Experimental examination of the phase transition of water on silica at 298 K.

The objective of this study was to investigate the prediction of the wetting characteristics obtained from the equilibrium adsorption analysis using the Zeta adsorption isotherm approach with an experimental study. Water vapor's adsorption and wetting characteristics on a hydroxylated and nano-polished silica substrate were studied in near-equilibrium conditions at temperatures near 298 K. Using a UV-visible interferometer, water vapor adsorbate film thicknesses were measured and converted into amount adsorbed per unit area. The current results show that the wetting transition occurred at an average subcooling value of 0.39 K, less than the predicted value of 0.49 K. All the different experimental observations showed growth of film thickness as a function of subcooling value with a maximum film thickness of 12.6 nm. The analysis of the results further showed that the maximum stable film was in a metastable state that then condensed in a dropwise manner, if perturbed by increasing the subcooling. The study further revealed that the adsorbate is unstable after transitioning. The solid surface energy calculated by including the near-equilibrium observations was comparable and close to that of the equilibrium studies, thus supporting solid surface energy as a material property.

Initiation of wetting, filmwise condensation and condensate drainage from a surface in a gravity field.

The zeta adsorption isotherm is based on the hypothesis that a vapour adsorbed on a solid surface consists of a collection of molecular clusters. We use this isotherm to propose a method for determining the wetting condition on a vertically oriented silicon surface exposed to heptane in a gravity field. Measurements indicate the amount adsorbed is larger at positions of smaller potential energy. The wetting condition is taken to be reached when the adsorbed vapour is transformed into the adsorbed liquid phase: adsorption lowers the surface tension of Si from the value in the absence of adsorption to that of liquid heptane at wetting, and then as the Si-heptane is cooled further it is reduced to zero, at a subcooling of 3.7 K. The expectation is that when this subcooling is reached, gravity would cause the larger molecular clusters to drain down the surface. This prediction is supported by experimental observations.

Vapour adsorption kinetics: statistical rate theory and zeta adsorption isotherm approach.

The equilibrium zeta adsorption isotherm for vapours indicates the amount adsorbed is finite for vapour-phase pressures approaching the saturation value, and is strongly supported by experimental measurements for a number of different vapour-solid surface systems. This isotherm assumes the adsorbate consists of differently sized molecular clusters in local equilibrium rather than the adsorbate being in layers. We use the local-equilibrium approximation and develop a method to determine the expression for chemical potential of the adsorbate in terms of the amount adsorbed, nA(t). This allows us to apply statistical rate theory to calculate nA(t) at five different vapour-phase pressures, xV (≡PV/Psat), in terms of a parameter, re. Statistical rate theory indicates that re describes the dynamics of a given isolated system under equilibrium conditions. We consider two methods for determining the value of re that give the best agreement with the measurements performed at each of five values of xV. In one method, we assume, re, is a function of both temperature, T, and pressure, xV, and determine the five best-fit values of re. In the experiments, xV changes by a factor of more than two, but the standard deviation in the re values is 6%. In the second method, we assume re is a function only of T; and find that the value of re is not changed significantly. In all cases, the calculated nA(t) agree with the measurements.

From adsorption to condensation: the role of adsorbed molecular clusters.

The adsorption of heptane vapour on a smooth silicon substrate with a lower temperature than the vapour is examined analytically and experimentally. An expression for the amount adsorbed under steady state conditions is derived from the molecular cluster model of the adsorbate that is similar to the one used to derive the equilibrium Zeta adsorption isotherm. The amount adsorbed in each of a series of steady experiments is measured using a UV-vis interferometer, and gives strong support to the amount predicted to be adsorbed. The cluster distribution is used to predict the subcooling temperature required for the adsorbed vapour to make a disorder-order phase transition to become an adsorbed liquid, and the subcooling temperature is found to be 2.7 ± 0.4 K. The continuum approach for predicting the thickness of the adsorbed liquid film originally developed by Nusselt is compared with that measured and is found to over-predict the thickness by three-orders of magnitude.

Nucleation and growth of condensate in nanoporous materials.

We consider the adsorption-desorption cycles of water and of three hydrocarbons on MCM-41 and on SBA-15. We show that during the desorption portion of a cycle, when the condensate is still at the mouth of the pores, in equilibrium, and the pressure, P, is the minimum value reached before pore-emptying begins, the contact angle is zero. This value of the contact angle is used with the Kelvin equation to calculate the pore radius of each of the mesoporous silicas considered. The standard deviations in the values are found to differ by only a few percent. We propose a method for predicting the size of adsorbed-molecular clusters that must be formed in the pores to initiate condensate formation there. Once formed, the condensate grows spontaneously to the pore mouth. If the vapour-phase pressure when this condition is reached is also P, the adsorption-desorption cycle is reversible. Three of the eight systems considered meet this condition and their adsorption-desorption cycles are experimentally reversible.

Prediction of the wetting condition from the Zeta adsorption isotherm.

We use the Zeta adsorption isotherm and propose a method for determining the conditions at which an adsorbed vapour becomes an adsorbed liquid. This isotherm does not have a singularity when vapour phase pressure, P(V), is equal to the saturation-vapour pressure, Ps, and is empirically supported by earlier studies for P(V) < Ps. We illustrate the method using water and three hydrocarbon vapours adsorbing on silica. When the Zeta isotherm is combined with Gibbsian thermodynamics, an expression for γ(SV), the surface tension of the solid-vapour interface as a function of x(V)(≡P(V)/Ps) is obtained, and it is predicted that adsorption lowers γ(SV) from the surface tension of the substrate in the absence of adsorption, γ(S0), to that at the wetting condition. The wetting hypothesis indicates that γ(SV) at wetting, x, is equal γ(LV), the surface tension of the liquid-vapour interface. For water vapour adsorbing on silica, adsorption lowers γ(SV) to γ(LV) at xVW equal unity, but for the hydrocarbons heptane, octane and toluene adsorbing on silica xVW is found to be 1.40, 1.30 and 1.32 respectively.

Contact angles and surface properties of nanoporous materials.

The validity of thermodynamics at the nanoscale has been questioned, but we demonstrate that it can be applied to determine surface properties of a nonporous and of a mesoporous silica from the measured adsorption isotherms of three hydrocarbon vapors. These measurements give the total amount of vapor adsorbed and liquid formed in the pores as a function of pressure. We compare these measurements with the thermodynamic predictions when the pressure dependence of the contact angle inside the pore is taken into account and determine the values of the surface properties by requiring the thermodynamically predicted isotherms to have the same pressure dependence as the measurements. We assess the procedure by considering the consistency of the property values obtained with each of the three vapors, and find the properties differ by only a few percent. This consistency indicates that thermodynamics is valid at least in pores down to a radius of 1.3 nm.

Effects of nonlinear interfacial kinetics and interfacial thermal resistance in planar solidification.

Large temperature discontinuities were recently measured at a solid-liquid interface during heat transport processes. These observations suggest that when heat flows between two phases, the interface is not well characterized by assuming thermal equilibrium. This can be of importance in rapid solidification processes. In this paper we consider a planar front model that solidifies from its undercooled melt. We use a generalized interfacial boundary condition that includes nonlinear kinetic effects and allows for a temperature discontinuity. The effects of the new boundary condition on the solidification rates and the temperature profile are reported as a function of time. Our analysis shows that the undercooling regime where constant phase-front velocities are observed at steady states (traveling-wave solutions) are unaffected by the new boundary conditions. These solutions arise when the Stephan number is larger than 1. On the other hand, the solidification rates and the steady-state velocities are greatly affected by the assumed conditions at the interface. Incorporation of an interface thermal resistance, or Kapitza resistance, generates temperature discontinuities at the interface, leads to reduced solidification rates and the Mullins-Sekerka instability arises at longer wavelengths deformation of the planar front.

Onset of Marangoni convection for evaporating sessile droplets.

We have generated stability parameters using a linear stability analysis to predict the onset criteria for Marangoni convection in evaporating sessile droplets for two types of substrates, insulating and conducting. The stability problem was formulated with boundary conditions that allow for a temperature discontinuity at the liquid-vapour interface and the inclusion of an expression for the evaporation flux that considers this temperature discontinuity. We introduce no fitting coefficients; therefore, the stability parameters we generate contain only physical variables. The results indicate that spherical sessile droplets evaporating on insulating substrates are predicted to have a similar onset criteria with sessile droplets evaporating on conducting substrates. The onset prediction for sessile droplets evaporating on insulating substrates is found to be considerably different than the case of liquids evaporating from conical funnels constructed of insulating materials owing to the modification of the boundary condition from the geometrical shift and the corresponding retention of modes in the solution. A parametric analysis demonstrates how the input variables impact the stability of evaporating sessile droplets.

Onset of Marangoni convection for evaporating liquids with spherical interfaces and finite boundaries.

We examine the stability of liquids with spherical interfaces evaporating from funnels constructed of different materials. A linear stability analysis predicts stable evaporation for funnels constructed of insulating materials and introduces a stability parameter for funnels constructed of conducting materials. The stability parameter is free of fitting variables since we use the statistical rate theory expression for the evaporation flux. The theoretical predictions are found to be consistent with experimental observations for H(2)O evaporating from a funnel constructed of poly(methyl methacrylate) and for H(2)O and D(2)O evaporating from a funnel constructed of stainless steel.

Statistical rate theory examination of ethanol evaporation.

A series of low-temperature (246 < T(I)(L) < 267 K) steady-state ethanol evaporation experiments have been conducted to determine the saturation vapor pressure of metastable ethanol. The measured interfacial conditions have been used with statistical rate theory (SRT) to develop an expression for the saturation vapor pressure as a function of temperature, f(srt)(eth). This expression is shown to be thermodynamically consistent because it gives predictions of both the evaporative latent heat and the liquid constant-pressure specific heat that are in agreement with independent measurements of these properties. In each experiment, the interfacial vapor temperature was measured to be greater than the interfacial liquid temperature, [triple bond]DeltaT(I)(LV). When f(srt)(eth) is used in SRT to predict DeltaT(I)(LV), the results are shown to be consistent with the measurements. Other expressions for the saturation vapor pressure that are in the literature are examined and found to be thermodynamically inconsistent and do not lead to valid predictions of DeltaT(I)(LV).

Stability of evaporating water when heated through the vapor and the liquid phases.

The stability of a water layer of uniform thickness held in a two-dimensional container of finite or semi-infinite extent is examined using linear stability theory. The liquid-vapor interface can be heated both through the liquid and through the vapor, as previously experimentally reported. The need to introduce a heat transfer coefficient is eliminated by introducing statistical rate theory (SRT) to predict the evaporation flux. There are no fitting or undefined parameters in the expression for the evaporation flux. The energy transport is parametrized in terms of the evaporation number, Eev, that for a given experimental circumstance can be predicted. The critical Marangoni number for the finite, Macf, and for the semi-infinite system, Mac(infinity), can be quantitatively predicted. Experiments in which water evaporated from a stainless-steel funnel and from a polymethyl methacrylate (PMMA) funnel into its vapor have been previously reported. Marangoni convection was observed in the experiments that used the stainless-steel funnel but not with the PMMA funnel even though the Marangoni number for the PMMA funnel was more than 27,000. The SRT-based stability theory indicates that the critical value of the Marangoni number for the experiments with the PMMA funnel was greater than the experimental value of the Marangoni number in each case; thus, no Marangoni convection was predicted to result from an instability. The observed convection with the stainless-steel funnel resulted from a temperature gradient imposed along the interface.

Energy transport by thermocapillary convection during Sessile-Water-droplet evaporation.

The energy transport mechanisms of a sessile-water droplet evaporating steadily while maintained on a Cu substrate are compared. Buoyancy-driven convection is eliminated, but thermal conduction and thermocapillary convection are active. The dominant mode varies along the interface. Although neglected in previous studies, near the three-phase line, thermocapillary convection is by far the larger mode of energy transport, and this is the region where most of the droplet evaporation occurs.

Adsorption at the solid-liquid interface as the source of contact angle dependence on the curvature of the three-phase line.

We review the thermodynamic approach to determining the surface tension of solid-fluid interfaces. If the pressure is in the narrow range where the contact angle, θ, can exist, then for isothermal systems, adsorption at the solid-liquid interface affects γ(SL) or θ, but γ(SV) is very nearly equal γ(LV), the surface tension of the adsorbing fluid. For a liquid partially filling a cylinder, the pressure in the liquid phase at the three-phase line, x(3)(L), depends on the curvature of the three-phase line, C(cl), but the line tension can play no role, since it acts perpendicular to the cylinder wall. C(cl) is decreased as the cylinder diameter is increased; x(3)(L) is increased; and θ increases. For a given value of C(cl), x(3)(L) can be changed by rotating the cylinder or by changing the height of the three-phase line in a gravitational field. In all cases, for water in borosilicate glass cylinders, the value of θ is shown to increase as x(3)(L) is increased. This behaviour requires the Gibbsian adsorption at the solid-liquid interface to be negative, indicating the liquid concentration in the interphase is less than that in the bulk liquid. For sessile droplets, the value of θ depends on both x(3)(L) and C(cl). If the value of θ for spherical sessile droplets is measured as a function of C(cl), the adsorption at the solid-liquid interface that would give that dependence can be determined. It is unnecessary to introduce the line tension hypothesis to explain the dependence of θ on C(cl). Adsorption at the solid-liquid interface gives a full explanation.

Surface tension of solids in the absence of adsorption.

A method has been recently proposed for determining the value of the surface tension of a solid in the absence of adsorption, gammaS0, using material properties determined from vapor adsorption experiments. If valid, the value obtained for gammaS0 must be independent of the vapor used. We apply the proposed method to determine the value of gammaS0 for four solids using at least two vapors for each solid and find results that support the proposed method for determining gammaS0.

Investigation of local evaporation flux and vapor-phase pressure at an evaporative droplet interface.

In the steady-state experiments of water droplet evaporation, when the throat was heating at a stainless steel conical funnel, the interfacial liquid temperature was found to increase parabolically from the center line to the rim of the funnel with the global vapor-phase pressure at around 600 Pa. The energy conservation analysis at the interface indicates that the energy required for evaporation is maintained by thermal conduction to the interface from the liquid and vapor phases, thermocapillary convection at interface, and the viscous dissipation globally and locally. The local evaporation flux increases from the center line to the periphery as a result of multiple effects of energy transport at the interface. The local vapor-phase pressure predicted from statistical rate theory (SRT) is also found to increase monotonically toward the interface edge from the center line. However, the average value of the local vapor-phase pressures is in agreement with the measured global vapor-phase pressure within the measured error bar.

Absence of Marangoni convection at Marangoni numbers above 27,000 during water evaporation.

Two mechanisms by which Marangoni convection can be produced at the interface of water with its vapor are: (1) by imposing a temperature gradient parallel to the water-vapor interface, and (2) by imposing a temperature gradient perpendicular to the interface that results in the liquid becoming unstable. A series of evaporation experiments conducted with H2O and with D2O maintained at the mouth of a stainless-steel funnel indicated the presence of Marangoni convection, but the mechanism producing the convection was unclear. We have investigated the mechanism using a funnel constructed with a polymethyl methacrylate that has a small thermal conductivity relative to that of water and repeating the evaporation experiments. Marangoni convection was eliminated with this funnel even though the Marangoni number, Ma, was in the range 8277< or =Ma< or =27 847 . A comparison of the assumptions made in the theories available to predict the onset of Marangoni convection with the observations made in this study indicates some of the assumptions are invalid: although generally neglected, energy transport through the vapor to the interface of evaporating water is significant; there is an interfacial temperature discontinuity, but it is in the opposite direction of that assumed in the existing theories: the interfacial-vapor temperature is greater than that of the liquid during evaporation; and the prediction of the critical Marangoni number is based on an arbitrarily chosen value of the heat-transfer coefficient. When the temperature gradient is perpendicular to the water-vapor interface, these invalid assumptions indicate present theories do not apply to volatile liquids.

Role of molecular phonons and interfacial-temperature discontinuities in water evaporation.

During steady-state water evaporation, when the vapor phase is heated electrically, the temperature on the vapor side of the interface has been reported to be as much as 27.83 degrees C greater than that on the liquid side. The reported interfacial temperatures were measured with thermocouple beads that were less than 50 microm in diameter and centered 35 microm from the interface in each phase. We examine the reliability of these measurements by using them with a theory of kinetics to predict the interfacial-liquid temperature. The predicted temperature discontinuities are found to be in agreement with those measured up to a temperature discontinuity of 15.69 degrees C , but larger discontinuities cannot be confirmed because of uncertainties in the vapor-phase pressure measurements. The theory of kinetics used in the analysis includes molecular phonons in the expression for the evaporation flux. We show it is essential to include these terms if the theory is to be used to predict the temperature discontinuities.

Effect of contact line curvature on solid-fluid surface tensions without line tension.

For sessile droplets partially wetting a solid surface, it has been observed experimentally that the value of the contact angle depends on the contact line curvature and this dependence has been attributed to tension in the contact line. But previous analyses of these observations have neglected adsorption at the solid-liquid interface and its effect on the surface tension of this interface. We show that if this adsorption is taken into account the relation between the contact angle and contact line curvature is completely accounted for without introducing line tension. Further, from the observed relation between the contact angle and contact line curvature, the adsorption at the solid-liquid interface can be determined, as can the surface tensions of the solid-liquid and solid-vapor interfaces.