation within the droplet volume is considered, and that within the texture of substrate by imbibition is neglected. Then, the energy balance approach is adopted to derive the evolution equation of the contact angle. When the equilibrium contact angle vanishes, the relaxation of dynamic contact angle θ of a droplet obeys a power-law decay θ∼t^{-α} except for the Newtonian and the non-Newtonian shear-thinning liquid of the Wenzel model on a spherical substrate. The spreading exponent α of the non-Newtonian shear-thickening liquid of the Wenzel model on a spherical substrate is larger than others. The relaxation of the Newtonian liquid of the Wenzel model on a spherical substrate is even faster showing the exponential relaxation. The relaxation of the non-Newtonian shear-thinning liquid of Wenzel model on a spherical substrate is fastest and finishes within a finite time. Thus, the topography (roughness) and the topology (flat to spherical) of substrate accelerate the spreading of droplet.

Publisher's Note: Spreading law of non-Newtonian power-law liquids on a spherical substrate by an energy-balance approach [Phys. Rev. E 96, 012803 (2017)].

This corrects the article DOI: 10.1103/PhysRevE.96.012803.

Size-dependent contact angle and the wetting and drying transition of a droplet adsorbed onto a spherical substrate: Line-tension effect.

The size-dependent contact angle and the drying and wetting morphological transition are studied with respect to the volume change for a spherical cap-shaped droplet placed on a spherical substrate. The line-tension effect is included using the rigorous formula for the Helmholtz free energy in the droplet capillary model. A morphological drying transition from a cap-shaped to a spherical droplet occurs when the substrate is hydrophobic and the droplet volume is small, similar to the transition predicted on a flat substrate. In addition, a morphological wetting transition from a cap-shaped to a wrapped spherical droplet occurs for a hydrophilic substrate and a large droplet volume. The contact angle depends on the droplet size: it decreases as the droplet volume increases when the line tension is positive, whereas it increases when the line tension is negative. The spherical droplets and wrapped droplets are stable when the line tension is positive and large.

Free-Energy Barrier of Filling a Spherical Cavity in the Presence of Line Tension: Implication to the Energy Barrier between the Cassie and Wenzel States on a Superhydrophobic Surface with Spherical Cavities.

The free-energy barrier of filling a spherical cavity having an inner wall of various wettabilities is studied. The morphology and free energy of a lens-shaped droplet are determined from the minimum of the free energy. The effect of line tension on the free energy is also studied. Then, the equilibrium contact angle of the droplet is determined from the generalized Young's equation. By increasing the droplet volume within the spherical cavity, the droplet morphology changes from spherical with an equilibrium contact angle of 180° to a lens with a convex meniscus, where the morphological complete drying transition occurs. By further increasing the droplet volume, the meniscus changes from convex to concave. Then, the lens-shaped droplet with concave meniscus spreads over the whole inner wall, resulting in an equilibrium contact angle of 0° to leave a spherical bubble, where the morphological complete wetting transition occurs. Finally, the whole cavity is filled with liquid. The free energy shows a barrier from complete drying to complete wetting as a function of droplet volume, which corresponds to the energy barrier between the Cassie and Wenzel states of the superhydrophobic surface with spherical cavities. The free-energy maximum occurs when the meniscus of the droplet becomes flat, and it is given by an analytic formula. The effect of line tension is expressed by the scaled line tension, and this effect is largest at the free-energy maximum. The positive line tension increases the free-energy maximum, which thus increases the stability of the Cassie superhydrophobic state, whereas the negative line tension destabilizes the superhydrophobic state.

Line tension and morphology of a sessile droplet on a spherical substrate.

The effects of line tension on the morphology of a sessile droplet placed on top of a convex spherical substrate are studied. The morphology of the droplet is determined from the global minimum of the Helmholtz free energy. The contact angle between the droplet and the spherical substrate is expressed by the generalized Young's formula. When the line tension is positive and large, the contact angle jumps discontinuously to 180^{∘}, the circular contact line shrinks towards the top of the substrate, and the droplet detaches from the substrate, forming a spherical droplet if the substrate is hydrophobic (i.e., the Young's contact angle is large). This finding is consistent with that predicted by Widom [J. Phys. Chem. 99, 2803 (1995)JPCHAX0022-365410.1021/j100009a041]; the line tension induces a drying transition on a flat substrate. On the other hand, the contact angle jumps to 0^{∘}, the circular contact line shrinks towards the bottom of the substrate, and the droplet spreads over the substrate to form a wrapped spherical droplet if the substrate is hydrophilic (i.e., the Young's contact angle is small). Therefore, not only the drying transition of a cap-shaped to a detached spherical droplet but also the wetting transition of a cap-shaped to a wrapped spherical droplet could occur on a spherical substrate as the surface area of the substrate is finite. When the line tension is negative and its magnitude increases, the contact line asymptotically approaches the equator from either above or below. The droplet with a contact line that coincides with the equator is an isolated, singular solution of the first variational problem. In this instance, the contact line is pinned and cannot move as far as the line tension is smaller than the critical magnitude, where the wetting transition occurs.

Line tension and morphology of a droplet and a bubble attached to the inner wall of a spherical cavity.

The effects of line tension on the morphology of a lens-shaped droplet and bubble placed on the inner wall of a spherical cavity are studied. The contact angle between the lens-shaped droplet and the concave spherical substrate is expressed by the generalized Young's formula. The equator of the spherical substrate is found to play a crucial role. Neither a droplet with its contact line on the upper hemisphere of the substrate nor one with its contact line on the lower hemisphere can transform into each other continuously. On a hydrophobic substrate, the contact angle jumps discontinuously to 180(∘), and the droplet is detached from the substrate to form a spherical droplet when the line tension is positive and large. This is similar to the drying transition on a flat substrate. On the other hand, on a hydrophilic substrate, the contact angle jumps discontinuously to 0(∘) when the line tension is positive and large. Then, the droplet spreads over the whole inner wall to leave a spherical bubble. Therefore, not only the drying transition but also the wetting transition is induced by positive line tension on a concave spherical substrate. There also exist stable as well as metastable droplets, whose phase diagrams can be complex. When the line tension is negative and its magnitude increases, the contact line approaches the equator infinitesimally from either above or below. However, it cannot cross the equator of a spherical cavity continuously. The droplet with a contact line that coincides with the equator is a singular droplet. The contact line is pinned and cannot move, irrespective of the magnitude of the line tension.

Line-tension-induced scenario of heterogeneous nucleation on a spherical substrate and in a spherical cavity.

Line-tension-induced scenario of heterogeneous nucleation is studied for a lens-shaped nucleus with a finite contact angle nucleated on a spherical substrate and on the bottom of the wall of a spherical cavity. The effect of line tension on the free energy of a critical nucleus can be separated from the usual volume term. By comparing the free energy of a lens-shaped critical nucleus of a finite contact angle with that of a spherical nucleus, we find that a spherical nucleus may have a lower free energy than a lens-shaped nucleus when the line tension is positive and large, which is similar to the drying transition predicted by Widom [B. Widom, J. Phys. Chem. 99, 2803 (1995)]. Then, the homogeneous nucleation rather than the heterogeneous nucleation will be favorable. Similarly, the free energy of a lens-shaped nucleus becomes negative when the line tension is negative and large. Then, the barrier-less nucleation with no thermal activation called athermal nucleation will be realized.

Line-tension effects on heterogeneous nucleation on a spherical substrate and in a spherical cavity.

The line-tension effects on heterogeneous nucleation are considered when a spherical lens-shaped nucleus is nucleated on top of a spherical substrate and on the bottom of the wall of a spherical cavity. The effect of line tension on the nucleation barrier can be separated from the usual volume term. As the radius of the substrate increases, the nucleation barrier decreases and approaches that of a flat substrate. However, as the radius of the cavity increases, the nucleation barrier increases and approaches that of a flat substrate. A small spherical substrate is a less active nucleation site than a flat substrate, and a small spherical cavity is a more active nucleation site than a flat substrate. In contrast, the line-tension effect on the nucleation barrier is maximum when the radii of the nucleus and the substrate or cavity become comparable. Therefore, by tuning the size of the spherical substrate or spherical cavity, the effect of the line tension can be optimized. These results will be useful in broad range of applications from material processing to understanding of global climate, where the heterogeneous nucleation plays a vital role.

Nucleation and growth by diffusion under Ostwald-Freundlich boundary condition.

The critical radius of a nucleus grown by diffusion in a solution is studied thermodynamically as well as kinetically. The thermodynamic growth equation called Zeldovich equation of classical nucleation theory and the kinetic diffusional growth equation combined with the Ostwald-Freundlich boundary condition lead to the same critical radius. However, it should be pointed out that the diffusional equation may lead to a kinetic critical radius that is different from the thermodynamic critical radius, thus indicating the possibility of kinetically controlling the critical radius of a nucleus.

Nucleation pathway of core-shell composite nucleus in size and composition space and in component space.

The kinetics of nucleation of a core-shell composite nucleus that consists of a core of stable final phase surrounded by a wetting layer of intermediate metastable phase is studied using the kinetic theory of binary nucleation not only in the size and composition space but also in the component space. The steady-state solution of the Fokker-Planck equation is considered. Various formulas for the critical nucleus at the saddle point as well as for the postcritical nucleus are derived. The kinetics of nucleation at the saddle point is more appropriately characterized in the size and composition space, while the kinetics of the postcritical nucleus is more appropriately described in the component space. Although both the free-energy landscape and the reaction rates play decisive role to determine the kinetics of nucleation at the saddle point, the details of the free-energy landscape are irrelevant to the kinetics of the postcritical nucleus.

Steady-state nucleation rate and flux of composite nucleus at saddle point.

The steady-state nucleation rate and flux of composite nucleus at the saddle point is studied by extending the theory of binary nucleation. The Fokker-Planck equation that describes the nucleation flux is derived using the Master equation for the growth of the composite nucleus, which consists of the core of the final stable phase surrounded by a wetting layer of the intermediate metastable phase nucleated from a metastable parent phase recently evaluated by Iwamatsu [J. Chem. Phys. 134, 164508 (2011)]. The Fokker-Planck equation is similar to that used in the theory of binary nucleation, but the non-diagonal elements exist in the reaction rate matrix. First, the general solution for the steady-state nucleation rate and the direction of nucleation flux is derived. Next, this information is then used to study the nucleation of composite nucleus at the saddle point. The dependence of steady-state nucleation rate as well as the direction of nucleation flux on the reaction rate in addition to the free-energy surface is studied using a model free-energy surface. The direction of nucleation current deviates from the steepest-descent direction of the free-energy surface. The results show the importance of two reaction rate constants: one from the metastable environment to the intermediate metastable phase and the other from the metastable intermediate phase to the stable new phase. On the other hand, the gradient of the potential Φ or the Kramers crossover function (the commitment or splitting probability) is relatively insensitive to reaction rates or free-energy surface.

A note on the nucleation with multiple steps: parallel and series nucleation.

Parallel and series nucleation are the basic elements of the complex nucleation process when two saddle points exist on the free-energy landscape. It is pointed out that the nucleation rates follow formulas similar to those of parallel and series connection of resistors or conductors in an electric circuit. Necessary formulas to calculate individual nucleation rates at the saddle points and the total nucleation rate are summarized, and the extension to the more complex nucleation process is suggested.

Analysis of molecular dynamics of colloidal particles in transported dilute samples by self-mixing laser Doppler velocimetry.

Colloidal particles in a liquid medium are transported with constant velocity, and dynamic light scattering experiments are performed on the samples by self-mixing laser Doppler velocimetry. The power spectrum of the modulated wave induced by the motion of the colloidal particles cannot be described by the well-known formula for flowing Brownian motion systems, i.e., a combination of Doppler shift, diffusion, and translation. Rather, the power spectrum was found to be described by the q-Gaussian distribution function. The molecular mechanism resulting in this anomalous line shape of the power spectrum is attributed to the anomalous molecular dynamics of colloidal particles in transported dilute samples, which satisfy a nonlinear Langevin equation.

Heterogeneous critical nucleation on a completely wettable substrate.

Heterogeneous nucleation of a new bulk phase on a flat substrate can be associated with the surface phase transition called wetting transition. When this bulk heterogeneous nucleation occurs on a completely wettable flat substrate with a zero contact angle, the classical nucleation theory predicts that the free-energy barrier of nucleation vanishes. In fact, there always exists a critical nucleus and a free-energy barrier as the first-order prewetting transition will occur even when the contact angle is zero. Furthermore, the critical nucleus changes its character from the critical nucleus of surface phase transition below bulk coexistence (undersaturation) to the critical nucleus of bulk heterogeneous nucleation above the coexistence (oversaturation) when it crosses the coexistence. Recently, Sear [J. Chem. Phys. 129, 164510 (2008)] has shown, by a direct numerical calculation of nucleation rate, that the nucleus does not notice this change when it crosses the coexistence. In our work, the morphology and the work of formation of critical nucleus on a completely wettable substrate are re-examined across the coexistence using the interface-displacement model. Indeed, the morphology and the work of formation changes continuously at the coexistence. Our results support the prediction of Sear and will rekindle the interest on heterogeneous nucleation on a completely wettable substrate.