LETTER pubs.acs.org/JPCL
Evaporation Length Scales of Confined Water and Some Common Organic Liquids Claudio A. Cerdeiri~na,†,‡ Pablo G. Debenedetti,*,† Peter J. Rossky,§ and Nicolas Giovambattista|| †
Department of Chemical and Biological Engineering, Princeton University, Princeton, New Jersey 08544, United States Departamento de Física Aplicada, Universidad de Vigo, Ourense 32004, Spain § Department of Chemistry and Biochemistry, University of Texas, Austin, Texas 78712, United States Department of Physics, Brooklyn College of the City University of New York, Brooklyn, New York 11210, United States
)
‡
ABSTRACT: When two macroscopic and repulsive surfaces are immersed in water, evaporation of the confined liquid is favored thermodynamically below a critical separation: the evaporation length scale. We use thermophysical property data to evaluate the evaporation length scale of water, and compare it to that of several common organic liquids over a broad range of temperatures, at atmospheric pressure. We show that water’s evaporation length scale is of the order of 1 μm, appreciably larger than generally thought. The evaporation length scale of several common organic liquids, although systematically smaller than water’s, is likewise macroscopic, attesting to the generality of the phenomenon. The only physical property that causes water’s evaporation length scale to be larger than that of other liquids is its surface tension. In the limit of small immersed surfaces, the evaporation length is proportional to the size of the immersed object, and does not depend on the confined liquid. SECTION: Statistical Mechanics, Thermodynamics, Medium Effects
C
onfined liquids are important in biology, geology, materials science, and numerous areas of engineering practice. Examples include catalysis and separations involving zeolites and clays,1 mineral inclusions,2 ion channels,3 the cell’s crowded interior,4 nanofluidics,5 and water transport in plants.6 The properties of water near hydrophobic surfaces7,8 have long elicited interest on account of their relationship to biological self-assembly and their role in applications such as the design of self-cleaning materials9 and anti-ice coatings.10 Kauzmann first pointed out that the watermediated tendency for hydrophobic units to aggregate is critical for protein folding and conformational stability.11 Subsequently Stillinger12 argued that the interface between liquid water and a sufficiently large hydrophobic object (∼1 nm at ambient conditions) should resemble a vaporliquid interface in the limit in which the interactions between water and the immersed object are purely repulsive. Lum et al. showed how such soft interfaces can emerge at the nanoscale.13 Water confined between two hard (impenetrable) walls is the simplest model of water-mediated interactions between large hydrophobic objects, and the model has been featured prominently in numerous theoretical and computational studies (see, e.g., refs 1320). It is also a natural reference point for the interpretation of numerous experimental observations.21,22 If two large hard objects are immersed in water under thermodynamic conditions where the liquid state is stable in bulk, the competition between bulk and surface energetics (which, as mentioned above, favors the formation of a thin vapor-like layer) will cause the confined liquid to become metastable with respect to a confined vapor phase at sufficiently small separations.1315,17 r 2011 American Chemical Society
We refer to this crossover separation below which a vapor-filled region between hard walls becomes thermodynamically stable as the evaporation length scale. In spite of this quantity’s importance, no comparison of water’s evaporation length scale to that of other common liquids exists, nor have accurate length scale calculations based on actual thermophysical property data been performed and analyzed. In this Letter, we use thermophysical property data to compare the evaporation length scales of water and eight common organic liquids over a broad range of temperatures, and we perform an analysis of the relative importance of each of the factors that contributes to this length scale. Consider two parallel square surfaces of dimensions L L, separated by a distance D, immersed in a bulk liquid whose chemical potential and temperature are μ and T, respectively.23 The free energy Ω (grand potential) of the liquid in the region between the plates is given by Ωl ¼ pl L2 D þ 2γls L2
ð1Þ
The corresponding expression for the vapor is Ωv ¼ pv L2 D þ 2γvs L2 þ 4γvl LD
ð2Þ
where pl and pv are the pressure of the liquid and of the metastable (in the bulk) vapor at the given chemical potential and temperature, and γls, γvs, and γvl denote the liquidsolid, Received: March 8, 2011 Accepted: April 7, 2011 Published: April 11, 2011 1000
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Table 1. Normal Melting Temperature Tm, Normal Boiling Temperature Tb, Liquid Number Density at Coexistence Gl,σ, and Surface Tension γlv at T = 298.15 K for the Studied Liquids Tm (K)
Tb (K)
Fl,σ (mol/cc)
γlv (Nm1)
water
273.2
373.2
0.0553
0.0720
methanol
176.1
337.8
0.0245
0.0221
ethylene glycol
260.6
470.7
0.0137
0.0480
glycerol
291.8
562.8
0.0179
0.0633
heptane
182.6
371.5
0.0068
0.0201
decane
243.3
447.2
0.0051
0.0233
cyclohexane
279.6
353.9
0.0092
0.0245
benzene toluene
278.6 178.1
353.3 383.8
0.0112 0.0094
0.0281 0.0279
liquid
vaporsolid, and vaporliquid interfacial tensions, respectively. The free energies of the confined liquid and vapor therefore become equal at a critical separation Dc: Dc ¼
"
2γvl
4γvl ðpl pv Þ 1 þ Lðpl pv Þ
#
2γvl ðpl pv Þ
ð3Þ
For D < Dc, Ωv < Ωl, and the confined vapor is thermodynamically favored relative to the liquid. Equation 3 is valid when the interactions between the fluid and the solid surface are purely repulsive (contact angle 180°), in which case γvl = γls γvs. If the immersed surfaces have a different geometry, the factor 4 in the denominator of eq 3 is replaced by a different, O(1) numerical constant c (e.g., c = 2 for circular plates of radius L). The second expression in eq 3 is valid for sufficiently large immersed objects [L . cγlv/(pl pv)]. For water at atmospheric pressure (pl = 1 bar) this limit corresponds to objects larger than 1 μm. In the opposite limit of small immersed objects, the quantity 4γlv/L(pl pv) becomes the dominant term in the denominator of eq 3 (e.g., this term is of order 103 for nanometer-sized objects in water at 1 bar). In this limit, Dc equals L/2 independently of the particular confined liquid, temperature, and pressure (2L/c for other geometries). Because our thermodynamic approach assumes that the confined fluid can be treated as homogeneous, and its thermodynamics is accurately described by eq 1, the small-L limit and the result Dc ∼ L/2 are valid for D appreciably larger than molecular dimensions. Simulations suggest7 that this limit is attained when D exceeds approximately 3 nm. Equation 3 can be expressed in another convenient form23 by expanding the pressure about the bulk saturation value at the given temperature, pσ pl pσ þ Fl, σ Δμ
ð4Þ
pv pσ þ Fv, σ Δμ
ð5Þ
where Δμ = μ μσ, and μσ is the chemical potential at saturation conditions at the given temperature. At sufficiently subcritical temperatures, Fl,σ . Fv,σ, hence pv ≈ pσ, and pl pv ≈ Fl,σΔμ. Hence, we have Dc
2γvl 2γvl pl pσ Fl, σ Δμ
ð6Þ
Figure 1. Evaporation length scale Dc as a function of distance from the boiling temperature Tb (in K), for water and eight common organic liquids, at atmospheric pressure.
Quantitative application of eqs 3 and 6 has been limited, to our knowledge, to water at ambient conditions. We now compare water’s evaporation length scale, Dc, and its interfacial and bulk contributions, γlv (free energy per unit area) and Fl,σΔμ (free energy per unit volume), respectively, to those of methanol, ethylene glycol, glycerol, heptane, decane, cyclohexane, benzene, and toluene. We focus in particular on the temperature dependence of the evaporation length scale, Dc(T), at atmospheric pressure, meaning that pl is set to 1 bar. To this end, γlv(T), Fl,σ(T), and pσ(T) values for each liquid were obtained from the NIST Chemistry WebBook24 and CDATA25 databases, from the normal melting temperature Tm to the normal boiling temperature Tb. Table 1 lists Tm, Tb, γlv (298 K) and Fl,σ (298 K) for the nine liquids. Figure 1 shows the evaporation length for the nine substances, between their normal boiling point (Tb T = 0, where Dc trivially diverges for all liquids) and Tm (in the case of glycerol and ethylene glycol, we show Dc over the temperature range for which γlv data are available). Although all values of Dc shown are macroscopic, the salient feature is the significantly larger evaporation length for water, ∼ 1500 nm, compared to that of methanol and the five nonpolar liquids investigated here, for which Dc ∼ 500 nm. The values for ethylene glycol (Dc ∼ 900 nm) and glycerol (Dc ∼ 1200 nm) fall in between. The magnitude of water’s evaporation length scale shown in Figure 1 agrees with the number quoted in ref 7 but departs significantly (by an order of magnitude) from that given in a number of literature references (e.g., refs 13 and 14). At low enough temperatures, the evaporation length decreases mildly with increasing temperature. This behavior mirrors the temperature dependence of γlv (viz., dγlv/dT < 0) because pl pσ is nearly constant (∼1 bar) at low T. The comparatively narrow liquid ranges of cyclohexane and benzene preclude negative values of dDc/dT. We now investigate the bulk and interfacial contributions to Dc. Figure 2a shows Δμ(T) for the nine liquids. It has been argued that the smallness of this quantity for water underlies its remarkably large evaporation length scale.14 While it is true that Δμ is indeed unusually small for water, its saturated liquid number density, Fl,σ, as shown in Figure 2b, is unusually large, and by a comparable factor, relative to that of the other eight liquids (see also Table 1). Accordingly, as shown in Figure 2c, the 1001
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Figure 3. Vaporliquid interfacial tension for the same liquids shown in Figure 1, as a function of temperature (given as distance in K from the boiling temperature Tb).
Figure 2. (a) Difference between the chemical potential at atmospheric pressure and the chemical potential at saturation, as a function of temperature (given as distance in K from the boiling temperature Tb) for the same liquids shown in Figure 1. (b) Saturated liquid number density as a function of temperature (given as distance in K from the boiling temperature Tb) for the same liquids shown in Figure 1. (c) Product of quantities shown in panels a and b; note the approach to the substance-independent limit of 1 bar at low enough temperatures.
relevant quantity, namely the product Fl,σΔμ, is quite similar, at any given Tb T, for the nine liquids investigated here, approaching the common value Fl,σΔμ ∼ 1 bar at low temperatures. It follows that water’s large evaporation length relative to the other liquids considered here originates solely in its high γlv. This is clearly shown in Figures 1 and 3. The macroscopic size of the evaporation length, for all liquids, is a consequence of the fact that plσ3/kT , 1 when pl ∼ 1 bar and σ is of the order of a typical molecular dimension (k is Boltzmann’s constant); in contrast, γlvσ2/kT is a quantity of the order of 1. From this it follows that Dc/σ .1 (see eq 3). In summary, we have shown, by comparing physical property data, that the evaporation length for water at atmospheric pressure is ca. 1500 nm. The corresponding values for common organic liquids are also macroscopic, attesting to the phenomenon’s generality,17 albeit systematically smaller, by as much as a factor of 3. Over the range Tm < T < Tb, every water property contributing to Dc is distinctive: its surface tension is unusually large, its chemical potential is notably close to the value at saturation, and its liquid density at saturation is remarkably large. However, the product of the latter two properties is necessarily equal to pl pσ, which in turn approaches the substanceindependent value pl (1 bar in this work) at low temperatures (i.e., when pσ , pl). The fact that water’s evaporation length is appreciably larger than that of the eight other common liquids
considered here is an exclusive consequence of its large liquidvapor surface tension. The present comparative study suggests several lines of future inquiry. It would be interesting to extend the calculations into the supercooled regime, where water’s anomalies become more pronounced (see, e.g., ref 26). Other than water, the substances investigated here are all common organic liquids. Extension of this type of analysis to ionic liquids and liquid metals, as well as to a broader range of polar and apolar organic liquids, appears worthwhile. Extension to high pressures and to the negative pressure regime is also of interest,18,27 as is the tuning of the surface tension by additives and the consequent effect on the evaporation length scale.27 Finally, it should be emphasized that the present analysis addresses exclusively the thermodynamics of surface-induced evaporation. The large value of water’s evaporation length suggests that the corresponding kinetic aspects28,29 are at least as deserving of attention.
’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
’ ACKNOWLEDGMENT We acknowledge helpful discussions with Professor David Chandler. C.A.C.’s research is supported by the Spanish Ministry of Education (Programa de Movilidad, #PR2009-0449). P.G.D. and P.J.R. gratefully acknowledge the support of the National Science Foundation (Collaborative Research in Chemistry Grants CHE-0908265 and CHE-0910615). Additional support from the R. A. Welch Foundation (F-0019) to P.J.R. is also gratefully acknowledged. ’ REFERENCES (1) Smit, B.; Maesen, T. L. M. Towards a Molecular Understanding of Shape Selectivity. Nature 2008, 451, 671–678. (2) Izraeli, E. S.; Harris, J. W.; Navon, O. Brine Inclusions in Diamonds: A New Upper Mantle Fluid. Earth Planet. Sci. Lett. 2001, 187, 323–332. 1002
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