25982
J. Phys. Chem. B 2006, 110, 25982-25993
Phase Transitions of Capillary-Held Liquids in a Slit-like Pore Nobuo Maeda† Department of Applied Mathematics, Research School of Physical Sciences and Engineering, Australian National UniVersity, Canberra ACT 0200, Australia ReceiVed: August 1, 2006; In Final Form: September 12, 2006
Dynamics of capillary held liquids plays important roles in a wide range of systems including adhesion, printing of paints and inks, the behavior of wet granular materials, and the mass transfer through porous media. Recent study suggested the presence of two distinct modes for the disappearance of capillary-held liquids in a slit-like pore of adjustable slit width that depended on the slit-opening rates. In contrast to the first mode that is well-documented in terms of the Young-Laplace equation, a novel and unexpected mode was observed when the liquid bridge was held in the vicinity of the thermodynamic phase boundary (equilibrium Kelvin length). Here we extended the study to three new compounds that have a wide range of vapor pressures. An evaporating liquid bridge developed large refractive index gradients that extended over a few micrometers from the edge of the meniscus once the slit width was increased beyond the equilibrium Kelvin length. This interfacial region with depleted refractive index retreated inward as the meniscus shrank with time, and the refractive index of the entire bridge subsequently fell from that of the liquid once the interfacial regions from the opposite sides of the shrinking bridge met at the center. The refractive index recovered to that of the liquid when the slit width was closed to below the Kelvin length and the vapor was allowed to recondense. The time scale of the evaporation and condensation depended on the size of the surface gap, and, when the surfaces were placed at a separation very close to the Kelvin length, it was possible to detect a stage in which the system was in an apparent kinetic equilibrium between two physical statesswith and without the liquid connecting the two surfaces.
Introduction The nucleation, condensation, evaporation, and flow of fluids in confined geometries play important roles in a wide range of phenomena, such as adhesion, the behavior of moist granular materials, and the mass transfer through porous media. Capillary action has received extensive scientific study since the days of Leonardo da Vinci, as extensively reviewed by Maxwell.1 Despite the long history of the subject,2-4 understanding of capillary action at all lengths and time scales is still incomplete. The Kelvin equation relates the equilibrium interfacial curvature of a meniscus to the undersaturation of a vapor.5 The rate of evaporation depends on the internal pressure of the liquid, which in turn depends on the curvature of the interface. The rate of condensation depends on the pressure of the surrounding vapor. Equilibrium can be established when the rate of evaporation equals the rate of condensation. There is sufficient experimental evidence for the validity of the Kelvin equation down to sub-micrometer scale for convex surfaces such as droplets.6 The validity for concave surfaces such as menisci was less obvious, but eventually verified by Fisher and Israelachvili for hydrocarbons7 and by Kohonen and Christenson for water.8 However, the Kelvin equation is essentially a thermodynamic equation and has little to tell about the kinetics of formation and/or disappearance of droplets or capillary-held liquids. For example, the classical homogeneous nucleation theory predicts the critical size of nuclei for droplet formation in a supersaturated vapor on the basis of the Kelvin equation, but it does not † Telephone: +61-2-6125-7584.Fax: +61-2-6125-0732.E-mail: nxm110@ rsphysse.anu.edu.au.
predict when and how nuclei that are larger than the critical size can suddenly form in an initially homogeneous vapor phase. Presence of finite (nonzero) hysteresis is a signature of a firstorder phase transition, such as superheating of liquids prior to boiling or undercooling of liquids prior to freezing. In many cases such hysteresis results from the nucleation process of an emerging phase. In the case of capillary condensation the hysteresis appears in the form of condensation hysteresis, i.e., pore-filling occurring at higher relative vapor pressures than pore-emptying. It has been shown that such condensation hysteresis can occur not only in complex porous media (which may contain so-called “ink-bottle” pores3) but also in a single pore of uniform geometry9-12si.e., arising from the first-order nature of the phase transition. There are two leading mechanisms of pore-filling in a slitlike pore:29 (1) the adsorbed films on pore walls act as nucleation sites of the condensing phase, and, as the relative vapor pressure mounts, there is a point when these films suddenly thickens and fill the remaining space in the pore (the film coalescence model);10,13 (2) nuclei of the condensing phase larger than the critical size form somewhere in the initially homogeneous vapor space in the pore and subsequently grow to fill the pore (the nucleation model).14 The film coalescence model is generally preferred to the nucleation model because the former is essentially a heterogeneous nucleation, whereas the latter is a homogeneous nucleation which generally requires a higher activation barrier to overcome. One mechanism of disappearance of capillary-held liquids (pore-emptying) in a slit-like pore has been well-established. Mechanical instability (the Laplace instability) is expected to occur that leads to the rupture of a capillary-held bridge when
10.1021/jp0649394 CCC: $33.50 © 2006 American Chemical Society Published on Web 11/30/2006
Capillary-Held Liquids in a Slit-like Pore the bridge is stretched rapidly. Everett and Haynes described the Laplace instability of macroscopic liquid bridges in detail.9 On the basis of numerical solutions of the Young-Laplace equation, Willett et al. derived a general equation that relates the rescaled (dimensionless) bridge volume to the rescaled rupture distance.15 We had recently shown that, under such conditions, liquid bridges as small as 10-16 m3 rupture due to the Laplace instability as predicted by the Young-Laplace equation.16 Very recently, Sirghi et al. further extended the validity of the Young-Laplace equation down to 10-23 m3 on the basis of force measurements using atomic force microscopy (AFM).17 The basic idea of the Laplace instability in the context of emptying of a slit-like pore is that the evaporation takes place at the already existing liquid-vapor interface and the liquid bridge gradually shrinks until the relative vapor pressure drops to the extent that the corresponding Kelvin diameter becomes smaller than the pore dimension. At this point the liquid bridge is effectively being stretched and the curvature of the interface becomes less negative, which effectively increases the liquid pressure and accelerates the rate of evaporation. The bridge would rapidly shrink laterally until its width becomes comparable to its height at which point the bridge becomes mechanically unstable and snaps. The resulting convex curvature of the two droplets on each pore wall after the snapping of the bridge further accelerates the evaporation of the liquid, and evaporation completes rapidly. A qualitatively different and unexpected behavior was observed when such liquid bridges were held in the vicinity of the phase boundaries (i.e., when the slit width was held in the vicinity of the equilibrium Kelvin diameter). We have previously reported the nucleation and phase transitions of three hydrocarbon fluids in a slit-like pore bounded by two mica walls of adjustable dimension, with special attention focused on the phase behavior of the fluids in the vicinity of the phase boundaries.16,18,30 It appeared that highly volatile cyclohexane exhibited very large nonuniform density gradient parallel to the slit (which we refer to as “lateral” direction hereafter) once the liquid bridge shrank to a certain size.18 Less volatile hexadecane showed less pronounced behavior.16 In either case, the density of the entire bridge apparently fell continuously from that of liquid to that of vapor, and the variation in the density was reversible until the density of the bridge virtually attained that of vapor. Importantly, the size of the bridges where such anomalous behavior was observed was unexpectedly large, essentially microscopic dimension (volume ≈ 1 µm3; it is worth noting that the lateral width of the bridge remained above a few micrometers during this apparent change in the density, whereas the length of the bridge was of the order of 50 nm). As discussed in ref 16, a few alternative interpretations of these surprising findings were invoked that did not assume any change in the density of the liquids. Among these, the most plausible interpretation at the time (Figure 6D of ref 16) assumed that the liquid “bridge” was instead two droplets on each mica wall that extended over a large lateral distance (>10 µm in the case of cyclohexane) and that these droplets were connected by a liquid “thread” that was below the resolution of our experimental technique.31 This “two droplets connected by a thread” picture could account for the apparently reversible and continuous behavior observed experimentally, although it remained unclear why a liquid bridge should assume such highly convoluted shapes or how such shapes could be maintained for prolonged periods of time. In this work, we extended the study to three new compounds that have very different vapor pressures from each other to
J. Phys. Chem. B, Vol. 110, No. 51, 2006 25983
Figure 1. Schematic illustration of the experimental setup. The conventional double-cantilever spring that supports the lower surface in a typical SFA was replaced with a rigid metal block (effective spring constant of ≈105 N/m) to suppress vibration of the surface.
extend the validity of the phenomenon and also to elucidate the nature of the anomalous behaviors observed in the vicinity of the phase boundaries. In particular, a few novel features were detected for cyclooctane, which has an intermediate vapor pressure of the other two compounds at room temperature, and we devote a large portion of this paper to the study using cyclooctane. Materials and Methods Figure 1 shows the main features of the experimental setup used in this study, which is virtually the same as that used in the previous SFA studies of capillary condensation and evaporation.11,12 The mica surfaces that constitute the slit-like pore had molecularly smooth areas over 1 cm2 each. One side of the mica sheets was coated with 50 nm of silver. The mica surfaces were mounted on cylindrical silica disks (radius of the cylinder ≈ 2 cm), with the silvered side glued to the disk with thermosetting epoxy, and these disks were mounted to the SFA with their cylindrical axes perpendicular to each other. The conventional double cantilever spring that supports the lower surface in a typical SFA was replaced with a rigid metal block (effective spring constant of ≈ 105 N/m) to suppress vibration of the surface. The upper surface was mounted at the end of a piezo transducer which allows precise control of surface separation. The local geometry of the mica surfaces at small surface separations corresponds to a sphere on a plane. Unless otherwise specified, we usually quote the value of surface separation, H, so measured between the closest point of approach between the sphere and the plane. The actual local geometry of the mica surfaces, including the details such as deformation due to surface forces and the presence of a meniscus, was monitored in situ and/or recorded using a CCD located at the exit of a spectrometer.
25984 J. Phys. Chem. B, Vol. 110, No. 51, 2006 When white light passes through a Fabry-Perot type interferometer, such as the one we have here with a pair of backslivered mica surfaces, and then dispersed by a spectrometer, fringes of equal chromatic order (FECO)19,20 emerge at the exit of the spectrometer. The details of multiple-beam interferometry can be found in ref 20, so here we only note a few important points. First, we used the three-layer interferometer equation20 in this study. When two neighboring FECOs were recorded simultaneously for a given image, the three-layer interferometer equation allows independent determination of the surface separation, H, and the average refractive index, n, between the two mica surfaces. This is possible because two neighboring FECOs, which we refer to as odd- and even-order FECOs, have different wavelength-separation relationships. Second, we still use the same old convention of the odd and even orders of the FECOs in this study as those in our previous relevant publications.16,18 In future publications the new convention21 will be followed. The intensity distribution of a peak of each recorded FECO spectra typically extended over several pixels. Some of the recorded images were analyzed using custom software from Digital Optics that fits a Gaussian curve to the intensity distribution of each spectral peak, which enabled the location of the peak position with the maximum precision of ≈0.01 pixels. However, there are some images for which the software could not fit a Gaussian curve to the spectral intensity peaks. In these cases the peak position was measured manually with the precision of 1 pixel. Independent calibration of the optical magnification of the system yielded the relationship that 1 pixel of the CCD corresponded to about 600 nm in real space. One of two gratings was used in the spectrometer for a given experiment, depending on the thickness of the mica sheets used, and the dispersion of each grating was calibrated using standard lines of a mercury lamp (≈0.0491 and ≈0.0727 nm/pixel). The chemicals were obtained from Sigma-Aldrich or Fluka and used as received. The purities of the as-received samples were 99.7% for n-hexane, 99+% for n-tetradecane, 99+% for cyclooctane, and 99% for squalane. Measurements were carried out in dry and prefiltered nitrogen gas and the vapor of interest at 25 °C (unless otherwise specified) with CaH2 as desiccant inside the SFA chamber. In some experiments using n-hexane or cyclooctane a small amount (typically 4 mol %) of squalane was added to the main component of the chemical to control the relative vapor pressure. The control experiments using 100 mol % squalane showed no measurable variation in the refractive index from that in dry and prefiltered nitrogen gas during the time scales of a typical experiment. Also, there was no measurable change in the size of the capillary condensates during a given experimental run for which squalane was used as a solute. For sparingly volatile n-tetradecane, no squalane was added for the purpose of controlling the vapor pressure, because the rate of evaporation and condensation of tetradecane is already slow and the evaporation of squalane cannot be neglected. Instead, measurements were made during the approach to saturation of the SFA chamber, which takes several days after the initial injection of the liquid into the chamber. We also note that up to three IR filters were used to control the heating of the surfaces by the white light and hence the undersaturation of vapor. The actual undersaturation of vapor of each compound was measured in situ for each measurement from the discontinuity of the FECO when the two mica surfaces were in adhesive contact. The discontinuity of FECO is an indication that there is a discontinuity in the optical path lengths between the mica
Maeda TABLE 1: Saturation Vapor Pressure (P0) and Viscosity (η) of Relevant Compoundsa
a
compound
P0 (Pa)
η (mPa s)
hexane cyclohexane cyclooctane tetradecane hexadecane
2.2 × 104 1.3 × 104 4.5 × 102 1.0 2.2 × 10-1
0.30 0.894 2.25 2.13 3.3
The data are from refs 22, 23, and 28.
Figure 2. Typical refractive index profile in nitrogen gas between two mica surfaces at the cross-section of the closest point of approach. The systematic errors rapidly increase at small surface separations ( 2rK, the refractive index falls farther away from that of liquid, and, if the surface gap was closed prior to the completion of the evaporation process, the refractive index value gradually recovered to that of liquid at the reduced H. Importantly, although only a finite number of data points is shown in Figure 4, the variation of the refractive index and of the surface gap was continuous for both. The situation is analogous to a ball on top of an apex of a hill where slight displacement of the center of mass leads to the fall of the ball, but appropriate upward push can keep the ball in the vicinity of the apex (unstable equilibrium). We also note that before the liquid bridge shrinks to these stages, it was possible to stretch the bridge to a larger surface separation and then bring it back again to below the equilibrium Kelvin diameter, without significant fall of the refractive index. The hysteresis loops, such as the one shown in Figure 4, were similar to those observed for hexadecane earlier. They were not unique and depended on the history of a particular bridge: e.g., how large the volume of the bridge was when it was stretched to the surface separation of 2rK, to which surface separation that is larger than 2rK the bridge was stretched, how long the liquid was allowed to evaporate at that surface separation before the surface separation was reduced, to which surface separation the gap was closed, and how quickly the gap was closed. Cyclooctane. Cyclooctane has an intermediate vapor pressure between the highly volatile cyclohexane/n-hexane and the sparingly volatile n-hexadecane/n-tetradecane at room temperature.22,23 The behavior of cyclooctane bridges resembles those
Figure 4. Typical refractive index profile of the entire liquid bridge between two mica surfaces in undersaturated vapor of n-tetradecane when the surfaces are held in the vicinity of the equilibrium Kelvin diameter. The corresponding FECO images are also shown. The time scale to go around one hysteresis loop could be varied by changing the size of the hysteresis loop (of the order of tens of seconds to minutes).
25986 J. Phys. Chem. B, Vol. 110, No. 51, 2006
Maeda
Figure 5. Typical lateral refractive index profile between two mica surfaces in undersaturated cyclooctane vapor at P/P0 ) 0.87. Four selected stages during the final stages of evaporation were shown together with the corresponding FECO images.
of cyclohexane or n-hexane for some aspects, although the kinetics was much slower. Figure 5 shows typical lateral refractive index profiles between two mica surfaces in undersaturated cyclooctane vapor at P/P0 ) 0.87, the lowest relative vapor pressure studied. Four selected stages during the final stages of evaporation (when the surface separation was held in the vicinity of the Kelvin diameter) are shown together with the corresponding FECO images. Much clearer profiles were obtained for cyclooctane than for n-hexane or cyclohexane reported previously whose kinetics was much faster. The images in A-C were recorded with 5 s exposure time to enhance the signal-to-noise ratio. The image in D was recorded with 1 s exposure time because of the much faster kinetics toward the end of the evaporation process. The “lateral dimension” in the FECO images appears vertical here (the horizontal direction representing the wavelengths of the fringes), so they need to be rotated by 90° to match the analyzed profiles. As the profile evolved from A to B to C to D, the refractive index gradient became more pronounced. In C, the refractive index gradually varied laterally from ≈1.4 to 1 over a surprisingly large distance
of ≈5 µm for each side of the liquid bridge, and the refractive index at the middle of the liquid bridge (≈1.4) is below the bulk value of 1.458. In D, the refractive index in the middle of the liquid bridge takes values around 1.2, again over a surprisingly large lateral distance of ≈3 µm. The decrease in the surface separation from stage D led to gradual regrowth or recondensation of the liquid bridge, with no evidence of sudden and irreversible Laplace instability. We note that when the liquid bridge was allowed to completely evaporate at H > 2rK instead, no gradual regrowth or reemergence of the bridge was observed as H is reduced, until van der Waals jumps eventually pulled the two mica walls (with thin adsorbed films) together at much smaller surface separations.10-12 We also note that the thicknesses of the adsorbed films in this study were typically 0.60.8 nm per surface, in agreement with the dispersion theory.24 Importantly, the three-layer interferometer equation allows independent calculations of the surface separation and the refractive index profiles. Thus we can reconstruct the surface geometry calculated from the FECO images that correspond to the four selected stages in Figure 5, as shown with the filled
Capillary-Held Liquids in a Slit-like Pore
J. Phys. Chem. B, Vol. 110, No. 51, 2006 25987
Figure 6. Surface geometry calculated from the FECO images that correspond to the four selected stages in Figure 5, shown as filled black circles (and a line to go through them to guide the eyes). The open squares at each stage show the profile of cyclooctane films calculated from the refractive index profile in Figure 5, with the assumption of the bulk liquid refractive index of 1.458 for the films and symmetry (equal film thickness at each surface for a given cross-section). Note the much larger lateral dimension than the vertical dimension.
black circles (and a line to go through them to guide the eyes) in Figure 6. The surface geometry at each stage is that of a sphere on a flat, as expected, and the closest point of approach (the minimum separation) between the two mica surfaces is 45 (A), 35 (B), 30 (C), and 46 nm (D), respectively. Moreover, we can reconstruct the profile of cyclooctane films on the surfaces if the refractive index profile in Figure 5 arose from unusual shapes of the films. The reconstructed film profiles are shown in Figure 6 with open squares. Here we assumed that such “film” had the bulk liquid refractive index of 1.458 and also assumed equal film thickness on each surface for a given cross-section. It can be seen that in A and B, the two “droplets” on the surfaces are connected near the center (“lateral dimension” ≈ 0), whereas in C and D there is a clear gap separating the two droplets. It is important to note the much larger lateral dimension than the vertical dimension. A typical aspect ratio is ≈500:1, and we cannot reconstruct the real shape on this paper. As can be seen, the profiles of such films or droplets at each stage have slightly convex shapes everywhere. We cannot rule out the presence of a thin thread of liquid bridge that connects the two droplets, presumably near the closest point of approach, that is below the optical resolution of the FECO technique ( 2rK in undersaturated cyclooctane vapor of P/P0 ) 0.92. The interfacial region (indicated by the arrows labeled “I”) first appears at the outer edge of the meniscus A. The interfacial region retreats inward as the inner liquid region (indicated by the arrow labeled “L”) shrinks laterally (A-H), until the interfacial regions from both ends join at the center (I and J). The lateral width and the refractive index gradient of these interfacial regions apparently increase steadily during the course of evaporation (A-H), and by the stage I, it results in a large continuous refractive index gradient that extends over several micrometers laterally. When the surfaces are held at H > 2rK after stage J, the bridge goes through stages similar to those shown in Figure 5. The time scale of the emergence and the growth of the interfacial region depended on the separation at which the surfaces were held (of the order of seconds to tens of seconds).
quantitative analyses because we cannot identify the fringe position with reasonable accuracy. We note that such blurring of the FECO images occurred only in the vicinity of the Kelvin diameter when the liquid region first reemerged as the surface gap was being closed. We also note that only the bridge region became blurred in Figure 9A, while the outer region (vapor region) apparently remained unaffected. Therefore this blurring effect is different from ambient vibrations which would result in blurring of the entire FECO images. Rather, it appears that some sort of “repeated transition” effect emerges at the first onset of recondensation of the vapor. Such repeated transition effect diminished either as the liquid region was allowed to grow
Capillary-Held Liquids in a Slit-like Pore
Figure 9. (A) Image showing “blurred” bridge region. It was often observed that the FECO images of both odd- and even-order fringes became blurred at the first onset of the regrowth of a cyclooctane bridge (P/P0 ) 0.92). Only the bridge region became blurred, while the outer region (vapor region) apparently remained unaffected. This blurring of the bridge region was only observed at the first onset of the bridge regrowth. (B) At the lowest relative pressure of cyclooctane studied (P/P0 ) 0.87), it was possible to detect the even-order FECO recorded at two different wavelengths within the exposure time of the CCD. Such images could be recorded one after another, suggesting that the system was apparently jumping back and forth between two distinct states, namely, with and without the liquid at the center of the interfacial region.
larger or as the emerging liquid region was made to evaporate away by slightly separating the surfaces. In light of Figure 9B, which is suggestive that the FECO keep jumping between two different wavelengths, the blurring of the FECO images in Figure 9A could also be the result of the same repeated transition effect. In light of the findings shown in Figure 8, it is reasonable to assume that the even-order FECO component at the shorter wavelengths in Figure 9B corresponds to that of the liquid region and that at the longer wavelengths corresponds to that of the interfacial region. Then it appears that the system was jumping back and forth between two distinct physical states, namely, with and without the liquid region in the background of the interfacial region, whatever its physical meanings may be. The puzzle is that the longer wavelength of the even-order FECO for the interfacial region apparently corresponds to a higher refractive index than the emerging liquid region at a shorter wavelength. Unfortunately, quantitative analyses are difficult for these stages because of the broadened odd-order FECO and the unknown amount of the deformation of the mica surfaces (the odd-order FECO at this stage typically shows a pointing shape which is indicative of the presence of the Laplace pressure pulling the mica surfaces together). We note that the middle section of the interfacial region immediately prior to the emergence of the liquid region already had a refractive index that of bulk liquid. Thus the reemerging liquid region apparently had a refractive index lower than that of bulk liquid (but higher than that of vapor). Discussion It is worth noting that the phenomena of the main interest in this study are clearly of nonequilibrium processes, and therefore they are highly rate-dependent. The time scales of evaporation, condensation, flow, and (if any) other route of material transfer
J. Phys. Chem. B, Vol. 110, No. 51, 2006 25989 must depend on the relative location of the surfaces with respect to the equilibrium Kelvin length of the system, and the response time of the system to a given change in an external condition (e.g., opening or closing the gap) depends on the physical properties (e.g., vapor pressure, viscosity) of the compound under study. As such, a given course of events (e.g., evaporation followed by condensation followed by evaporation again) and the final fate of the system after a particular experimental run (whether a liquid totally disappears with the gap open or grows to an equilibrium size with the gap closed) depend on the path the system has taken both in time and in space. Keeping these general features in mind, we discuss various aspects of the results below. Lateral Refractive Index Gradient and the Size of the Hysteresis Loop. The refractive index gradient that extends laterally over several micrometers from the edge of the meniscus has so far been detected for n-hexane, cyclohexane, and cyclooctane, whereas no such lateral gradient has been detected for n-hexadecane or n-tetradecane. For the former compounds, there is a stage that a bridge exhibits both a large lateral refractive index gradient and a reduced refractive index at the center of the bridge (Figure 5C,D). Figure 8 suggests that the development of the large lateral refractive index gradient is the direct consequence of the development of the interfacial region. In contrast, for the latter compounds, a bridge shrinks laterally first while maintaining the refractive index of the bulk, and then the refractive index of the entire bridge falls toward that of vapor. The lateral gradient and the interfacial region observed for highly volatile compounds become somewhat less pronounced as the relative vapor pressure (P/P0) increases and the behavior resembles more that of the sparingly volatile compounds. Thus the first guess is that the rapid rate of evaporation is responsible for the emergence of the lateral gradient. Both a higher P0 and a lower P/P0 are expected to increase the net rate of evaporation. Then, one possibility is that the rapid evaporation leads to enhanced cooling of the interface, which could effectively reduce the rate of evaporation locally and also induce Marangoni type flows. Another possibility is that the narrow slit geometry hinders timely material transfer (diffusion) and that the rapid rate of evaporation induces some sort of local concentration or pressure gradients in the vapor in the vicinity of the interface. The hysteresis loops such as the one shown in Figures 4 and 7 could be manipulated for the moderately volatile cyclooctane and the sparingly volatile tetradecane and hexadecane, but not for the highly volatile hexane and cyclohexane. There is limitation in our experimental setup as to how quickly we can vary the surface gap in response to the change in the refractive index. Therefore the reason for the absence of large hysteresis loops for the highly volatile compounds is likely to be the manifestation of our inability to close the surface gap in a timely manner once the refractive index at the center of the bridge starts falling significantly below that of the liquid. The same experimental limitation hindered detailed study of the highly volatile compounds at a high P/P0 (≈0.99), because the large corresponding Kelvin diameter required quick and large reduction in the surface separation when the refractive index of the entire bridge started to fall, where we could only record the first onset of the refractive index fall. We note that vapor pressure may not be the only factor that separates these two groups. For example, the volatile compounds also happen to have smaller viscosities than the sparingly volatile compounds. Then the observed difference in the disappearance
25990 J. Phys. Chem. B, Vol. 110, No. 51, 2006 behavior could be related to the flow properties of these liquids on the mica walls. Fall of the Refractive Index of an Entire Liquid Bridge at the Final Stage of Evaporation. Continuous fall of refractive index of an entire bridge toward the end of the evaporation process was observed for all the compounds studied so far. For volatile compounds such as n-hexane, cyclohexane, and cyclooctane, the liquid bridge develops large lateral refractive index gradients and then the refractive index at the middle section (which corresponds to the highest value) begins to fall while the extent of the lateral gradient continues to shrink simultaneously. For sparingly volatile long-chain alkanes, the liquid bridge first shrinks laterally to a few micrometers while retaining the refractive index of the bulk liquid and then the refractive index of the entire bridge falls continuously from that of bulk liquid to that of vapor with comparatively little change in the lateral width of the bridge. It is important to acknowledge that the width of the bridge remains a few micrometers wide laterally for all the liquids studied so far, which is above the optical resolution of the FECO technique. The optical resolution is generally limited by the uncertainty principle of the quantum mechanics to the order of the wavelength, i.e., a fraction of a micrometer for the visible wavelengths used in this study. We note at this stage that the larger sampling width by the entrance slit of a spectrometer in a typical SFA setup does not worsen the optical resolution. For a typical slit width of ≈70 µm and the optical magnification of our microscope of ×24, the entrance slit is typically sampling information within a ≈3 µm wide stripe at the surfaces. Then one may conceive that once a liquid bridge shrinks to below 3 µm every point of a FECO is “averaged” over ≈3 µm, which could include both the liquid bridge and the surrounding vapor. If this were the case, one may expect that, for a bridge of lateral diameter below 3 µm, its refractive index would appear taking values below that of the bulk liquid, and the decrease in the refractive index would continue as the lateral size shrinks because the relative proportion of the surrounding vapor within the 3 µm window would increase. However, this is not the case. To the contrary, such “optical averaging” to the direction perpendicular to the slit of the spectrometer (or the absence thereof) does not depend on the slit width, at least in the range used in the experiments. For the sake of simplicity we consider a 1 µm wide liquid bridge surrounded by vapor. When the incident white light comes out of the interferometer formed by a pair of back-silvered mica sheets, the out-coming light consists of two distinct wavelengths, one corresponding to the inside and the other corresponding to the outside of the bridge. There is no physical reason to suspect that these two packs of photons of two distinct wavelengths interfere with each other before they reach the diffraction grating inside the spectrometer, where they are diffracted to two distinct directions and form intensity peaks at two different locations on the face of the CCD. Emergence of the “Interfacial Region” at the Edge of the Cyclooctane Menisci once the Surface Separation Exceeds the Equilibrium Kelvin Diameter. The detailed sequence of events during the course of evaporation shown in Figure 8 provides us important clues as to the mechanisms behind the anomalous behaviors. The evaporating cyclooctane bridge in fact appears to consist of two regions, which we referred to as the liquid region and the interfacial region. When the liquid bridge was held above the equilibrium Kelvin diameter, the interfacial region receded toward the center of the bridge as the inner liquid region shrank and eventually disappeared. After the interfacial regions from both sides joined at the center, large
Maeda lateral refractive index gradients developed. This corresponds to the stage which we previously termed “the final stage of evaporation” or “once the bridge shrinks below a certain size” in refs 16 and 18. The highly volatile hexane and cyclohexane developed large interfacial regions and subsequently developed large refractive index gradients that extended over several micrometers. The sparingly volatile tetradecane and hexadecane, on the other hand, did not develop the interfacial regions that could be detected with confidence, and consequently the refractive index of the entire bridge dropped continuously from that of liquid to that of vapor. Even-Order FECO Recorded at Two Different Wavelengths within the Exposure Time of the CCD. When the mica surfaces were located at a “right” surface separation in the vicinity of the Kelvin diameter, FECO images such as those shown in Figure 9 could be recorded repeatedly one after another. The even-order FECO component which corresponds to the liquid region apparently emerges at a shorter wavelength than that which corresponds to the interfacial region. The corresponding odd-order FECO was broadened or blurred, suggesting that the odd-order FECO may also have taken two wavelengths. These findings suggest that the system was placed right at the phase boundary at the first onset of recondensation. Then, thermal/Brownian type motion may put the system locally (somewhere within the lateral extent of the interfacial region) above or below the Kelvin diameter, which led to evaporation or condensation of the liquid or vapor. The concomitant formation or disappearance of the liquid region could lead to the presence or absence of the Laplace pressure between the two surfaces, which could result in pulling or releasing of the surfaces, i.e., blurring of the FECO. This kind of discontinuous behavior (“jumps”) was in contrast to the behavior of the sparingly volatile tetradecane and hexadecane for which the refractive index of the entire liquid bridge apparently varied continuously. We note that the refractive index of the entire liquid bridge of the sparingly volatile tetradecane and hexadecane at the first onset of recondensation (when the size of the bridge apparently started growing) was often below that of the bulk liquid. This result may be related to the observation in Figure 9B in which the growing liquid region apparently had the refractive index lower than that of the bulk liquid. Possible Scenarios That Could Account for the Observed Phenomena. We previously presented four leading scenarios in an attempt to account for the experimental observations (Figure 6 of ref 16). These were (A) development of nonuniform density of liquids in pores, in particular monotonic density gradients from the edge of the meniscus, (B) formation of nanobridges and/or nanobubbles, presumably due to cavitation within the liquid bridge, (C) formation of a droplet on each of the mica walls, (D) similar to C but the two droplets are connected by a liquid “thread”. We believe that these four scenarios exhaust all the possibilities, and examine each of them in light of the new results presented in this work. The first scenario (A) assumes nonuniform density or spatial concentration gradients of fluids presumably due to the geometry that hinders timely material transfers (diffusion) as the surface gap is opened or closed across the Kelvin diameter. The Lorentz-Lorenz equation25 directly relates the refractive index of a given material to its density. The systematic difference observed between the highly volatile hexane/cyclohexane and the sparingly volatile tetradecane/hexadecane could be a result
Capillary-Held Liquids in a Slit-like Pore of limited diffusion of some sort of liquid clusters in vapor within the narrow gap bounded by the mica walls. The results shown in Figure 9 suggest that for volatile compounds when the system was placed at a “right” surface separation the system could jump back and forth between two distinct states, i.e., with and without the emerging liquid region in the middle of the evaporating interfacial region. It may be tempting to conclude that the vapor has started condensing at the closest point of approach of the surfaces while the liquid is still evaporating at the outer region where the surface separation is slightly larger. The puzzle is that the measured refractive index of the emerging liquid region at the closest point of approach of the surfaces is apparently less than that of the bulk liquid. The physical meaning behind this phenomenon is not clear. Here we note that it may be related to the reduced refractive index of the entire bridge on the growing path of the hysteresis loop that was typically observed for the sparingly soluble compounds (Figure 4; also Figure 2 of ref 16). Unlike cyclooctane or other more volatile compounds, the refractive index of the sparingly volatile compounds apparently does not jump back and forth between these two “states”. A hysteresis loop of the sparingly volatile compounds is typically larger than that of cyclooctane, and it could be that the refractive index gap between the evaporating and the growing branches in the hysteresis loop of these compounds may be too large to be bridged by the “repeated transition”. Scenario B assumes cavitation of some sort occurring inside the liquid bridge subjected to negative pressure, which may lead to formation of a number of “nanobridges” and/or “nanocavities” between the mica walls. Figure 9 suggests that the refractive index of the emerging liquid region is lower than that of the bulk liquid. The apparently depleted refractive index suggests that, even though the liquid region in Figure 9B appears to extend over many micrometers laterally, the space cannot be totally filled with liquid. The presence of vapor cavities in the liquid next to the mica walls could account for the depleted refractive index of the liquid region and the negative Laplace pressure pulling the mica surfaces together (the pointing shape of the odd-order FECO in Figure 9B). This scenario could in principle account for other experimental observations such as reversible and highly continuous behavior of the liquid bridge, large lateral refractive index gradients for highly volatile compounds, and large hysteresis loops for the sparingly volatile compounds. However, it remains unclear as to how such a high surface energy configuration is sustainable. We here merely note that the absence of anomalous refractive index profiles at H < 2rK suggests that such “cavities” did not preexist on mica walls. Thus the cavitation must be induced by the separation of the surfaces but is not likely due to a viscous effect of the liquid. If it were induced by the viscous effect, one would expect such cavitation to occur more readily at faster slit-opening rates because the volume between the mica walls is expanding more rapidly. However, there is no evidence to suggest that. To the contrary, the Laplace instability was found to occur at a surface separation predicted by the Young-Laplace equation for liquid bridges down to submicroscopic dimensions.16,17 Scenarios C and D invoke highly complex distribution of the liquid that is consistent with the measured refractive index profiles. Evaporation usually causes cooling of the interface, and the increased local surface energy by the cooling could provide the driving force for Marangoni flows. The greater width of the interfacial region and the larger lateral refractive index gradient for the highly volatile compounds could be due to the increased rate of interfacial cooling. We note that the heating
J. Phys. Chem. B, Vol. 110, No. 51, 2006 25991 effect of the surfaces by the white light in the SFA setup is not relevant here because the illuminated area is larger than the entire mica surfaces. Marangoni convection flow induced by self-cooling due to the latent heat of vaporization studied by Buffone and Sefiane is the most relevant to our systems,26 although the scale of their system is much larger than ours and the rate of evaporation in their systems is much faster. In a macroscopic capillary tube, the rate of evaporation of the liquid is greater near the capillary wall than in the middle of the capillary, and the liquid is replenished to this region from the middle of the capillary by Marangoni flow.26 Thus, if we assume a similar mechanism applies to our submicroscopic systems, then the apparent fingering of the vapor shown in Figure 6 could result from Marangoni flow continuing to replenish the liquid to the edge of the meniscus from the middle of the pore bounded by the mica walls. Then, once the rate of material transfer via Marangoni flow approaches the rate of evaporation at the edge of the meniscus, it may be possible to develop extraordinary fingering of the vapor without much retreat of the three-phase line. The major difficulty of scenario C is that the bridge apparently has a refractive index lower than that of the bulk liquid when it first regrows at H < 2rK. The possible “refractive index jumps” for volatile compounds shown in Figure 9 suggest that the central cross-section of the evaporating interfacial region at this stage has the refractive index value of that of bulk liquid, whereas the reemerging liquid region apparently corresponds to a lower refractive index. If the refractive index jumps for volatile compounds were the result of the physical snapping of the liquid bridge and coalescence of the droplets on each mica wall, it remains unclear why the liquid bridge should snap and form two droplets on the mica walls when the width of the bridge was ≈500 times larger than its height, or why a regrowing bridge had a refractive index less than that of the liquid. A related issue is that the bridge of the sparingly volatile tetradecane and hexadecane at the first onset of regrowth had refractive index values significantly lower than that of the bulk liquid, which subsequently gradually recovered the bulk value. These observations suggest that the two droplets/films on the mica walls may have been connected by a thin liquid thread which grew or shrank laterally with the variation of H, scenario D. The complex shapes of the liquid profile (slightly positive curvature of the droplet region and highly negative curvature of the thread region) would induce pressure gradients inside the liquid domain, and concomitant counter flows from the droplet region to the thread region may feed and stabilize the thread. However, scenario D still shares some difficulty with scenario C. In addition to the results shown in Figure 9, which appears difficult to explain using any of the scenarios invoked here, the shape of the odd-order FECO during the course of regrowth at H < 2rK often appears pointing (indication of the negative Laplace pressure pulling the mica walls together) while the refractive index is apparently still below that of the bulk liquid. This is especially clear for n-hexadecane studied earlier (Figures 2I,J of ref 16). Does this mean that the liquid pressure inside the droplet region, next to the mica walls, was negative despite its slightly positive curvature? Or does this mean that the liquid distribution profile is not as simplistic as that of “two droplets connected with a thread”? Or maybe equilibrium concepts such as the pressure difference across a curved interface are not valid here because the entire process is clearly nonequilibrium. Nevertheless, we can roughly estimate the extent of the interfacial cooling by calculating the effective local P/P0 from
25992 J. Phys. Chem. B, Vol. 110, No. 51, 2006 the liquid distribution profile shown in Figure 6C if some sort of quasi-equilibrium can be assumed. Here we assume quasiequilibrium between a given small section of the film/droplet and the local vapor next to it and also assume that the pressure of the vapor (P) is uniform over the entire liquid distribution profile. Then the effective local P/P0 that corresponds to the thickness of a given section of the film/droplet was calculated from the dispersion theory,24 and the variation in the effective local P/P0 was solely attributed to the variation in the effective local P0. Then the interfacial temperature that would yield the same effective local P0 was calculated using the tabulated data of the vapor pressure of cyclooctane at different temperatures.22 The calculations show that the interfacial cooling of cyclooctane which yields the profile in Figure 6C increases from the edge of the droplet (meniscus) toward the center and is up to ≈3 °C at the thickest part of the droplet.34 This temperature differential is large, given the moderate evaporation rate of cyclooctane at the relative vapor pressures used in this study and the large interfacial area between the droplets and the mica walls which would dissipate the heat. Yet another possibility is the combined effects of different scenarios. Recent theoretical study by Li et al. showed that the critical temperature in confined spaces is always smaller than in bulk, and consequently the density of the liquid phase at a given temperature is lower in a slit-like pore than in bulk and the density of the vapor phase becomes higher in pores than in open space.27 The study also showed occurrence of layering transitions between multiple metastable states of adsorbed liquid films on the walls of the slit-like pore. It appears that both the variation in the density of fluids and the layering transitions of liquid films on pore walls occur simultaneously in a given system. Then we may need to combine more than one of the scenarios envisioned above in order to fully account for the broad range of experimental observations. We can conceive many different ways to combine multiple scenarios. For example, the cavitation effect (scenario B) may be influenced by the cooling effect of the evaporating interface (scenario C or D) and hence may not be uniform throughout the evaporating bridge: it is conceivable that cavitation may be easier near the center of the evaporating bridge, whereas it may be suppressed near the edge of the meniscus because of the cooling effects. We will leave this for future study. Conclusions Experimental evidence is mounting that strongly suggests the presence of anomalous and unexpected behavior of fluids in the vicinity of the phase boundaries. This behavior is in stark contrast to the Laplace instability of involatile liquids that can be adequately described by the Young-Laplace equation down to submicroscopic dimensions. The recent findings by Sirghi et al. indicated that a liquid bridge should be mechanically stable when stretched rapidly (i.e., the rate of evaporation can be neglected) until the bridge width becomes comparable to the bridge height even for bridges as small as 10-23 m3 (that correspond to the height/width of the order of 20 nm). What we observed in the vicinity of the phase boundary is qualitatively different. Even in the simplest systems such as those presented here (one-component vapor in a single slit-like pore of molecularly smooth and chemically inert walls), the behavior of confined fluids in the vicinity of the phase boundaries turned out to be highly complex. It appears that two distinct modes of disappearance of a liquid bridge exist, depending on the rate of evaporation of the liquid with respect to the rate of change of
Maeda the external conditions such as surface separation. The first mode is based on the Laplace instability, and it can be adequately described by the Young-Laplace equation down to submicroscopic scales. We are yet to have a clear picture for the second mode, but we hope the newly found results will eventually help in the elucidation of the observed phenomena. This second mode is expected to be important when porous media are filled or emptied by gradual variation of the external vapor pressure. Therefore the remaining outstanding questions ought to be of considerable scientific merit. Acknowledgment. The author gracefully acknowledges helpful discussions with Prof. Jacob Israelachvili and Dr. Mika Kohonen. This work was supported by CRC Smartprint. References and Notes (1) Maxwell, J. C. Encycl Britannica 1875, 56-71. (2) Gelb, L. D.; Gubbins, K. E.; Radhakrishnan, R.; SliwinskaBartkowiak, M. Rep. Prog. Phys. 1999, 62, 1573. (3) Adamson, A. W.; Gast, A. P. Physical Chemistry of Surfaces, John Wiley & Sons. Inc.: New York, 1997. (4) Defay, R.; Prigogine, I.; Bellemans, A.; Everett, D. H. Surface Tension and Adsorption; Longmans Green: London, 1966. (5) Thomson, W. Proc. R. Soc. Edinburgh 1870, 7, 63. (6) Blackman, M.; Lisgarte, Nd; Skinner, L. M. Nature 1968, 217, 1245. (7) Fisher, L. R.; Israelachvili, J. N. J. Colloid Interface Sci. 1981, 80, 528. (8) Kohonen, M. M.; Christenson, H. K. Langmuir 2000, 16, 7285. (9) Everett, D. H.; Haynes, J. M. J. Colloid Interface Sci. 1972, 38, 125. (10) Derjaguin, B. V.; Churaev, N. V. J. Colloid Interface Sci. 1976, 54, 157. (11) Christenson, H. K. Phys. ReV. Lett. 1994, 73, 1821. (12) Curry, J. E.; Christenson, H. K. Langmuir 1996, 12, 5729. (13) Forcada, M. L. J. Chem. Phys. 1993, 98, 638. (14) Restagno, F.; Bocquet, L.; Biben, T. Phys. ReV. Lett. 2000, 84, 2433. (15) Willett, C. D.; Adams, M. J.; Johnson, S. A.; Seville, J. P. K. Langmuir 2000, 16, 9396. (16) Maeda, N.; Israelachvili, J. N.; Kohonen, M. M. Proc. Natl. Acad. Sci. U.S.A. 2003, 100, 803. (17) Sirghi, L.; Szoszkiewicz, R.; Riedo, E. Langmuir 2006, 22, 1093. (18) Maeda, N.; Israelachvili, J. N. J. Phys. Chem. B 2002, 106, 3534. (19) Tolansky, S. Multiple-Beam Interferometry of Surfaces and Films; Oxford University Press: London, 1949. (20) Israelachvili, J. N. J. Colloid Interface Sci. 1973, 44, 259. (21) Briscoe, W. H.; Horn, R. G. J. Opt. A: Pure Appl. Opt. 2004, 6, 112. (22) CRC Handbook of Chemistry and Physics; CRC Press: Boca Raton, FL, 1999-2000. (23) Bradley, R. S.; Shellard, A. D. Proc. R. Soc. London, Ser. A 1949, 198, 239. (24) Mahanty, J.; Ninham, B. W. Dispersion Forces; Academic Press: London, 1976. (25) Born, M.; Wolf, E. Principles of Optics; University Press: Cambridge, U.K., 1999. (26) Buffone, C.; Sefiane, K. Int. J. Multiphase Flow 2004, 30, 1071. (27) Li, Z. D.; Cao, D. P.; Wu, J. Z. J. Chem. Phys. 2005, 122. (28) Levien, B. J.; Mills, R. Aust. J. Chem. 1980, 33, 1977. (29) A nucleation process is not required for wedge-like pores, and the condensation hysteresis in these systems is believed to arise, in part, from the contact angle hysteresis of the condensed phase on the pore walls. (30) Even though we vary the separation of the confining pore walls instead of the relative vapor pressure of the vapor under study, it is assumed to be equivalent to varying the relative vapor pressure around a pore of fixed geometry. Thus, this sort of situation arises when porous media are filled or emptied by gradual variation of the external vapor pressure. (31) The term “thread” may be misleading given that its lateral width can be much larger than its length. Nonetheless we refer to this geometry as thread in this paper so as to distinguish it from a “bridge”, which we refer to as the entire fluid entity between the two mica walls. (32) Because of the small contact angles, θ, of the liquids used in this study (
