Positive Line Tension Research Articles

Microscopic visualization of heterogeneous condensation of water vapor on hydrophilic and hydrophobic particles

Heterogeneous condensation of water vapor on particles is widely used in the fields of atmospheric cloud physics and industrial particulate abatement. Particles surface wettability plays a fundamental role in water vapor condensation, but a microscopic insight of the impact of particles surface wettability on vapor condensation is lacked. Therefore, the Environmental Scanning Electron Microscope (ESEM) is adopted to directly visualize vapor condensation on particles with different surface wettability at a microscopic scale. Firstly heterogeneous condensation of water vapor on the hydrophilic and hydrophobic particles is obtained by the ESEM. The results show that when water vapor condenses on the hydrophilic particles, a spherical cap-shaped embryo first appears and subsequently develops into a spherical droplet to finally wrap the single particle. While on the hydrophobic particles, a spherical cap-shaped embryo will first appear and continue to grow into a spherical droplet to finally detach from the particle. Then the wetting coefficient is introduced to characterize the wetting degree of the droplet on the particle. The wetting coefficient will increase with the decrease of the contact angle, so the wettability of water vapor on the hydrophilic particles is better than that on the hydrophobic particles. Meanwhile based on classical nucleation theory, the geometrical factor and the critical supersaturation will decrease when the contact angle decreases, making it easier for water vapor to condense on the hydrophilic particles. Finally the line tension is investigated to explain the condensation mechanism. On the hydrophilic particles, when the droplet has crossed the equatorial line of the particle, the positive line tension promotes the spreading movement of the droplet over the particle to form a wrapped spherical droplet. Therefore, there is a transition from a spherical cap-shaped embryo before crossing the equatorial line of the particle to a wrapped spherical droplet after crossing the equatorial line. While on the hydrophobic particles, the negative line tension suppresses the spreading movement of the droplet over the particle to form a detached spherical droplet. So there is a transition from a spherical cap-shaped embryo before crossing the equatorial line of the particle to a detached spherical droplet after crossing the equatorial line.

Different pathways for engulfment and endocytosis of liquid droplets by nanovesicles

During endocytosis of nanoparticles by cells, the cellular membranes engulf the particles, thereby forming a closed membrane neck that subsequently undergoes fission. For solid nanoparticles, these endocytic processes have been studied in some detail. Recently, such processes have also been found for liquid and condensate droplets, both in vitro and in vivo. These processes start with the spreading of the droplet onto the membrane followed by partial or complete engulfment of the droplet. Here, we use molecular dynamics simulations to study these processes at the nanoscale, for nano-sized droplets and vesicles. For both partial and complete engulfment, we observe two different endocytic pathways. Complete engulfment leads to a closed membrane neck which may be formed in a circular or strongly non-circular manner. A closed circular neck undergoes fission, thereby generating two nested daughter vesicles whereas a non-circular neck hinders the fission process. Likewise, partial engulfment of larger droplets leads to open membrane necks which can again have a circular or non-circular shape. Two key parameters identified here for these endocytic pathways are the transbilayer stress asymmetry of the vesicle membrane and the positive or negative line tension of the membrane-droplet contact line.

Free energy of hydrophilic and hydrophobic pores in lipid bilayers by free energy perturbation of a restraint.

The free energy of pore formation in lipid bilayers has been previously calculated using a variety of reaction coordinates. Here, we use free energy perturbation of a cylindrical lipid exclusion restraint to compute the free energy profile as a function of pore radius in dimyristoylphosphatidylcholine (DMPC) and dioleoylphosphatidylcholine (DOPC) bilayers. Additionally restraining the headgroups to lie on the membrane surface allows us to also calculate the free energy profile of hydrophobic pores, i.e., cylindrical pores lined by acyl chains. For certain pore radii, the free energy of wetting of hydrophobic pores is calculated using the density bias method. It is found that wetting of hydrophobic pores becomes thermodynamically favorable at 5.0 Å for DMPC and 6.5 Å for DOPC, although significant barriers prevent spontaneous wetting of the latter on a nanosecond time scale. The free energy of transformation of hydrophilic pores to hydrophobic ones is also calculated using free energy perturbation of headgroup restraints along the bilayer normal. This quantity, along with wetting and pore growth free energies, provides complete free energy profiles as a function of radius. Pore line tension values for the hydrophilic pores obtained from the slope of the free energy profiles are 37.6 pN for DMPC and 53.7 pN for DOPC. The free energy profiles for the hydrophobic pores are analyzed in terms of elementary interfacial tensions. It is found that a positive three-phase line tension is required to explain the results. The estimated value for this three-phase line tension (51.2 pN) lies within the expected range.

Dynamic wettability alteration for combined low salinity brine injection and surfactant flooding on silica surface

Experimental results of dynamic contact angle measurements of low salinity brine and combined low salinity water and surfactant flooding (LSW–SF) on silica surface are presented in this study. Dynamic contact angle experiments were carried out for four crude oil samples with low salinity brines and combined low salinity and sodium dodecylbenzene sulfonate (SDBS) solutions. Similar measurements with the same ion strength in presence of different cations, Na+ and Ca2+, for low salinity injections and combined low salinity–SDBS solutions were also carried out to study the change in dynamic contact angles and wetting behavior. The presences of different cations change the initial contact angles in low salinity solution injections and show different dynamic behaviors in presence of different crude oils. The signs and values of the line tension to oil/water interfacial tension ratios have been calculated from the size dependence of the dynamic contact angle measurements on the silica coated quartz crystal microbalance sensors. Analysis indicates positive line tension values for low salinity brine systems and negative values for LSW–SF systems. Injection of surfactant solutions in presence of electrolyte prompts the spreading of the oil droplet over the surface, which is induced by interfacial tension gradient from the top of the oil droplet toward the contact line. The results indicated that spreading time, which is the required time for oil drop to gradually flatten out, is dependent on type of electrolytes and is a function of surface excess concentration of the surfactant.

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.

Line tensions of galena (001) and sphalerite (110) surfaces: A molecular dynamics study

We used molecular dynamics (MD) simulation to determine the line tension of two sulfide mineral surfaces, namely galena (PbS)(001) and sphalerite (ZnS)(110) at 298K. Density functional theory (DFT) calculations were used to estimate the partial charges, and the force field parameters were derived based on the nature of the surfaces and the governing potential terms. The computed bulk structures, surface energies and water adsorption energies of the minerals using such partial charges and force field parameters agree well with experiment or DFT results. Contact angles of a series of nano-sized water clusters with different sizes were determined from the MD simulations. Line tensions and macroscopic contact angles of the two mineral surfaces were then determined using the modified Young's equation. The macroscopic contact angles (~84° and 40° for galena and sphalerite surfaces) agree well with the experimental data. The resultant line tensions are on the order of 10−11J/m for both surfaces but with different signs. Here, the line tension for PbS(001) with very low wettability is positive, and for the hydrophilic ZnS(110) surface is negative. Examination of the orientation of the water molecules on the two mineral surfaces showed that the tetrahedron structure present in bulk water (order parameter (q)=0.61) was significantly perturbed on the ZnS(110) surface (q=0.11) compared to the PbS(001) surface (q=0.47). In particular, the average angle between water dipole moment and the surface normal and the radial distribution functions of the water molecules show that the oxygen in water molecules tends to orient itself to the cationic sites (Zn atom) in the sphalerite surface, while such orientation to the Pb atom in the galena surface is minimal. The above results explain that the favorable interaction between water molecules and sphalerite causes the microscopic contact angles to be smaller than the macroscopic value, which in turn yields a negative line tension. On the contrary, stronger affinity of water molecules to the bulk region rather than to galena surface results in bigger microscopic contact angles compared to the macroscopic value and a positive line tension.

Thermostability analysis of line-tension-associated nucleation at a gas-liquid interface.

The influence of line tension on the thermostability of a droplet nucleated from an oversaturated vapor at the interface of the vapor and another immiscible liquid is investigated. Along with the condition of mechanical equilibrium, the notion of extremization of the reversible work of formation is considered to obtain the critical parameters related to heterogeneous nucleation. From the energetic formulation, the critical reversible work of formation is found to be greater than that of homogeneous nucleation for high value of the positive line tension. On the other hand, for high value of the negative line tension, the critical reversible work of formation becomes negative. Therefore, these thermodynamic instabilities under certain substrate wettability situations necessitate a free-energetics-based stability of the nucleated droplet, because the system energy is not minimized under these conditions. This thermostability is analogous to the transition-based stability proposed by Widom [B. Widom, J. Phys. Chem. 99, 2803 (1995)]10.1021/j100009a041 in the case of partial wetting phenomena along with the positive line tension. The thermostability analysis limits the domain of the solution space of the present critical-value problem as the thermodynamic transformation in connection with homogeneous and workless nucleation is considered. Within the stability range of the geometry-based wetting parameters, three limiting modes of nucleation, i.e., total-dewetting-related homogeneous nucleation, and total-wetting-associated and total-submergence-associated workless nucleation scenarios, are identified. Either of the two related limiting wetting scenarios of workless nucleation, namely, total wetting and total submergence, is found to be favorable depending on the geometry-based wetting conditions. The line-tension-associated nucleation on a liquid surface can be differentiated from that on a rigid substrate, as in the former, the stability based on mechanical equilibrium and a typical case of workless nucleation with complete submergence are observed.

Mesoscale pattern formation of self-propelled rods with velocity reversal.

We study self-propelled particles with velocity reversal interacting by uniaxial (nematic) alignment within a coarse-grained hydrodynamic theory. Combining analytical and numerical continuation techniques, we show that the physics of this active system is essentially controlled by the reversal frequency. In particular, we find that elongated, high-density, ordered patterns, called bands, emerge via subcritical bifurcations from spatially homogeneous states. Our analysis reveals further that the interaction of bands is weakly attractive and, consequently, bands fuse upon collision in analogy with nonequilibrium nucleation processes. Moreover, we demonstrate that a renormalized positive line tension can be assigned to stable bands below a critical reversal rate, beyond which they are transversally unstable. In addition, we discuss the kinetic roughening of bands as well as their nonlinear dynamics close to the threshold of transversal instability. Altogether, the reduction of the multiparticle system onto the dynamics of bands provides a unified framework to understand the emergence and stability of nonequilibrium patterns in this self-propelled particle system. In this regard, our results constitute a proof of principle in favor of the hypothesis in microbiology that velocity reversal of gliding rod-shaped bacteria regulates the transitions between various self-organized patterns observed during the bacterial life cycle.

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 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.

Inclusion of line tension effect in classical nucleation theory for heterogeneous nucleation: A rigorous thermodynamic formulation and some unique conclusions.

A rigorous thermodynamic formulation of the geometric model for heterogeneous nucleation including line tension effect is missing till date due to the associated mathematical hurdles. In this work, we develop a novel thermodynamic formulation based on Classical Nucleation Theory (CNT), which is supposed to illustrate a systematic and a more plausible analysis for the heterogeneous nucleation on a planar surface including the line tension effect. The appreciable range of the critical microscopic contact angle (θc), obtained from the generalized Young's equation and the stability analysis, is θ∞ < θc < θ' for positive line tension and is θM < θc < θ∞ for negative line tension. θ∞ is the macroscopic contact angle, θ' is the contact angle for which the Helmholtz free energy has the minimum value for the positive line tension, and θM is the local minima of the nondimensional line tension effect for the negative line tension. The shape factor f, which is basically the dimensionless critical free energy barrier, becomes higher for lower values of θ∞ and higher values of θc for positive line tension. The combined effect due to the presence of the triple line and the interfacial areas (f(L) + f(S)) in shape factor is always within (0, 3.2), resulting f in the range of (0, 1.7) for positive line tension. A formerly presumed appreciable range for θc(0 < θc < θ∞) is found not to be true when the effect of negative line tension is considered for CNT. Estimation based on the property values of some real fluids confirms the relevance of the present analysis.

Growth mechanism of metal-oxide nanowires synthesized by electron beam evaporation: a self-catalytic vapor-liquid-solid process.

We report the growth mechanism of metal oxide nanostructures synthesized by electron beam evaporation. The condensed electron beam can easily decompose metal oxide sources that have a high melting point, thereby creating a self-catalytic metal nanodot for the vapor-liquid-solid process. The metal oxide nanostructures can be grown at a temperature just above the melting point of the self-catalyst by dissolving oxygen. The morphology of nanostructures, such as density and uniformity, strongly depends on the surface energy and surface migration energy of the substrate. The density of the self-catalytic metal nanodots increased with decreasing surface energies of the substrate due to the perfect wetting phenomenon of the catalytic materials on the high surface energy substrate. However, the surfaces with extremely low surface energy had difficulty producing the high density of self-catalyst nanodot, due to positive line tension, which increases the contact angle to >180°. Moreover, substrates with low surface migration energy, such as single layer graphene, make nanodots agglomerate to produce a less-uniform distribution compared to those produced on multi-layer graphene with high surface migration energy.

Line tension and reduction of apparent contact angle associated with electric double layers

The line tension of an electrolyte wetting a non-polar substrate is computed analytically and numerically. The results show that, depending on the value of the apparent contact angle, positive or negative line tension values may be obtained. Furthermore, a significant difference between Young's contact angle and the apparent contact angle measured several Debye lengths remote from the three-phase contact line occurs. When applying the results to water wetting highly charged surfaces, line tension values of the same order of magnitude as found in recent experiments can be achieved. Therefore, the theory presented may contribute to the understanding of line tension measurements and points to the importance of the electrostatic line tension. Being strongly dependent on the interfacial charge density, electrostatic line tension is found to be tunable via the pH value of the involved electrolyte. As a practical consequence, the stability of nanoparticles adsorbed at fluid-fluid interfaces is predicted to be dependent on the pH value. The theory is suited for future incorporation of effects due to surfactants where even larger line tension values can be expected.

The Effect of Electrostatics on the Line Tension at the Edge of a Bilayer

For bilayers composed of charged lipids placed in an aqueous medium, Betterton and Brenner (Phys Rev. Lett., 2000) showed that the line tension λ at the bilayer edge comprises two contributions: a positive hydrophobic line tension, λh, corresponding to the elastic deformation of lipids in the line, and a negative electrostatic line tension, λe, corresponding to the capacitive energy of double layer charging. They showed that λe depends on a single dimensionless parameter, P, which is the ratio of λh and a characteristic scaling of λe. For values of P below a critical value of 2, the net line tension becomes negative, implying that fluid bilayers with high surface charge densities should be unable to close up to form vesicles, and that vesicles should be unstable structures. However, we have prepared and imaged stable vesicles with surface charge densities and pH/salt conditions corresponding to P<<2; this is inconsistent with the above prediction.To understand this discrepancy, we revisit the calculations of Betterton and Brenner, with the new, key inclusion of details of the geometry of the bilayer. This results in two additional dimensionless parameters in the problem: α, the ratio of the bilayer thickness to the Debye length, and f, the surface charge density in the edge of the bilayer relative to the planar region of the bilayer. If the surface charge density is uniform over the entire bilayer (f=1) and α is O(1) or higher, we show that λh can be positive, i.e. even electrostatics would favor a decrease in the perimeter of an open hole in the bilayer! We delineate conditions under which the net line tension becomes positive, and show that this can happen even when P<<2, in agreement with our experimental observations.

Post-Tanner stages of droplet spreading: the energy balance approach revisited

The spreading of a circular liquid drop on a solid substrate canbe described in terms of the time evolution of its base radiusR(t). In complete wetting, the quasistationary regime (far away from initialand final transients) typically obeys the so-called Tanner law, withR∼tαT,αT = 1/10. Late-time spreading may differ significantly from the Tanner law: in some cases the dropdoes not thin down to a molecular film and instead reaches an equilibrium pancake-likeshape; in other situations, as revealed by recent experiments with spontaneously spreadingnematic crystals, the growth of the base radius accelerates after the Tanner stage. Here wedemonstrate that these two seemingly conflicting trends can be reconciled within a suitablyrevisited energy balance approach, by taking into account the line tension contribution tothe driving force of spreading: a positive line tension is responsible for the formation ofpancake-like structures, whereas a negative line tension tends to lengthen the contactline and induces an accelerated spreading (a transition to a faster power law forR(t) than in the Tanner stage).

On the stability of liquid droplets with line tension

Prompted by recent works on line tension effects upon the stability of liquid droplets laid on a rigid substrate, we prove existence and stability of equilibrium configurations when the substrate is either planar or spherical. For positive line tension our argument involves only set symmetrization, while for negative line tension a stability criterion is proposed which makes variational problems well-posed in the framework of sets with finite perimeter. Several applications illustrate the role of the criterion in providing a natural cutoff length that selects destabilizing modes.

Specific wetting probed with biomimetic emulsion droplets

We have produced emulsion droplets of controlled size and composition coated by ligands, and studied the adhesion of these drops on a solid substrate coated by receptors and polymers. Using transmission, RICM and fluorescence microscopy we assess the size, contact angle and ligand density for each drop. We first show that non-specific interactions significantly enhance the proteins density within the adhesive patch. Then we show that binding within the patch is partially inhibited in good agreement with the hypothesis of an absence of translational diffusion. We confirm that the density of specific bonds sets the adhesive energy and therefore the final contact angle, and finally show that specific binding in our system is always associated with the existence of a positive line tension, which linearly increases with the density of receptors. These experiments describe a new scenario for specific wetting which raises the importance of the coupling between non-specific interactions and specific binding.

Molecular dynamics simulations of water droplets on polymer surfaces

Molecular dynamics simulations were used to study the wetting of polymer surfaces with water. Contact angles of water droplets on crystalline and two amorphous polyethylene (PE) and poly(vinyl chloride) (PVC) surfaces were extracted from atomistic simulations. Crystalline surfaces were produced by duplicating the unit cell of an experimental crystal structure, and amorphous surfaces by pressing the bulk polymer step by step at elevated temperature between two repulsive grid surfaces to a target density. Different-sized water droplets on the crystalline PE surface revealed a slightly positive line tension on the order of 10(-12)-10(-11) N, whereas droplets on crystalline PVC did not yield a definite line tension. Microscopic contact angles produced by the simple point charge (SPC) water model were mostly a few degrees smaller than those produced by the extended SPC model, which, as the model with lowest bulk energy, presents an upper boundary for contact angles. The macroscopic contact angle for the SPC model was 94 degrees on crystalline PVC and 113 degrees on crystalline PE. Amorphicity of the surface increased the water contact angle on PE but decreased it on PVC, for both water models. If the simulated contact angles on crystalline and amorphous surfaces are combined in proportion to the crystallinity of the polymer in question, simulated values in relatively good agreement with measured values are obtained.

Residual stability of sessile droplets with negative line tension

We study the local stability of a sessile droplet with nonvanishing line tension along the contact line, where three phases are in equilibrium. We confirm Widom's results [J. Phys. Chem. 99, 2803 (1995)] on the local stability of a droplet with positive line tension in a larger class of perturbations. When the line tension is negative, we prove that the restricted class of perturbations employed by Widom fails to capture the instability of equilibria. A notion of residual stability is introduced, which makes quantitative the condition under which equilibrium of droplets with negative line tension are likely to be observed.

Aspects of the stabilisation of emulsions by solid particles: Effects of line tension and monolayer curvature energy

It is well-documented that solid particles can be effective stabilisers of emulsions, and that the type of the emulsion formed is related, inter alia, to the particle wettability. In general it is to be expected that the free energy change accompanying the formation of a solid-stabilised emulsion will be positive and that emulsion stability will therefore be kinetic in nature. Recently work has appeared showing that very small solid particles, with radius around say 15 nm, can be excellent stabilisers of emulsions. This raises the possibility that line tension acting in the three-phase contact lines around adsorbed particles could have an effect on particle adsorption and hence on their effectiveness as emulsion stabilisers. We explore the effects of both positive and negative line tension on the free energy of adsorption of spherical particles at spherically curved liquid interfaces and on the free energy of drop formation. For small particles it is shown that physically realistic values of positive line tension can lead to exclusion of particles from drop interfaces, either by rendering the adsorption free energy positive or by creating energy barriers to adsorption. We also consider the effects of lateral interactions between adsorbed particles on particle adsorption, particularly strong Coulombic repulsion mediated through the oil phase. It is known that the preferred type of emulsions stabilised by surfactants can be accounted for by the existence of a curvature energy possessed by close-packed surfactant monolayers. We show here that close-packed monolayers of spherical particles at a liquid surface also possess curvature energies, and we calculate bending elastic moduli (κ) as a function of particle size, oil/water interfacial tension, line tension and contact angle. For a monolayer of (hypothetical) particles with radius 0.5 nm (similar to that of a low molar mass surfactant), the value of κ is expected to be of the order of kT, just as for surfactant monolayers. It has been assumed in our work that equal spherical particles are hexagonally close-packed around spherical drops. It is well known however that such packing is not possible, but we show in an Appendix that the assumption of hexagonal packing leads to only small errors (in the context) in calculated free energies of emulsion formation and in the calculated bending elastic moduli of particle monolayers.