Premelting, fluctuations, and coarse-graining of water-ice interfaces David T. Limmer and David Chandler Citation: The Journal of Chemical Physics 141, 18C505 (2014); doi: 10.1063/1.4895399 View online: http://dx.doi.org/10.1063/1.4895399 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/141/18?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Coarse-grained simulations of DNA overstretching J. Chem. Phys. 138, 085101 (2013); 10.1063/1.4792252 Mixing coarse-grained and fine-grained water in molecular dynamics simulations of a single system J. Chem. Phys. 137, 044120 (2012); 10.1063/1.4739068 Hydrophobic interactions with coarse-grained model for water J. Chem. Phys. 134, 234509 (2011); 10.1063/1.3602217 Liquid-vapor interfaces of water-acetonitrile mixtures of varying composition J. Chem. Phys. 123, 184706 (2005); 10.1063/1.2102892 Computer modeling of melting of ionized ice microcrystals J. Chem. Phys. 119, 10237 (2003); 10.1063/1.1617276 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.36.169.53 On: Wed, 01 Oct 2014 02:31:28 THE JOURNAL OF CHEMICAL PHYSICS 141, 18C505 (2014) Premelting, fluctuations, and coarse-graining of water-ice interfaces David T. Limmer1,a) and David Chandler2 1Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08540, USA 2Department of Chemistry, University of California, Berkeley, California 94609, USA (Received 14 July 2014; accepted 25 August 2014; published online 16 September 2014) Using statistical field theory supplemented with molecular dynamics simulations, we consider premelting on the surface of ice as a generic consequence of broken hydrogen bonds at the boundary between the condensed and gaseous phases. A procedure for coarse-graining molecular configu- rations onto a continuous scalar order parameter field is discussed, which provides a convenient representation of the interface between locally crystal-like and locally liquid-like regions. A number of interfacial properties are straightforwardly evaluated using this procedure such as the average premelting thickness and surface tension. The temperature and system size dependence of the premelting layer thickness calculated in this way confirms the characteristic logarithmic growth expected for the scalar field theory that the system is mapped onto through coarse-graining, though remains finite due to long-ranged interactions. Finally, from explicit simulations the existence of a premelting layer is shown to be insensitive to bulk lattice geometry, exposed crystal face, and curvature. © 2014 AIP Publishing LLC.[http://dx.doi.org/10.1063/1.4895399] I. INTRODUCTION case of water and ice, this driving force can be easily ratio- nalized from a microscopic perspective. Figure 1 shows rep- A premelting layer refers to thermodynamically stable resentative configurations of the surface of ice at conditions disordered structure at the interface of an otherwise ordered far away from, and close to, the melting temperature. To cap- crystalline solid. It appears when temperature is near but be- ture these configurations, we have carried out molecular sim- low the bulk melting temperature.1 First proposed by Michael ulations of the TIP4P/2005 model of water.11 At very cold Faraday in 1842 to explain the low friction of the surface of conditions, i.e., temperature T much below the melting tem- ice,2 definitive experimental evidence for surface melting was perature T , the surface is ordered and molecules at the sur- not observed on ice until 1987.3 Since then advances in sur- m face are forced to break one hydrogen bond on average. At face selective experimental techniques have provided pow- higher temperatures, T → T , hydrogen bonding is still dis- erful tools for direct atomic-resolution observations of this m rupted but this energetic loss is balanced by an entropic gain in surface phase transition.4 These experimental studies have restoring translational invariance parallel to the surface. The been complemented by a number of detailed atomistic sim- concomitant enhancement of fluctuations at the interface in- ulations that also find a premelting layer on the surface of creases in scale as the melting temperature is approached. Ex- models of ice.5–9 Despite these previous studies, at present no actly at T , the degeneracy in the free energy dictates that for microscopic description exists that both establishes the pre- m an infinite system the thickness of the interface may diverge cise nature of the transition and connects this level of de- as it is equally likely that the bulk is liquid or crystal. tail with a more coarse-grained picture necessary to explain How the thickness of the premelting layer changes with experimental observations. Using a simple field theory de- temperature reflects the interplay between bulk free energies scribed previously,10 complimented with molecular dynamic that favor order and the boundary condition that excludes it. simulations, we address this deficiency. As reviewed in Ref. 4, experimental estimates of the thick- Most liquids at ambient conditions are close to their triple ness of the premelting layer at a prescribed temperature, typ- point and as a consequence, at ambient conditions the chem- ically T ≈ T − 1 K, vary by over two orders of magnitude ical potential differences between the liquid, solid, and vapor m depending on the technique and interpretation. Partially, this phase are all very small. Therefore, near the melting temper- variability occurs because different techniques probe differ- ature, T , a thermodynamic criterion for the existence of the m ent physical properties with differing correlations to structural premelting layer is given by balancing the different surface disorder. X-ray and proton scattering are typically more sen- terms, sitive to long ranged order, and other surface selective tech- = − − 12 γ γs,v γ,s γ,v , (1) niques, such as sum frequency generation spectroscopy and atomic force microscopy,13 have also been used with suc- where γ ,γ , and γ are the solid-vapor, liquid-solid, and s,v ,s ,v cess. In all cases, however, the strong temperature depen- liquid-vapor surface tensions, respectively. When γ > 0, dence of the thickness adds to uncertainties due to contam- there is a thermodynamic driving force for premelting. In the ination and surface preparation. Further, in all cases it is not straightforward to connect the observations to molecular level a)Electronic mail: [email protected] details. 0021-9606/2014/141(18)/18C505/8/$30.00 141, 18C505-1 © 2014 AIP Publishing LLC This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 68.36.169.53 On: Wed, 01 Oct 2014 02:31:28 18C505-2 D. T. Limmer and D. Chandler J. Chem. Phys. 141, 18C505 (2014) of the mW model.14 The system size is characterized by two lengths, the width of the crystal slab, W, and its length, L.The width is approximately 3 nm and we vary the length between 3 and 13 nm. The temperature is controlled with a Nose- Hoover thermostat, with a time constant of 1 ps. Depending on the proximity to the melting temperature, simulations were run between 10 and 1000 ns in order to obtain converged es- timates of the surface properties. The long timescales are re- quired because the size of fluctuations in the interface become as large as the system as Tm is approached. Use of the mW model allows us to easily access these timescales. Molecular dynamics calculations are performed with LAMMPS.23 As expected from experiment, the average configuration of the ice slab depends sensitively on temperature. Figure 2 shows the mean density projected along the direction perpen- dicular to the plane of the surface. The mean density is cal- culated by binning the particle positions along the z axis. The density profile is normalized by its mean value in the center of FIG. 1. Characteristic snapshots of the basal surface of Ice Ih taken from our the slab, ρxtl, as averaged over many unit cells. Near the center molecular dynamics simulations of the TIP4P/2005 model. of the slab, the density exhibits the expected oscillations char- acteristic of crystal structure. At low temperatures, shown in panels (a) and (b), these oscillations persist to the end of the Theory and simulation offer a way to remove this am- → slab. At high temperatures, T Tm, shown in panels (c) and biguity. Specifically, using sufficiently general statistical me- (d), these oscillations decay upon approaching the exposed chanical arguments, which relate experimental observables crystal surface. The uniformity of the density near the crystal to emergent behavior with assumptions tested with explicit surface is an indication that the surface is disordered. molecular simulations, the phenomenology of the premelt- For water, locally liquid-like states can be distinguished ing layer can be understood and quantified. In what follows, from more ordered crystal-like states in terms of an or- we will show that a simple extension of the field theory we der parameter, q, that is real and a scalar. There are many 10 have used previously accurately predicts the existence and such measures suitable for this purpose that have been used scaling of the premelting layer as a function of both tempera- previously.24–26 As with our previous work,10 we chose ture and system size. The accuracy of our field theory is con- the local order parameter from Steinhardt, Nelson and firmed with efficient molecular dynamics simulations of the Ronchetti,27 mW model.14 N + = (i) − q(r) qliq q δ(r ri ) , (2) II. LOCAL ORDER PARAMETER PROFILES i=1 We begin to examine the premelting layer using molec- where ri is the position of the ith oxygen among N water ular dynamics simulations of a minimal model of water, the molecules, and 14 ⎛ ⎞ mW model.