Letter Cite This: J. Phys. Chem. Lett. 2018, 9, 5179−5182
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Why Is It So Difficult to Identify the Onset of Ice Premelting? Yuqing Qiu and Valeria Molinero*
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Department of Chemistry, The University of Utah, Salt Lake City, Utah 84112-0580, United States ABSTRACT: Premelting of ice at temperatures below 0 °C is of fundamental importance for environmental processes. Various experimental techniques have been used to investigate the temperature at which liquid-like water first appears at the ice−vapor interface, reporting onset temperatures from −160 to −2 °C. The signals that identify liquid-like order at the ice−vapor interface in these studies, however, do not show a sharp initiation with temperature. That is at odds with the expected first-order nature of surface phase transitions, and consistent with recent large-scale molecular simulations that show the first premelted layer to be sparse and to develop continuously over a wide range of temperatures. Here we perform a thermodynamic analysis to elucidate the origin of the continuous formation of the first layer of liquid at the ice−vapor interface. We conclude that a negative value of the line tension of the ice−liquid−vapor three-phase contact line is responsible for the continuous character of the transition and the sparse nature of the liquid-like domains in the incomplete first layer.
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remelting at the ice−vapor interface1−3 is important in areas as diverse as atmospheric chemistry, the microstructure of snow, and the motion of glaciers.2,4,5 Much work has been devoted to understand the properties of the premelted ice surface and the conditions at which it forms, using experiments and molecular simulations.2,6−29 Nevertheless, there has not yet been agreement on the temperature at which premelting starts at the ice−vapor interface. Experimental studies have reported the onset of liquid-like order at the ice−vapor interface at temperatures ranging from 113 to 271 K.7,30−33 This variability can be partly attributed to the different properties, resolutions and depths probed by the experimental techniques.34 An analysis of the data reported in these studies, however, evidence a more fundamental reason: the properties signaling liquid-like order at the ice surface do not display a sharp, clear-cut initiation with temperature. Likewise, molecular simulations of the ice−vapor interface with the SPC/E, TIP4P, TIP4P/2005, TIP4P/ice, and mW water models that identify liquid-like water using various measures concur that the first quasi-liquid layer grows gradually with temperature.6,11−13,29 Recent large-scale molecular simulations11,12 with the mW water model reveal that the first premelted layer develops gradually with temperature, forming sparse liquid domains (Figure 1). The simulations indicate that the fraction of liquid in the outermost layer of ice increases from ∼0.4 at 250 K to 0.65 at 270 K11,12 (Figure 2). We note that mW accurately reproduces the increase in average thickness of the premelted layer with t = 1 − T/Tm of TIP4P/ice,6 the experimental structure of liquid water and ice,35−39 the melting temperature Tm = 273 K,37 difference in chemical potential between ice and liquid40,41 and entropy of melting39,41 of water, as well as the experimental surface tensions of the ice−liquid6,42,43 and liquid−vapor39,44 interfaces. The continuous development of the first liquid layer in the simulations with mW,6,11−13 SPC/E, and the TIP4P family © XXXX American Chemical Society
Figure 1. Premelting of the ice−vapor interface at 260 and 270 K from simulations of ref 11 with the mW water model.39 Liquid-like (quasi-liquid) water is shown with blue balls, ice with gray sticks. The fraction of liquid-like water in the outermost layer is 0.38 at 260 K and 0.65 at 270 K.11 Ice and liquid were identified using the CHILL+ algorithm.48 The ice exposed the basal plane to the vapor, with area 10 nm × 10 nm. Left panel is reproduced from ref 11. Copyright 2017 American Chemical Society.
of models29 mirrors the increase in the amplitude of the waterlike component in recent sum frequency generation (SFG) spectroscopy experiments from 0.34 ± 0.03 at 245 K to 0.62 ± 0.02 at 270 K45 (Figure 2). Taken together, the experimental and simulation results suggest that premelting at the ice−vapor interface is not initiated by a first order transition with a welldefined onset temperature, the common scenario in other surface transitions,46,47 but a continuous transformation over a range of temperatures. To our knowledge, the origin of this Received: July 18, 2018 Accepted: August 27, 2018 Published: August 27, 2018 5179
DOI: 10.1021/acs.jpclett.8b02244 J. Phys. Chem. Lett. 2018, 9, 5179−5182
Letter
The Journal of Physical Chemistry Letters
where Tm is the equilibrium melting temperature of bulk ice, ΔSm its entropy of melting, and ρ the density of bulk liquid water. To estimate Δγ for the mW ice−vapor interface, we first determine Tpremelting = −5 °C by interpolation of the high temperature linear regime of the width of the premelted layer, δQLL,vs ln(1 − T/Tm) in the simulations of the mW ice− vapor interface of ref 6 to the width of a water monolayer, δQLL = H = 0.35 nm. We then use that value of Tpremelting and the bulk thermodynamic data of mW to derive Δγ = 2 mJ m−2 using eq 2. The Δγ we derive for the first quasi-liquid layer is significantly smaller than the γice−vapor − (γliquid−vapor + γice−liquid) = 82 mJ m−2 deduced from the scaling of the thickness of the premelting layer with ∼3 to 6 liquid layers.6 A crude estimation of the free energy of the mW ice−vapor interface from the enthalpy of breaking water−water bonds, neglecting the entropic contribution, would give γice−vapor = 105 mJ m−2 at 273 K, whichusing γliquid−vapor = 68 mJ m−2 44 and γice−liquid = 35 mJ m−2 49 for mW at 273 Kwould result in Δγ = 2 mJ m−2. We expect that the entropic contribution would decrease γice−vapor. While the agreement of this estimation with the Δγ computed from Tpremelting is probably a coincidence, and the use of bulk liquid properties to compute Δγ using eq 2 cannot be accurate, the estimation above suggests that Δγ may be smaller than determined in ref.6. It would be interesting to use thermodynamic integration42 to independently compute γice−vapor and Δγ for mW and other water models and assess how these properties depend on the width of the premelted layer. Although the thermodynamic model predicts that at 270 K the first quasi-liquid layer is complete, the simulations indicate that at that temperature only 69% of the first layer and 6% of the second are molten.12 The incomplete melting results in a certain roughness of the ice−liquid interface, which−we conjecture− is required to satisfy the value of the ice−liquid surface tension. As the surface tension of mW and water are comparable, 35 mJ m−2 for mW6,43,49 and 29 to 33 mJ m−2 for water,50,51 we expect both to have comparable roughness of the premelted liquid-ice interface. The line tension τ has no bearing on the value of Tpremelting. The sign of τ, however, determines whether the transition at Tpremelting is first order or continuous. If the line tension is positive, it opposes the formation of liquid nuclei at the surface, creating a barrier for their formation. This results in a free energy landscape with two minima as a function of the size of the nuclei and a first order transition for the formation of the first liquid layer. If the line tension of the ice−liquid−vapor line is negative, the free energy as a function of the size of the liquid-like clusters has a single basin that evolves continuously with temperature. This results in the continuous equilibrium development of a first premelted layer over a wide range of temperatures below Tpremelting. Another consequence of τ < 0 is a high ratio of perimeter to area of the quasi-liquid domains, the signature of the liquid-like clusters in Figure 1. The line tension of the ice−liquid−vapor contact line must be negative to explain the continuous development of liquid-like domains at temperatures below Tpremelting in the simulations.11,12 The agreement in Figure 2 between the increase in the fraction of liquid-like water in the first layer in the simulations and in the area of the liquid-water-like band at 3400 cm−1 in SFG experiments45 of the ice−vapor interface suggests that between 245 to 270 K the spectra may be probing the continuous development of the first layer of liquid. This would
Figure 2. Liquid-like water at the ice−vapor interface exposing the basal plane from simulations using the mW water model and detected by SFG experiments. Blue circles show the fraction of liquid in the first layer from simulations with controlled vapor pressure12 and the blue crosses from simulations of ice in contact with vacuum.11 At any temperature, the simulations predict the same fraction of liquid-like water for the basal and prismatic planes.12 Red circles indicate the areas of the water-like SFG band at 3400 cm−1 in the experiments of ref 45. The continuous increase in the area of the liquid-like band and decrease in the area of the crystalline band from 245 to 270 K reported in ref 45 suggest that the shift in frequency of the liquid-band around 257 K reported in ref 25 does not correspond to a stepwise development of a new liquid-like layer as proposed in that study. We conjecture that the frequency shift observed around 257 K in ref 25 may signal an increase in the average coordination among liquid-like water molecules in the incomplete layer (compare, for example, the liquid-like domains at 260 and 270 K in Figure 1).
anomalous behavior of the ice−vapor interface has not been addressed in the literature. Here we perform a thermodynamic analysis of the equilibrium formation of the first quasi-liquid layer at the ice−vapor interface using data from experiments and simulations with the mW water model, with the aim of understanding why it forms continuously and not through a first order transition. The free energy required to create a quasi-liquid nucleus at the ice−vapor interface is ΔG = ΔμHAρ − ΔγA + τl
(1)
where Δμ = μliquid −μice is the difference in chemical potential between liquid-like water and ice, which is positive (unfavorable) in the premelting region, H is the width of a single quasi-liquid layer, which we take to be 0.35 nm (the first minimum of the radial distribution function of liquid water), A is the area of the quasi-liquid nucleus in contact with ice and vapor, ρ is the density of the quasi-liquid water, τ is the line tension of the ice−liquid−vapor contact line and l is the length of this three-phase interface. The driving force for premelting is given by Δγ, which measures the difference in stability between the ice−vapor interface and the sum of the free energies of the ice−liquid and liquid−vapor interfaces. Premelting requires Δγ > 0. The chemical potentials for zero and one layer of quasiliquid at the ice−vapor interface are equal when Δγ = ΔμρH. It is possible to derive Δγ from knowledge of the temperature Tpremelting for that equilibrium, assuming that the properties of the premelted layer are those of bulk liquid water: Δγ = (Tm − Tpremelting)HρΔSm
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DOI: 10.1021/acs.jpclett.8b02244 J. Phys. Chem. Lett. 2018, 9, 5179−5182
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The Journal of Physical Chemistry Letters
(2) Dash, J.; Rempel, A.; Wettlaufer, J. The Physics of Premelted Ice and Its Geophysical Consequences. Rev. Mod. Phys. 2006, 78, 695− 741. (3) Wilen, L.; Wettlaufer, J.; Elbaum, M.; Schick, M. DispersionForce Effects in Interfacial Premelting of Ice. Phys. Rev. B: Condens. Matter Mater. Phys. 1995, 52, 12426−12433. (4) Bartels-Rausch, T.; et al. A Review of Air-Ice Chemical and Physical Interactions (Aici): Liquids, Quasi-Liquids, and Solids in Snow. Atmos. Chem. Phys. 2014, 14, 1587−1633. (5) Kahan, T. F.; Wren, S. N.; Donaldson, D. J. A Pinch of Salt Is All It Takes: Chemistry at the Frozen Water Surface. Acc. Chem. Res. 2014, 47, 1587−1594. (6) Limmer, D. T.; Chandler, D. Premelting, Fluctuations, and Coarse-Graining of Water-Ice Interfaces. J. Chem. Phys. 2014, 141, 18C505. (7) Kouchi, A.; Furukawa, Y.; Kuroda, T. X-Ray Diffraction Pattern of Quasi-Liquid Layer on Ice Crystal Surface. J. Phys. Colloq. 1987, 48, C1-675−C1-677. (8) Brochard-Wyart, F.; Di Meglio, J. M.; Quére, D.; De Gennes, P. G. Spreading of Nonvolatile Liquids in a Continuum Picture. Langmuir 1991, 7, 335−338. (9) Elbaum, M.; Schick, M. Application of the Theory of Dispersion Forces to the Surface Melting of Ice. Phys. Rev. Lett. 1991, 66, 1713− 1716. (10) Murata, K.-i.; Asakawa, H.; Nagashima, K.; Furukawa, Y.; Sazaki, G. Thermodynamic Origin of Surface Melting on Ice Crystals. Proc. Natl. Acad. Sci. U. S. A. 2016, 113, E6741−E6748. (11) Hudait, A.; Allen, M. T.; Molinero, V. Sink or Swim: Ions and Organics at the Ice−Air Interface. J. Am. Chem. Soc. 2017, 139, 10095−10103. (12) Pickering, I.; Paleico, M.; Sirkin, Y. A. P.; Scherlis, D. A.; Factorovich, M. H. Grand Canonical Investigation of the Quasi Liquid Layer of Ice: Is It Liquid? J. Phys. Chem. B 2018, 122, 4880− 4890. (13) Shepherd, T. D.; Koc, M. A.; Molinero, V. The Quasi-Liquid Layer of Ice under Conditions of Methane Clathrate Formation. J. Phys. Chem. C 2012, 116, 12172−12180. (14) Paesani, F.; Voth, G. A. Quantum Effects Strongly Influence the Surface Premelting of Ice. J. Phys. Chem. C 2008, 112, 324−327. (15) Conde, M.; Vega, C.; Patrykiejew, A. The Thickness of a Liquid Layer on the Free Surface of Ice as Obtained from Computer Simulation. J. Chem. Phys. 2008, 129, 014702. (16) Li, Y.; Somorjai, G. A. Surface Premelting of Ice. J. Phys. Chem. C 2007, 111, 9631−9637. (17) Bluhm, H.; Ogletree, D. F.; Fadley, C. S.; Hussain, Z.; Salmeron, M. The Premelting of Ice Studied with Photoelectron Spectroscopy. J. Phys.: Condens. Matter 2002, 14, L227−L233. (18) Benet, J.; Llombart, P.; Sanz, E.; MacDowell, L. G. PremeltingInduced Smoothening of the Ice-Vapor Interface. Phys. Rev. Lett. 2016, 117, 096101. (19) Bishop, C. L.; Pan, D.; Liu, L.; Tribello, G. A.; Michaelides, A.; Wang, E. G.; Slater, B. On Thin Ice: Surface Order and Disorder During Pre-Melting. Faraday Discuss. 2009, 141, 277−292. (20) Pan, D.; Liu, L.-M.; Slater, B.; Michaelides, A.; Wang, E. Melting the Ice: On the Relation between Melting Temperature and Size for Nanoscale Ice Crystals. ACS Nano 2011, 5, 4562−4569. (21) Michaelides, A.; Slater, B. Melting the Ice One Layer at a Time. Proc. Natl. Acad. Sci. U. S. A. 2017, 114, 195−197. (22) Johnston, J. C.; Molinero, V. Crystallization, Melting, and Structure of Water Nanoparticles at Atmospherically Relevant Temperatures. J. Am. Chem. Soc. 2012, 134, 6650−6659. (23) Moore, E. B.; Allen, J. T.; Molinero, V. Liquid-Ice Coexistence Below the Melting Temperature for Water Confined in Hydrophilic and Hydrophobic Nanopores. J. Phys. Chem. C 2012, 116, 7507− 7514. (24) Limmer, D. T.; Chandler, D. Phase Diagram of Supercooled Water Confined to Hydrophilic Nanopores. J. Chem. Phys. 2012, 137, 044509.
require a negative line tension of the ice−liquid−vapor contact line for water in experiments. A recent analysis of experimental ice nucleation rates concluded that the line tension of water decreases on cooling, becoming negative at 233 K.52 The analysis of ref 52 is based on classical nucleation theory using a parametrization of the experimental rates of ice nucleation53 that underestimates the contribution of stacking disorder to the crystallization driving force.38 It has been demonstrated that this leads to an underestimation of the ice−liquid surface tension38 (γice−liquid is predicted to be 25.8 mJ m−2 at 273 K in the parametrization53 vs the 29 to 33 mJ m−2 measured at that temperature50,51). The underestimation of γice−liquid must translate into an overestimation of the cost of τ in ref 52, as the nucleation barrier is not a variable but fixed by the rate of nucleation in the experiment. This implies that the experimental line tension of the ice−liquid−vapor interface must become negative at temperatures closer to the melting point. In summary, our analysis suggests that a negative value of the ice−liquid−vapor line tension in water and the mW model is responsible for the sparse structure of the incomplete first premelted layer and its continuous development starting at very low temperatures. It would be interesting to extract in future work the value of the ice−liquid−vapor line tension of mW and other water models from the fluctuations of the length l of the contact line and to explore whether this or other information about the shape of the sparse liquid-like domains can be gleaned from diffraction or spectroscopic studies of the ice−vapor interface. Ice premelting in the literature focuses almost exclusively on the ice−vapor interface. It is straightforward to show, using Classical Nucleation Theory,54 that surfaces that have Δγ > 0 cannot heterogeneously nucleate ice.55 This implies that any surface that is unable to heterogeneously nucleate ice must produce a premelted water layer when in contact with ice crystals. The study of premelting at these ice interfaces may shed light on the universality of the results here discussed for the initiation of premelting at the ice−vapor interface.
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AUTHOR INFORMATION
Corresponding Author
*(V.M.) E-mail:
[email protected]. ORCID
Valeria Molinero: 0000-0002-8577-4675 Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank Arpa Hudait for preparing the snapshots of Figure 1 from simulations of ref 11, Huib J. Bakker for sharing the data from ref 45 and discussions, Matı ́as Factorovich for sharing data from ref 12, and David T. Limmer for discussions. This work was supported by the National Science Foundation through Award CHE-1305427 “Center for Aerosols Impacts on Climate and the Environment”. We thank the Center for High Performance Computing at the University of Utah for technical support and a grant of computer time.
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