VOLUME 82, NUMBER 12 PHYSICAL REVIEW LETTERS 22MARCH 1999 Impurity Effects in the Premelting of Ice 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 inter- actions 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 experi- mental 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 sdd ­ 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]. 2516 0031-9007y99y82(12)y2516(4)$15.00 © 1999 The American Physical Society VOLUME 82, NUMBER 12 PHYSICAL REVIEW LETTERS 22MARCH 1999 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 ~ d1y2. Only at low tem- distribution of ions of number density nszd 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 Zenszd Zeno 2Zec k T ­ 2 ­ 2 e y 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 sT, P, d, Nid ­ r,m,sT, Pdd 1misT,PdNi to the plane of the film, and no is the ion density at Ni 1RTNi ln 1mssT,PdNs its midplane. At very short ranges, the theory must be r,d modified to account for steric effects [17], but we will 1F sdd, (4) not deal with these complications here (Israelachvili [15] T provides a thorough review and discusses the limitations wherein, r,, m,, and N, ­ r,d 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 1y2 T 2kz 21 o b interfacial free energy as discussed above. For simplicity cszdøcse , where k ­ 2 (2) e nb of presentation, the entropic term, RTN ln Ni , is written µ ∂ i r,d 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 e2ksd2sd ness increases the free energy. In this limit, the excess FT sdd ­ 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 ø e , (3) Dg ­ gs, 1g,yswd 2gd is the difference in free energy keeo between the dry and wet interfaces, where s, ,, 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 FDHsdd and FvdW sdd 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.