Thermodynamic and Kinetic Supercooling of Liquid in a Wedge

THE JOURNAL OF CHEMICAL PHYSICS 129, 154509 ͑2008͒

7KHUPRG\QDPLF DQG NLQHWLF VXSHUFRROLQJ RI OLTXLG LQ D ZHGJH SRUH ͒ Dominika Nowak,1 Manfred Heuberger,2,3 Michael Zäch,2,4 and Hugo K. Christenson1,a 1School of Physics and Astronomy, University of Leeds, Leeds LS2 9JT, United Kingdom 2Laboratory for Surface Science and Technology, Department of Materials, Swiss Federal Institute of Technology, ETH, CH-8093 Zürich, Switzerland 3Empa Advanced Fibers, Swiss Federal Laboratories for Materials Testing and Research, CH-9014 St. Gallen, Switzerland 4Department of Applied Physics, Chalmers University of Technology, S-412 96 Göteborg, Sweden ͑Received 1 July 2008; accepted 15 September 2008; published online 21 October 2008͒

Cyclohexane allowed to capillary condense from vapor in an annular wedge pore of mica in a ͑ ͒ surface force apparatus SFA remains liquid down to at least 14 K below the bulk melting-point Tm. This is an example of supercooling of a liquid due to conﬁnement, like melting-point depression in porous media. In the wedge pore, however, the supercooled liquid is in equilibrium with vapor, and the amount of liquid ͑and thereby the radius of curvature r of the liquid-vapor interface͒ depends on ␥ the surface tension LV of the liquid, not the interfacial tension between the solid and liquid. At ⌬ coexistence r is inversely proportional to the temperature depression T below Tm, in accordance with a recently proposed model ͓P. Barber, T. Asakawa, and H. K. Christenson, J. Phys. Chem. C 111, 2141 ͑2007͔͒. We have now extended this model to include effects due to the temperature dependence of both the surface tension and the enthalpy of melting. The predictions of the improved model have been quantitatively veriﬁed in experiments using both a Mark IV SFA and an extended surface force apparatus ͑eSFA͒. The three-layer interferometer formed by the two opposing, backsilvered mica surfaces in a SFA was analyzed by conventional means ͑Mark IV͒ and by fast spectral correlation of up to 40 fringes ͑eSFA͒. We discuss the absence of freezing in the outermost ͑ ͒ region of the wedge pore down to 14 K below Tm and attribute it to nonequilibrium kinetic supercooling, whereas the inner region of the condensate is thermodynamically supercooled. © 2008 American Institute of Physics. ͓DOI: 10.1063/1.2996293͔

, ,1752'8&7,21 single pore, either an annular wedge pore or a slit pore of variable slit width. The wedge pore is formed when two Surface effects on the phase behavior of matter are com- ͑ Ϸ ͒ mon in nature and occur throughout the industrial and tech- curved radius of curvature R 2cm mica surfaces are in nological spheres. The flow of granular media in humid at- flattened contact and the slit pore when the surfaces are kept mospheres is impeded by capillary condensation of water apart at small separations. between the grains,1 and frost heave is ultimately due to the The single-pore experiments have mainly examined con- migration of unfrozen water in subzero soils.2 Sintering of ditions in a wedge pore, which differs fundamentally from a metallic and ceramic powders is caused by a reduced melting slit pore or a cylindrical pore. Equilibrium between phases 3 temperature at the contact points of the particles, and the occurs over a range of vapor pressures for a fixed tempera- capillary condensation of dissolved water from oils influ- ture or for varying temperatures at a fixed vapor pressure ences the properties of greases with thickeners consisting of because a wedge pore can accommodate any radius of cur- 4 hydrophilic particles. vature of the interface. In a uniform cylindrical pore and in a Most of our empirical knowledge of surface effects on uniform slit pore, however, equilibrium occurs at a single phase behavior5,6 comes from experiments on the extended value of the pressure for each temperature, and vice versa. networks of interconnected pores that constitute porous me- Furthermore, a wedge pore also shows less hysteresis than a dia, such as silica gels, porous glasses such as Vycor, and activated carbon fibers. Recently, materials with isolated cy- slit pore or a cylindrical pore. There is no hysteresis of cap- lindrical pores with pores sizes varying from nanometers illary condensation as any interfacial radius of curvature ͑ ͒ ͓e.g., MCM-41 ͑Ref. 7͔͒ to hundreds of nanometers ͑etched from zero to effectively infinite for a macroscopic wedge alumina8͒ have become available. However, only a few types may be accommodated, and there is hence no nucleation bar- of experiment have examined conditions in a single pore, for rier. Only cylindrical pores with a conical taper at the closed example, early work on melting-point depression in a end would have this property, and in general both open- wedge,9 studies of capillary condensation in wedges10 and, in ended as well as closed cylindrical pores give rise to hyster- particular, a number of experiments using the surface force esis of capillary condensation.12 Liquid-solid equilibria, 11 apparatus ͑SFA͒. This instrument allows the study of a however, are subject to hysteresis in all pores as nucleation of solid is associated with a considerable energy barrier, ͒ a Electronic mail: h [email protected]. whether it occurs at the pore wall or in the bulk of the pore.

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As a result, freezing-melting hysteresis in interconnected, 1 ⌬H ⌬T = fus , ͑4͒ porous networks is a complex phenomenon that is not always ␥ r 2VML SLTm completely understood.13 The first capillary-condensation experiments with the except that the interfacial tension between solid and liquid ␥ ␥ SFA verified the Kelvin equation above the bulk melting SL is replaced by LV and that the factor 2 in the denomi- 14 temperature Tm of the liquid for radii of curvature of the nator is absent. liquid-vapor interface down to 3–4 nm. In a more recent Unfortunately, the number of substances that may be series of experiments Christenson and co-workers15–19 fo- used for accurate measurements of the radius of curvature of cused on capillary condensation below Tm using alcohols and capillary condensates below the melting point in a SFA is hydrocarbons, chosen for having melting points at or just severely restricted. Suitable substances must be inert so as above room temperature. The results show that liquid capil- not to dissolve the glue holding the mica surfaces to their lary condensates invariably form between mica surfaces for supports, the vapor pressure must be high enough that equi- ͑ ͒ ⌬ ͑ moderate 10–15 K temperature depressions T below Tm librium can be achieved in a reasonable time hours rather and that their size, i.e., the wedge width at the liquid-vapor than days or longer͒, and the melting point must be in a ϳ interface, is inversely proportional to the temperature depres- range that permits measurements down to 10 °C below Tm. ⌬ 19 sion T below Tm. In a few cases solid has been observed to The temperature-control system used in the previous study ͑ form either after the wedge pore is turned into a slit pore by could only take T down to 7 °C, so cyclo-octane Tm separating the mica surfaces from contact17 or as a result =14 °C͒ could be studied over a reasonable temperature of nucleation in the wedge pore at sufficiently low range. However, the saturated vapor pressure ps of cyclo- 18 ͑ ͒ 21 temperatures. octane is of the order of only 3 mm Hg 410 Pa at Tm, The capillary condensation of cyclo-octane19 has been leading to equilibration times of many days near coexistence. studied both in a mica wedge pore ͑where the contact angle ␪ Consequently, measurements could only be carried out of the cyclo-octane is small͒ and in a wedge pore of mica slightly off coexistence and it was often impossible to be coated with a fluorocarbon surfactant monolayer ͑where ␪ certain that equilibrium had been achieved, limiting experi- Ϸ60°͒. It was found that the size of the condensates de- mental accuracy and the certainty of the conclusions. pended on ␪ while the radius of curvature of the liquid-vapor We have now managed to improve greatly the experi- interface was independent of ␪, leading one of us to suggest mental conditions so that we have been able to study cyclo- ͓ ͑ ͔͒ that capillary condensation below Tm may be explained hexane ps =40 mm Hg or 5300 Pa at T=Tm Ref. 22 down quantitatively by combining the Kelvin equation with the to 14 °C below the melting point, which is 6.6 °C. To do Clausius–Clapeyron equation to obtain the vapor pressure this, better temperature control has been achieved by carry- over the curved interface of “supercooled,” capillary-held ing out measurements with an extended surface forces appa- 23–25 liquid. This was based on a model put forward in a 1944 ratus ͑eSFA͒ at ETH in Zurich and by improving the paper by Batchelor and Foster20 and yielded a relationship Mark IV SFA system at Leeds. In addition to providing ac- between the magnitude of the radius of curvature, r,ofthe curate verification of the model, these experiments have al- liquid-vapor interface and the temperature depression ⌬T be- lowed us to make comparisons between results obtained by / 26 ͑ low Tm for different relative vapor pressures p ps, application of conventional three-layer interferometry the SFA Mark IV͒ and fast spectral correlation23 ͑the eSFA͒. 1 ⌬T⌬H + RTT ln͑p /p͒ = fus m s . ͑1͒ ␥ ,, (;3(5,0(17$/ r VML LVTm The experiments at Leeds were carried out with a sim- ⌬ Here Hfus is the enthalpy of fusion, VML the molar volume, plified version of the Mark IV SFA ͑Ref. 11͒ equipped with a ␥ ͓ and LV the surface tension of the liquid. This equation is a large-volume ͑ϳ1l͒ chamber. The experimental procedure is simplification of Eq. ͑11͒ in Ref. 19, obtained by replacing given in detail in Ref. 19. Cleaved mica sheets ͑thickness of ⌬ ͔ ⌬ ͑ ͒ Tm in the last term by T+ T. For T=0, Eq. 1 reduces to 2–3 ␮m͒ that had been cut with a hot platinum wire were ͑ the Kelvin equation at Tm where r is the magnitude of the backsilvered ͑ϳ50 nm͒ by vacuum evaporation and glued to negative radius of curvature͒, silica disks ͑radius of curvature R=2 cm͒ with an epoxy resin ͑Epikote 1004͒ and the disks then mounted as crossed 1 RT p = m lnͩ s ͪ, ͑2͒ cylinders, with one surface on a rigid support to minimize ␥ r VML LV p spring deflection. After measurement of mica-mica contact in dry nitrogen a large excess ͑6ml͒ of liquid cyclohexane ͑Al- ͑ ͒ and at saturation where p=ps, Eq. 1 reduces to drich, 99.9+%͒ was injected into a dish on the bottom of the chamber at 16 °C, and the system was then allowed to reach ⌬ ⌬ 1 Hfus T equilibrium. The white light beam that generates the interfer- = . ͑3͒ ␥ r VML LVTm ence fringes did not pass through the cyclohexane reservoir. The equilibrium size of the annular capillary condensate This relationship is reminiscent of the Gibbs–Thomson that forms around the flattened contact zone of the surfaces equation5 for the melting-point depression in a cylindrical was measured by recording the interference fringes with a pore of radius r ͑which is identical to that of a wedge pore if charge coupled device ͑CCD͒ camera and analyzing the the width of the wedge at the liquid-solid interface is 2r͒, fringe shifts using the three-layer interferometry equations.26

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FIG. 1. The crossed-cylinder configuration of the mica surfaces is equiva- lent to a sphere on a flat. The adhesion due to the Laplace pressure in the capillary condensate leads to extensive surface flattening in contact. R is the mean radius of curvature of the undeformed surfaces and r1 and r2 are the principal radii of curvature of the interface of the annular capillary condensate. ␪ is the contact angle of the liquid on the mica surfaces and t is the thickness of adsorbed films on the surface. Note that t is expected to increase in proximity to the capillary condensate ͑see text͒.

͑ At each temperature in the range of −4.2 to 5.8 °C the sur- FIG. 2. Representative cross-sectional profiles of surface separation circles, left axis͒ and refractive index ͑triangles, right axis͒ in an annular cyclohex- faces were brought slowly into contact to note the presence ane condensate at T=−7.2 °C as measured using the eSFA. The lines be- of an inward jump due to capillary condensation. The sur- tween data points have no physical meaning and are plotted to guide the eye faces flattened due to the negative Laplace pressure in the only. Error bars ͑plotted only if larger than the symbol size͒ were calculated ͑ ͒ ͑ ͒ capillary condensate ͑see Fig. 1͒ and were left in contact according to Eqs. 8a and 8b in Ref. 23. The condensate size r was derived from the surface separation h at a refractive index of n=1.212 until the size of the condensate as determined from the inter- ͑dotted line͒, i.e., in the middle of the transition from liquid cyclohexane ference fringes no longer increased ͑up to 2.5 h for the ͑dashed line͒ to air ͑nϷ1͒. Note that the refractive-index transition is higher temperatures͒. The total radius of curvature, r,ofthe broader than expected due to various experimental artifacts, including dif- fractionlike effects occurring at the interface between the capillary conden- interface of the condensate is given by sate and the surrounding vapor and a small rotational misalignment of the CCD camera with respect to the focal plane of the spectrometer. Since fast 1 1 1 1 ͑⌬␭Ͼ = + Ϸ , ͑5͒ spectral correlation detects fringes in a large spectral range 225 nm, see Ref. 23͒, a tiny rotational misalignment of Ͻ0.1° is estimated to smear r r1 r2 r1 out the transition over several micrometers. where r1 is the curvature in a plane perpendicular to the two mica surfaces and r is the radius of the annulus of con- 2 using fast spectral correlation.23 In subsequent measurements densed liquid. Typically, r1 is 10–100 nm and r2 is about ␮ / carried out at temperatures T different from these reference 20 m, so the error introduced by neglecting 1 r2 is unimportant. The contact angle ␪ of cyclohexane on mica is about temperatures, the apparent variation in mica thickness Dm 6°27 and r is hence to a very good approximation equal to due to thermal expansion/contraction was accounted for by ͑ ͒ half the surface separation h at the perimeter of the conden- extracting the respective mica thickness Dm T from a linear sate ͑see Fig. 1͒. By extrapolating the fringes from outside fit to the three reference measurements. Additionally, we and inside the condensate to the middle of the gap, two sepa- made control measurements of the refractive index of nitro- rate values of h were obtained. Corrections due to the ther- gen and cyclohexane as a function of surface separation be- ͑ mal expansion of mica were taken from previous work.19 fore and after injecting cyclohexane p.a. grade, Fluka ͒ Theoretically, the proximity of the condensate is expected to Chemicals into the eSFA chamber, respectively. Condensate increase the thickness of the adsorbed film to one-and-a-half sizes were determined at various temperatures after the sur- 28,29 ͑ times that found at an isolated surface. A correction for faces had been brought together slowly typically this was made by subtracting three times the film thickness 0.1–0.2 nm/s͒ to the point where an inward jump marked t measured at a single surface ͑with the eSFA, see below͒ the onset of capillary condensation and after the condensate from h, giving the radius of curvature of the condensate as size had equilibrated. To this end, we used the scanning abil- r=͑h−3t͒/2. The difference between the r values obtained ity of the eSFA ͑Ref. 23͒ to acquire surface separation and from the two measurements of h is the experimental error in refractive index profiles across the flattened contact zone and each measurement. After separation the surfaces were left the condensate ͑see Fig. 2͒, where a transition of the refrac- apart to allow the condensate to evaporate; the temperature tive index from Ϸ1 to 1.424 ͑i.e., the bulk refractive index of was then changed and the system left to equilibrate overnight liquid cyclohexane͒ marks the periphery of the condensate. at the new temperature. The condensate size h was measured in the middle of this The work at Zurich was performed with an eSFA whose transition region where n=͑1.424+1͒/2. chamber has a volume of about 75 cm3. A mixture of glucose The adsorption of cyclohexane to the mica surfaces was and galactose ͑1:1͒ heated to 144 °C on a hotplate was used determined in the absence of a meniscus, with the surfaces to glue mechanically ͑with scissors͒ cut and backsilvered separated by DϽ100 nm and by fitting the transmission mica sheets of 2–4 ␮m thickness onto silica supports with spectrum of a five-layer interferometer ͑mica-adsorbate-gas- R=2 cm, which were then mounted in the eSFA. After the adsorbate-mica͒ to the experimentally determined fringe po- instrument had been purged with nitrogen for several hours, sitions using a fast spectral correlation algorithm,23 where the mica thickness was determined accurately in mica-mica the ͑total͒ surface separation and the total adsorbed film contact at three different temperatures ͑22, 12, and 2 °C͒ thickness, 2t, were fitting parameters. The mica thickness as

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FIG. 4. Interference fringes observed with capillary condensate of cyclohex- aneat+2.9°C͑⌬T=3.7 K͒, as measured in the Mark IV SFA. The wave- length increases to the left. The vertical lines are the Hg yellow doublet and the Hg green line used for calibration. Each fringe is a doublet due to the birefringence of mica. At the location of the interface between the capillary condensate and the surrounding vapor there are discontinuities in the fringe patterns. FIG. 3. Film thickness of cyclohexane adsorbed on an isolated mica surface as a function of temperature, as measured in the eSFA. Error bars in t correspond to an estimated error of Ϯ0.2 nm. temperatures above Tm, the range and magnitude of the sol- vation forces between two surfaces should be insensitive to 34 well as the refractive indices of the cyclohexane and nitrogen small temperature changes. gas layers were fixed as obtained by the control measure- Figure 4 shows the appearance of the interference ments described above. fringes when an annular condensate of cyclohexane has reached its equilibrium size, after 40 min at 2.9 °C, or 3.7 K ͑ ͒ / ,,, 5(68/76 below Tm Mark IV results . The inverse radius 1 r has been plotted as a function of ⌬T in Fig. 5. The error bars denote The adsorbed film thickness as a function of the tem- the difference between the two values of h and thereby r. perature depression ⌬T is shown in Fig. 3 ͑results with the The importance of correcting for the adsorbed layer thick- eSFA͒, and as may be seen the film thickness appears to ness is evident from a comparison of the uncorrected and Ϯ ͑⌬ Ͻ ͒ decrease from 2.3 0.3 nm above Tm T 0 to corrected data points. Two separate series of measurements Ϯ 1.2 0.3 nm at temperatures below Tm. Since the mean layer thickness of cyclohexane confined between two surfaces is 0.55–0.6 nm,30 this suggests a decrease from a thickness corresponding to four molecular diameters to one of only two. A decrease in adsorbed film thickness of about two mean molecular diameters at the bulk melting point has been measured previously with tert-butanol31 and cyclo-octane32 and is consistent with the expected transition in wetting at the bulk melting or triple point. Although the film thick- / Ϸ nesses were determined for p pS 0.99, it is likely that the film thickness is rather insensitive to small changes in relative vapor pressure near coexistence, as found for tert-butanol31 and cyclo-octane.32 Theoretically, any dra- / matic dependence of wetting on p pS is unlikely below the melting point.33 On approach the mica surfaces are pulled into contact from surface separations in the range of 12–20 nm. Al- though this was not measured accurately in these experiments, the presence of a clear inward jump that did not change upon repeat measurements was taken as an indication FIG. 5. Inverse radius of curvature of cyclohexane condensates 1/r as a that no accumulation of contamination in the capillary con- function of the temperature depression ⌬T. Filled symbols denote conden- densates occurred. Moreover, the values are close to what sate sizes, which were corrected for the adsorbed film thickness t,asmea- ͑ ͒ has been found previously for tert-butanol below its melting sured in two independent runs with the Mark IV SFA triangles and circles 31 and one run with the eSFA ͑squares͒. Open symbols correspond to the re- point. spective uncorrected data. The solid lines are the predictions of Eq. ͑6͒ for / ͑ ͒ / ͑ ͒ After the inward jump the surfaces came into contact p ps =1.000 lower line and p ps =0.991 upper line , representing the ex- separated by two to three statistical layers of cyclohexane perimental conditions with the Mark IV and the eSFA, respectively. For / comparison, the dashed lines are the theoretical predictions for p ps molecules, as found in previous studies at higher tempera- / =1.000 and p ps =0.991 if neither the temperature dependence of the sur- tures. This is in agreement with the expectations that while face tension nor the temperature dependence of the enthalpy of fusion are ͓ ͑ ͒ ␥ ␥ ͑ ͔͒ adsorption from vapor at a single surface should increase for taken into account Eq. 1 , with LV =const= LV T=Tm .

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The lateral extension of this high- condensate radius determined as described in Sec. II, and index zone indicates that it persists to a surface separation of these values are also plotted in Fig. 5. some 10 nm. The magnitude of the refractive index measured here ͑1.45–1.50͒ is close to that expected for solid ,9 ',6&866,21 cyclohexane, while the flow of the condensates as the contact diameter is changed shows clearly that they are liquid. In previous work with cyclo-octane the experimental re- 37 ͑ ͒ The mica used in the eSFA was mechanically cut, sults were compared with the predictions of Eq. 1 , without whereas the mica in the Mark IV had been melt cut with a correcting for the temperature dependence of the surface ten- white hot Pt wire. From a comparison of the results we may sion or the enthalpy of fusion. In view of the much more conclude ͑not unexpectedly͒ that the likely presence of nano- accurate data obtained with cyclohexane we have derived a 38,39 ͑ ͒ particles of Pt on the melt-cut mica sheets has no influ- correction to Eq. 1 by considering the heat-capacity correc- ence on the equilibrium size of capillary condensates. tion to the Clausius–Clapeyron equation. The derivation is Although we did not observe freezing of the cyclohex- given in the Appendix, and the result is ane condensates in the mica wedge, it is interesting to specu- T T p late on the thermodynamic and kinetic factors that would ⌬Tͩ⌬H − T ⌬Cͩ1+ lnͩ ͪͪͪ + RTT lnͩ s ͪ fus m ⌬ m lead to the possible coexistence of solid and liquid in the 1 T Tm p = , wedge, as found in studies of melting-point depression in a r V ␥ ͑T͒T ML LV m wedge.9 The Gibbs–Thomson equation for the annular ͑6͒ wedge ͓Eq. ͑4͔͒ gives the thermodynamic condition for the where ⌬C=C −C , the difference in the heat capacity be- coexistence of liquid and solid in a wedge. If the condensing L S ␪ ͑ tween liquid and solid cyclohexane. ␥ has been replaced liquid has a small contact angle on the pore wall as for LV ␪Ϸ ͒ with ␥ ͑T͒ to emphasize the inclusion of the temperature cyclohexane, so that cos 1 the width h of the liquid- LV ⌬ ␥ dependence of the surface tension, obtained by extrapolation vapor interface for a given T is proportional to LV accord- ␥ ing to Eq. ͑3͒, and if the contact angle ␾ of the liquid against of data from Ref. 36, according to which LV increases from 24.2 mN m−1 at 25 °C to 27.9 mN m−1 at −7.2 °C. its solid phase on the pore wall is small, the width h of the ␥ The solid lines in Fig. 5 are theoretical expectations from solid-liquid interface is proportional to 2 SL according to Eq. ͑ ͒ ␥ Ͼ ␥ Eq. ͑6͒, with the relative vapor pressure estimated from the 4 . Hence, if LV 2 SL, a capillary condensate below Tm is metastable towards freezing in the outer part of the wedge Kelvin equation and the measured condensate size above Tm. The enthalpy and surface tension corrections at ⌬T=13 K where are about 3% and 6%, respectively, with both increasing the 4V ␥ T predicted r values relative to that of the more approximate h Ͼ ML SL m . ͑7͒ ⌬ ⌬ Eq. ͑1͒. These corrections decrease the slope of the theoret- Hfus T ical line, whereas the correction for the adsorbed layer thick- In porous media studies it is often assumed that freezing ness, which is of a similar magnitude, shifts the point up- via homogeneous nucleation may occur when a pore of di- wards. ameter h can accommodate a minimum-size spherical The greater experimental accuracy obtained with cyclo- nucleus given by5,13 hexane compared to previous measurements with cyclo- octane and the corrections for temperature dependence allow 6V ␥ T h = ML SL m . ͑8͒ us to show convincingly that the curvature of the liquid- ⌬ ⌬ Hfus T vapor interface below the melting point is determined by Eq. ͑6͒. The results also highlight how the equilibrium between This is not the size of the critical nucleus, for which the free vapor and liquid in a wedge pore leads to complete revers- energy of formation has a maximum, but the size at which ibility with increased condensation occurring on increase in the free energy of the phase change first becomes negative the temperature towards Tm and evaporation on decrease in and actual freezing occurs. Equation ͑8͒ is the same for the the temperature. annular mica wedge, except that the “pore” diameter is now Parallel measurements with both an established experi- the width at the liquid-vapor interface, which greatly restricts mental setup in Leeds and the recently developed eSFA setup the volume available for nucleus formation. Heterogeneous in Zurich have allowed us to compare results obtained at nucleation of solid from liquid by the crystalline mica sur- different locations and using different methodologies to ac- face would appear to be extremely unlikely. Freezing via quire and evaluate multiple-beam interference spectra. From homogeneous nucleation thus appears possible for liquids ␥ Ͼ ␥ the good agreement between the Leeds and Zurich data we which have LV 3 SL. This certainly includes cyclohexane, ␥ −2 40,41 conclude that multiple-beam interferometry of capillary con- for which estimates of SL are about 5 mJ m . The fact densates can be carried out equally well using traditional that we did not observe freezing suggests that the rate of

$XWKRUFRPSOLPHQWDU\FRS\5HGLVWULEXWLRQVXEMHFWWR$,3OLFHQVHRUFRS\ULJKWVHHKWWSMFSDLSRUJMFSFRS\ULJKWMVS 154509-6 Nowak et al. J. Chem. Phys. , 154509 ͑2008͒ homogeneous nucleation in the mica wedge is prohibitively ing point, to pL, the saturated vapor pressure at a temperature slow. This is not surprising given that the volume of the T, gives condensates in this study ͓given approximately by V=␲Rh2 p ⌬H ͑T ͒ − T ͑C − C ͒ 1 1 ͑Ref. 17͔͒ are of the order 10−10–10−8 cm3 or close to the lnͩ L ͪ =−ͩ vap m m G L ͪͩ − ͪ volume of 10 ␮m droplets. pm R T Tm The contact angle of most liquids on mica ͑and other ͑ ͒ CG − CL T high-energy surfaces͒ are close to zero, and most liquids + lnͩ ͪ. ͑A3͒ R T would be expected to wet most substrates in the presence of m their own solid, so the above should be of general applica- Similarly, for the saturated vapor pressure over solid pS, bility. Furthermore, for most substances one expects ␥ LV ⌬ ͑ ͒ ͑ ͒ Ͼ ␥ ͑ ͒ pS Hsub Tm − Tm CG − CS 1 1 3 SL, with water a possible exception, since at Tm 0°C lnͩ ͪ =−ͩ ͪͩ − ͪ ␥ −1 ␥ LV=76 mNm , and reported SL values range from pm R T Tm 42 26 to 35 mN m−1. ͑ ͒ CG − CS T Most liquids condensed in a mica wedge below T thus + lnͩ ͪ. ͑A4͒ m R T show both thermodynamic or equilibrium supercooling due m ⌬ ͑ ͒ to the surface free energy terms as well as kinetic or non- Here Hsub Tm is the enthalpy of sublimation at Tm and CS equilibrium supercooling due to a nucleation barrier and/or the heat capacity of the solid. Subtraction of Eq. ͑A3͒ from restrictions on the available volume. Nevertheless, solid has Eq. ͑A4͒ gives been found to nucleate, whether from liquid or directly from 15,18 ͑⌬ ⌬ ͒ ͑ ͒ vapor, in two cases, neo-pentanol and tetrabromo- ͩ pL ͪ Hsub − Hvap − Tm CL − CS ͩ 1 1 ͪ 43 ln =− − methane. It is to be hoped that future experiments will help pS R Tm T explain more completely the freezing or absence thereof of ͑C − C ͒ T liquids in wedge pores. − L S lnͩ ͪ R Tm $&.12:/('*0(176 ⌬H − T ⌬C 1 1 ⌬C T =− fus m ͩ − ͪ − lnͩ ͪ, The work carried out at Leeds University was funded by R Tm T R Tm EPSRC Grant No. GR/R55450/01. The work in Zürich was ͑A5͒ funded by the Swiss National Science Foundation ͑SNSF͒. ⌬ ⌬ ͑ ͒ ⌬ ͑ ͒ ⌬ ͑ ͒ We would like to thank N. D. Spencer for scientiﬁc and where Hfus= Hfus Tm = Hsub Tm − Hvap Tm is the la- ⌬ ﬁnancial support and J. Vanicek ͑Zurich͒ and S. Weston and tent heat of fusion at the melting point, and C=CL −CS. A. Price ͑Leeds͒ for technical help with the temperature con- Since these are condensed phases, any difference between Cv trol. and Cp is unimportant. The vapor pressure p over a curved ͑ surface at temperature T can be related to pL the saturated $33(1',; (17+$/3< &255(&7,21 72 7+( vapor pressure over a planar surface͒ at T with the Kelvin ͓ ͑ ͔͒ &21'(16$7( &859$785( %(/2: Tm equation Eq. 2 , The derivation below extends that presented in Ref. 19 p V ␥ lnͩ L ͪ =− ML LV , ͑A6͒ by including the heat-capacity difference between the phases. p r RT ⌬ ͑ ͒ L The enthalpy of vaporization Hvap T as a function of temperature is given by where VML is the molar volume of the condensing liquid and r is the curvature of the liquid-vapor interface ͑negative for ⌬ ͑ ͒ ⌬ ͑ ͒ ͑ ͒ ͑ ͒ L Hvap T = Hvap Tm − CG Tm − T + CL Tm − T a wetting liquid͒. For a liquid capillary condensate in equi- ⌬ ͑ ͒ ͑ ͒ ͑ ͒ = Hvap Tm − Tm CG − CL + T CG − CL , librium with vapor at a pressure p we can write ͑A1͒ p p p V ␥ p lnͩ L ͪ =lnͩ L ͪ −lnͩ S ͪ =− ML LV −lnͩ S ͪ, ͑A7͒ ⌬ ͑ ͒ p p p r RT p where Hvap Tm is the enthalpy of vaporization at the S L melting-point T , and C and C are the heat capacities of m G L which with Eq. ͑A6͒ gives the vapor ͑gas͒ and liquid, respectively ͑assumed to be inde- ͒ ⌬ ⌬ ⌬ pendent of temperature . This is used to derive a heat- Hfus − Tm Cͩ 1 1 ͪ C ͩ T ͪ capacity correction to the Clausius–Clapeyron equation for − − − ln R Tm T R Tm vapor pressure over a liquid below T , m ␥ VML LV pS pL T ⌬ =− −lnͩ ͪ. ͑A8͒ dp 1 HvapdT ͵ = ͵ rLRT p p R T2 pm Tm ⌬ ͉ ͉͑ Introducing T=Tm −T and r= rL since the net curvature is T ⌬ ͑ ͒ ͑ ͒ ͑ ͒ 1 Hvap Tm − Tm CG − CL + T CG − CL negative͒ gives = ͵ dT. R T2 Tm ͑⌬ ⌬ ͒⌬ ⌬ ␥ Hfus − Tm C T C ͩ T ͪ VML LV ͩ pS ͪ ͑A2͒ − ln = −ln , RTTm R Tm rRT p ͑ ͒ Integration from pm, the saturated vapor pressure at the melt- A9

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