Impurity Effects in the Premelting of Ice

VOLUME 82, NUMBER 12 PHYSICAL REVIEW LETTERS 22MARCH 1999

J. S. Wettlaufer Applied Physics Laboratory and Department of Physics, University of Washington, Box 355640, Seattle, Washington 98105 (Received 25 November 1998) The effect of impurities on the surface and interfacial melting of ice is investigated in the context of the Derjaguin-Landau-Verwey-Overbeek theory by calculating van der Waals and Coulombic interactions within interfacial solution ﬁlms. At high temperatures, the classical solute effect dominates the melting behavior. However, depending on the amount of impurity, as temperature decreases the slope of the ﬁlm-thickness versus temperature curve changes in a manner that depends on the relative strengths of van der Waals and Coulombic interactions. The results explain the wide range of experimental discrepancies and hence impact a host of environmental phenomena. [S0031-9007(99)08716-5]

PACS numbers: 64.70.Dv, 68.15.+e, 68.45.Gd, 82.65.Dp

The ice-water conversion is one of the most common Therefore, the discrepancies in the existing experimen- first order transitions experienced by humanity and ob- tal data on ice, and the fact that essentially all environ- served in the natural environment. Our basic understand- mentally relevant phenomena will involve contaminants ing of the melting of ice involves the modern theory of in some form, motivate an attempt to gain a simple under- premelting [1,2] which refers to the existence of liquid standing of how impurities can influence surface melting. at temperatures and pressures in the solid region of the In this Letter, a phenomenological model is proposed to bulk phase diagram. The environmental importance of gain insight into the problem. premelting in ice is great, playing a fundamental role in We consider the thermodynamic conditions necessary frost heave [3–5], the slip at glacier beds [6], and chemi- for the stable existence of an impure film disjoining a solid cal uptake on atmospheric ice surfaces [7]. Here, we from its vapor, from air, or from a foreign wall at tempera- investigate the previously unquantified effect of soluble tures below the bulk melting temperature. The film is an impurities on the interfacial premelting of ice. electrolyte solution containing, for example, a monovalent Premelting occurs in most materials including metals, ionic species such as NaCl. In either surface or interfacial semiconductors, solid rare gases, and molecular solids and melting, the interfaces can be considered to have a finite can be caused by a number of mechanisms. A familiar charge density qs, and hence any ions present in the film example is the Gibbs-Thomson shift in bulk coexistence screen the Coulomb interaction with an efficiency that de- due to curvature of the interface. Surface (interfacial) pends on their number density. Long ranged van der Waals premelting refers to a less familiar, but no less common, interactions are always present, and they may influence process whereby a liquid, or a liquidlike, film is present the melting behavior [14]. Essentially the same physical at the surface of a crystal in contact with its vapor phase phenomena underlie the forces that act between charged (a rigid wall) at temperatures below the bulk melting tran- surfaces in solution, examples of which pervade techno- sition [8]. The process begins gradually at T , 0.9 Tm, logical and biological problems [15], and provide the basis where Tm is the bulk transition temperature, with mono- of Derjaguin-Landau-Verwey-Overbeek (DLVO) theory layer films that thicken with increasing T. Another variant which assesses the competition between the screened of surface melting is grain boundary melting wherein pre- Coulomb and attractive van der Waals interactions [16]. melting occurs at junctions between crystals of the same For example, DLVO theory explains the flocculation or material. During complete melting the film thickens with- stabilization of charged colloidal particles. An essential out limit as the melting line is approached from the solid distinction with the case of surface melting concerns the region of the bulk phase diagram. In incomplete melting nature of the nonretarded van der Waals contribution the film growth is truncated at finite undercooling, usually to the total excess surface free energy per unit area, AH by retarded potential effects. Melting is driven by the ten- denoted by FvdW ͑d͒ ෇ 2 12pd2 , where AH is the Hamaker dency to reduce the interfacial free energy. The asymme- constant. For colloidal substrates in solution, d is their try of the surface phase change about Tm is the hallmark of separation distance, while for interfacial melting d is the surface melting: Most fluids can be deeply supercooled, water film thickness. For identical colloids the Hamaker but there is a complete absence of superheating of their constant is positive producing an attraction. The screened solid phase. Coulomb interactions are repulsive and suppress the Although the surface melting of ice is well documented flocculation of charged colloids. For dissimilar materials, (recent reviews are given in [2,9]), so is the great variation the van der Waals contribution can be both attractive and in the magnitude and temperature dependence of the melt- repulsive. Such is the case of the interfacial melting of ing behavior (see Figs. 1 of [10,11] and see also [12,13]). ice [14].

The screening of surface charges is described by thin, increase in impurity concentration, and decrease the calculating the electrostatic potential c created by the Debye screening length: k21 ~ d1͞2. Only at low tem- distribution of ions of number density n͑z͒ throughout the peratures, where the films are very thin and many of our film, viz. the Poisson-Boltzmann equation approximations are violated, does the variation in k21 it- 2 self begin to be important. However, what is important d c Zen͑z͒ Zeno 2Zec k T ෇ 2 ෇ 2 e ͞ b , (1) here is how this variation influences the total free energy dz2 ee ee o o of the system, which is written as where e is the dielectric constant of the film, eo is the free space permittivity, z measures the distance normal GT ͑T, P, d, Ni͒ ෇ rᐉmᐉ͑T, P͒d 1mi͑T,P͒Ni to the plane of the film, and no is the ion density at Ni 1RTNi ln 1ms͑T,P͒Ns its midplane. At very short ranges, the theory must be rᐉd modified to account for steric effects [17], but we will 1F ͑d͒, (4) not deal with these complications here (Israelachvili [15] T provides a thorough review and discusses the limitations wherein, rᐉ, mᐉ, and Nᐉ ෇ rᐉd are the molar density, of the Poisson-Boltzmann equation). For a monovalent chemical potential per mole, and the number of moles per unit area of the solvent, m is the chemical potential of electrolyte, and a surface potential cs less than 25 mV, i Eq. (1) can be linearized in the so-called Debye-Hückel the impurity, R is the gas constant, ms and Ns are the limit which yields chemical potential per mole and the number of moles per unit area of the solid, and F is the total effective ee k T 1͞2 T 2kz 21 o b interfacial free energy as discussed above. For simplicity c͑z͒ഠcse , where k ෇ 2 (2) e nb of presentation, the entropic term, RTN ln Ni , is written µ ∂ i rᐉd is called the Debye length, which describes the character- in the dilute limit of ideal solution theory, but the analysis istic falloff of the ion field adjacent to the surface, where can proceed without these approximations if we replace nb is the bulk ion density. A repulsive force between the molar ratio by the activity coefficient for the impurity two charged surfaces originates in the restriction of the in the solvent. Such complications do not change the entropy of the ions as the surfaces are brought closer: Al- essential behavior of the system. The interfacial term though ions of charge opposite to those of the surface are captures the asymptotic behavior of the system within the attracted to it, they are repelled from each other, and the context of DLVO theory as follows: increased proximity induced by decreasing the film thick- s2 e2k͑d2s͒ ness increases the free energy. In this limit, the excess FT ͑d͒ ෇ gd 1Dg 12 2 2 , (5) interfacial free energy per unit area across a film of thick- µ 2d 2 ! ness d is written in which the dry interfacial energy, whether it be the 2 solid/vapor or solid/wall interface, is represented by gd, 2qs 2kd FDH ഠ e , (3) Dg ෇ gsᐉ 1gᐉy͑w͒ 2gd is the difference in free energy keeo between the dry and wet interfaces, where s, ᐉ, y, w and the total excess interfacial free energy of relevance denote solid, liquid, vapor, and wall, respectively. The to the interfacial melting of ice in the presence of second term describes the sum of the DLVO contributions electrolytes is the sum of FDH͑d͒ and FvdW ͑d͒ which can to the free energy as a function of film thickness, where be considered to be a special case of DLVO theory. s is a short range cutoff of the order of a molecular The experimental paradigm of surface force measure- diameter. The coefficients can be related to those of ments [15] provides the film under scrutiny an infinite DLVO theory [4]. supply of solution at fixed concentration. Hence, an ex- Interfacial adsorption of impurities shifts the interfacial periment can be prepared in which the Debye length is free energy in a manner described by the classical Gibbs- fixed for all values of the film thickness. In contrast, adsorption isotherm [18]. In contrast to the case of during surface melting we envisage the following experi- nonelectrolyte solutions, gᐉy increases with electrolyte mental situation. A surface is prepared at a temperature concentration due to an image charge effect. Although where, in the absence of impurities, a film is present. Ni it is known that the difficulty of impurity incorporation moles per unit area of a single species of monovalent influences surface structural phase transitions by reducing nonvolatile impurities are deposited in the film. Because the surface energy [19], we only have experimental of their low volatility, and the fact that the ice-solution information for the water/vapor interface. Because the equilibrium segregation coefficient is extremely small, of impurities are nonvolatile and insoluble in ice, each of the order 1026 (which we take to be zero), there are a fixed interfacial coefficients in Dg will be modified in a manner number of electroyte ions in the film, maintaining equilib- that depends on, among other things, the magnitude rium with the interfacial charge. An increase (decrease) of the surface charge density and the bulk electrolyte in film thickness is accommodated by melting (freezing) concentration. An increase in jDgj will enhance surface of the solid so that, up to the solubility limit, the impurity melting relative to the pure case, and there are some concentration is simply inversely proportional to the film experimental indications that this occurs in ice against volume. Hence, at low temperatures, a surface film will air [10,11]. Nonetheless, the exact dependencies of the 2517 VOLUME 82, NUMBER 12 PHYSICAL REVIEW LETTERS 22MARCH 1999 constituents of Dg on Ni are presently unknown to us and the second and third terms are the nonretarded van der hence to introduce further theoretical treatment risks an Waals and the Coulombic contributions, respectively. extension beyond the realm of firm evidence. The form of Figure 1 shows the film thickness versus undercooling at our result is amenable to an experimental estimate of this the interface between ice and a substrate of (a) silicon, (b) effect. A more sensitive impurity dependent effect resides polyvinylidene chloride, and (c) water vapor, with differ- in the film thickness dependence of the Debye length ent values of Ni. The Hamaker constant decreases from as discussed above. Equation ͑2͒ leads to k ෇ c Ni͞d, silicon to the water vapor interface. The dominance of where c 7.237 3 107 m1͞2 mol21͞2 is a constant. the surface Raoult’s law at high temperatures is clearly ෇ p As the temperature is raised or lowered isobarically, demonstrated: As Ni increases, (i) the film thickens,as material is exchanged between the solid and the liquid, expected, and (ii) the temperature at which this power and the condition for equilibrium is law (d ~DT21) begins to dominate the melting behavior decreases. It is particularly sensitive in the case of the ≠GT ≠GT ෇ , (6) vapor interface where it may be strong enough to over- ≠Nᐉ ≠Ns whelm the incomplete melting controlled by van der Waals which leads to the following relationship between the interactions in the pure case [10,20] and drive complete undercooling and the film thickness: surface melting, thereby reconciling the difference between experimental observations made by different groups 2 Tm RTmNi Dgs Dgc Ni [10–13]. For large enough contamination at any inter- Tm 2 T ෇ 2 3 2 rᐉqm " d d 2 s d face, the entire range of melting behavior is dominated by s p the surface Raoult’s law. As the temperature decreases, 3 1 1 e2c Ni ͞d ͑d2s͒ . (7) the first crossover occurs to interfacial melting controlled √ d ! # by van der Waals interactions (d ~DT21͞3) appearing at The first term is the surface version of Raoult’s law, and higher temperatures as the Hamaker constant increases.

FIG. 1. Film thickness versus undercooling at the interface between ice and a substrate of (a) silicon, (b) polyvinylidene chloride, and (c) water vapor. In (a) and (b) Ni ෇ 0.06 (solid line), 0.18 (small dashed line), and 6 (dotted line) 24 mM NaCl per square meter. In (c) Ni ෇ 6 3 10 (long dashed line), 6 3 1023 (dash-dot line), 0.06 (solid line), 0.18 (small dashed line), and 6 (dotted line) mM NaCl per square meter. Depending on the sign convention taken for FvdW ͑d͒, the Hamaker constant can be related 2 to the interfacial coefficient Dg, viz. 12ps Dg ෇ AH [4]. Hamaker constants can be calculated using the complete theory of dispersion forces [14,20], which requires as input, spectral data for the dielectric functions of ice, water, and the substrate of interest, or they can be taken from experimental measurements. In the former case, the results are often sensitive to the data set and fitting model used to incorporate them into dispersion force theory. Elbaum and Schick [20] used two data sets to calculate film thickness versus temperature, and here the data which produced the largest Hamaker constant [22] for the vapor interface was 222 used, giving AH ෇ 23.07 3 10 J. For polyvinylidene chloride we use fits to experimental data AH ෇ 21.50 3 10220 J [5] and increase this by 1 order of magnitude to describe Silicon [14]. The vapor interface is most sensitive to the impurity content. As its temperature increases, the lowest concentrations change slope twice (in this range), and above about a mK, the lower dopant level yields a thicker film. As the temperature increases further, there is another crossover (not seen on this scale). Clearly, the thickness is sensitive to small amounts of impurities, and hence it is quite dependent on factors that are extremely difficult to control in an experimental apparatus.

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For sufficiently low concentrations the range of undercool- Conversations with J. G. Dash, S. C. Fain, and Y. ing of this van der Waals plateau is limited, and a Coulom- Furukawa are gratefully acknowledged, as are helpful bic shoulder appears which rapidly steepens the thinning; comments on the manuscript made by A. W. Rempel the slope exceeding that of the purely solutal effect. As and the referees. K. M. Cuffey’s queries were largely noted above, for sufficiently large Ni the surface Raoult’s responsible for resurrecting my studies of this effect. This law may dominate the melting behavior, but it should be work has been generously supported by NSF Grant No. stressed that there is a combination of effects that occur. OPP 95-23513 and ONR Grant No. N00014-94-1-0120. First, the increasing strength of the solute effect extends the surface Raoult’s law to lower temperature; second, it decreases in the range of the Coulombic interaction. In other words, as Ni decreases, the range of the Coulombic [1] J. G. Dash, in Ice Physics and the Natural Environment, interaction increases and its strength can begin to dominate edited by J. S. Wettlaufer, J. G. Dash, and N. Untersteiner, that of the van der Waals interaction of a film of a higher NATO ASI, Ser. I, Vol. 56 (Springer-Verlag, Heidelberg, concentration. Hence, for a given material, we see the pos- 1999), pp. 11–22. [2] J. G. Dash, H-Y. Fu, and J. S. Wettlaufer, Rep. Prog. Phys. sibility of a crossover, within the van der Waals plateau, 58, 115 (1995). between films with two Ni values, the film of larger Ni [3] L. A. Wilen and J. G. Dash, Phys. Rev. Lett. 74, 5076 having a smaller thickness at the same undercooling. This (1995). behavior should be unique to interfacial melting, wherein [4] J. S. Wettlaufer and M. G. Worster, Phys. Rev. E 51, 4679 the Debye length depends on the film thickness. (1995). Clearly, it is important to examine the nature of melting [5] J. S. Wettlaufer, M. G. Worster, L. A. Wilen, and when the van der Waals interactions are attractive (non- J. G. Dash, Phys. Rev. Lett. 76, 3602 (1996). melting) and the Coulombic interactions (which can be [6] K. M. Cuffey, H. Conway, B. Hallet, A. M. Gades, and short or long range) are repulsive, but we leave this, and C. F. Raymond, Geophys. Res. Lett. (to be published). additional effects [15], including the facet dependence of [7] T. Peter, Annu. Rev. Phys. Chem. 48, 785, (1997). interfacial melting, for future research when the experi- [8] Surface and interfacial melting are used interchangeably here. mental data are refined. Finally, we note that Beaglehole [9] D. W. Oxtoby, in Ice Physics and the Natural Envi- [21] studied the effects of size and impurities on the melt- ronment (Ref. [1]), pp. 23–38; J. S. Wettlaufer, ibid., ing of ice particles, but restricted interfacial interactions to pp. 39–68. a very short range and did not consider how the impurities [10] M. Elbaum, S. G. Lipson, and J. G. Dash, J. Cryst. Growth extend the range of these interactions. 129, 491 (1993). We have derived a phenomenological treatment of the [11] Y. Furukawa and H. Nada, in Advances in the Un- role of impurities in surface and interfacial melting in the derstanding of Crystal Growth Mechanisms, edited by context of DLVO theory and have applied it to the phase T. Nishinaga et al. (Elsevier B.V., North-Holland, 1997), behavior of ice doped with NaCl. Upon addition of a pp. 559–573. small amount of impurity three different dependencies [12] D. Beaglehole and D. Nason, Surf. Sci. 96, 357 (1980). of film thickness on temperature result. Within a given [13] H. Dosch, A. Lied, and J. Bilgram, Surf. Sci. 366,43 (1996). type of dependence, the magnitude of the film thickness is [14] L. A. Wilen, J. S. Wettlaufer, M. Elbaum, and M. Schick, extremely sensitive to the amount of dopant, and at low Phys. Rev. B 52, 12 426 (1995). concentrations the Coulomb interaction can effectively [15] J. N. Israelachvili, Intermolecular and Surface Forces compete with the van der Waals interaction at long range. (Academic Press, New York, 1992). The surface melting of ice is well documented [2,9–13], [16] The original work can be found in B. V. Derjaguin and as is the variation in the magnitude and slope of the L. Landau, Acta Physicochim. URSS 14, 633 (1941); film-thickness versus temperature relations wherein a E. J. W. Verwey and J. Th. G. Overbeek, Theory of the given experiment often exhibits changes in slope [10–13]. Stability of Lyophobic Colloids (Elsevier, Amsterdam, Both the typical magnitude of the interexperimental 1948). variations and the changes in slope are easily bracketed [17] I. Borukhov, D. Andelman, and H. Orland, Phys. Rev. by the behavior shown in Fig. 1. Therefore, the results Lett. 79, 435 (1997). [18] W. J. Moore Physical Chemistry (Prentice Hall, Engle- provide a reasonable explanation for the differences wood Cliffs, N.J., 1955). observed among various experimental groups. Rather [19] J. P. Van der Eerden, J. Cryst. Growth. 128, 62 (1993). than attempting to prepare absolutely pure samples, it is [20] M. Elbaum and M. Schick, Phys. Rev. Lett. 66, 1713 hoped that the magnitude of the impurity sensitivity can (1991). be used as a guide for future experimental work toward [21] D. Beaglehole, J. Cryst. Growth. 112, 663 (1991). an approach of systematic doping. [22] J. Daniels, Opt. Commun. 3, 240 (1971).

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