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Thermodynamic and Kinetic Supercooling of Liquid in a Wedge

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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. 0021-9606/2008/129͑15͒/154509/7/$23.00, 154509-1 © 2008 American Institute of Physics $XWKRUFRPSOLPHQWDU\FRS\5HGLVWULEXWLRQVXEMHFWWR$,3OLFHQVHRUFRS\ULJKWVHHKWWSMFSDLSRUJMFSFRS\ULJKWMVS 154509-2 Nowak et al. J. Chem. Phys. , 154509 ͑2008͒ 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 .
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