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Thermodynamics of Ice Nucleation in Liquid Water Jianguo Mi J. Phys. Chem. B, Just Accepted Manuscript • DOI: 10.1021/jp512280p • Publication Date (Web): 29 Dec 2014 Downloaded from http://pubs.acs.org on January 3, 2015
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Thermodynamics of Ice Nucleation in Liquid Water
Journal: Manuscript ID: Manuscript Type: Date Submitted by the Author: Complete List of Authors:
The Journal of Physical Chemistry jp-2014-12280p Article 10-Dec-2014 Wang, Xin; Beijing University of Chemical Technology, Dept. of Chem. Eng. Wang, Shui; Beijing University of Chemical Technology, College of Chemical Engineering Xu, Qinzhi; Institute of Microelectronics, Chinese Academy of Sciences Mi, Jianguo; Beijing University of Chemical Technology, Dept. of Chem. Eng.
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Thermodynamics of Ice Nucleation in Liquid Water Xin Wang,† Shui Wang,† Qinzhi Xu,§ and Jianguo Mi†* †
State Key Laboratory of Organic-Inorganic Composites, Beijing University of Chemical Technology, Beijing 100029, China
§
Institute of Microelectronics of Chinese Academy of Sciences, Beijing, 100029, China
We present a density functional theory approach to investigate the thermodynamics of ice nucleation in supercooled water. Within the theoretical framework, the free-energy functional is constructed by the direct correlation function of oxygen−oxygen of the equilibrium water, and the function is derived from the reference interaction site model in consideration of the interactions of hydrogen−hydrogen, hydrogen−oxygen, and oxygen−oxygen. The equilibrium properties, including vapor−liquid and liquid−solid phase equilibria, local structure of hexagonal ice crystal, and interfacial structure and tension of water−ice are calculated in advance to examine the basis for the theory. The predicted phase equilibria and the water−ice surface tension are in good agreement with the experimental data. In particular, the critical nucleus radius and free-energy barrier during ice nucleation are predicted. The critical radius is similar to the simulation value, suggesting that the current theoretical approach is suitable in describing the thermodynamic properties of ice crystallization.
Keywords: density functional theory, phase equilibria, surface tension, free-energy barrier
*
Corresponding author. E-mail:
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INTRODUCTION Ice nucleation in supercooled water is a common natural process, and is probably the most relevant to biology, physics, geology, and atmospheric science.1-3 The phase transition from supercooled water into ice begins with the formation of a nucleus. In order to freeze, water molecules must initially form icelike aggregates. Spontaneous fluctuations give rise to ice-like regions that remain unstable unless they exceed a critical size, enabling them to overcome the unfavorable interfacial energy. In supercooled water, the nucleus is thought to have a hexagonal shape, 4 and once the activation barrier is surmounted, the stable crystal phase will continue to grow. Despite its important role in natural environments and refrigeration technologies, freezing of water is far from being understood in terms of the structure, dynamics, and the dependence on the degree of supercooling.5, 6 This is because the intermediate state in nucleation process, or the critical nucleus, is very short-lived. A better understanding of ice nucleation on a microscopic level is thus a topic of current research in theory, simulation, and experiment. Due to its importance to develop crystallization technology, ice nucleation was extensively studied in experiments. Several experiments focused on the measurement of homogeneous nucleation rate of supercooled water.7-10 These experiments were performed with small droplets to avoid heterogeneous nucleation. Meanwhile, Tabazadeh et al.11 found that in atmospheric droplets most of phase transformations could be surface-based rate processes. On the other hand, Bauerecker et al.12 investigated the formation of ice nucleus in the interfacial region, which coincides with a previous study.13 Nevertheless, measurements on supercooled water are remarkably difficult because of the relative small critical nuclei and uncontrolled ice nucleation. Thus, there are very large uncertainties even in the limited experimental data. Compared with experiment, computer simulation can yield descriptive information at best. More importantly, computer simulation provides a microscopic description of the process to investigate the thermodynamic and dynamic characteristics of nucleation.14-16 In this respect, at high supercooling, Moore and coworkers17, 18 proposed a coarse-grained water model by means of accelerated dynamics simulation. At moderate supercooling, Reinhardt et al.19 investigated the free-energy profile with an allACS Paragon Plus Environment
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atom water model for homogeneous ice nucleation. Radhakrishnan and Trout,20 Quigley and Rodger,21 and Brukhno et al.22 applied the TIP4P model to calculate the free-energy barrier to ice nucleation. Very recently, Sanz et al.23 calculated the nucleation rate and the critical cluster size with the TIP4P/2005 and TIP4P/ice water models at moderate supercoolings. These works provide the overall landscape toward comprehensive understanding of ice nucleation. However, simulation has the disadvantage of being computationally intensive. For crystal nucleation, the structural construction and reformation is particularly slow since the large-scale molecular motion is hardly proceeded. Development of a crystal nucleation theory that is computationally convenient but can still capture all the atomistic details of freezing thermodynamic and dynamics is always an attractive goal. From the point of microscopic view, statistical mechanics theories are good candidates to investigate crystallization thermodynamics. For examples, Harrowell and Oxtoby24 proposed an order parameter theory of freezing using a square gradient approximation to investigate the homogeneous crystal nucleation. Curtin25 predicted the interfacial properties by a new flexible parameterization. Rosenfeld et al.26 modified the fundamental-measure theory for three-dimensional hard spheres. Tarazona27 and Oettel et al.28 introduced tensorial weighted densities to predict crystal free-energy. For water and heavy water, Fu et al.29 formulated the excess Helmholtz free-energy functional with a local density approximation combined with the statistical associating theory. As such, two empirical parameters were inevitably included. So far, a fully microscopic understanding of ice crystal growth with the intra- and inter-molecular interactions and the thermodynamic boundary conditions as the only input is still a great challenge. For the crystal nucleation, phase field crystal (PFC) model is an ideal theoretical approach.30 In recent years, PFC was derived from microscopic dynamical density functional theory (DDFT) to solve crystallization and freezing problem.31 Its free-energy functional was deduced from the RamakrishnanYussouff32 type perturbative DFT after some simplifications that lead to a Brazowskii/Swift-Hohenberg form,33, 34 while the time evolution was governed by an overdamped conservative equation of motion.
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However, PFC model is limited for crystallization in model or simple fluids. It cannot be extended to water/ice system due to its deficiency in description of hydrogen-bond association in water. In fact, quantitative calculation the contribution of hydrogen bonding to ice nucleation is quite difficult due to its long-ranged characteristics. The free-energy expression for the long-ranged electrostatic interactions is still unavailable. Fortunately, the reference interaction site model (RISM) integral equation theory provides an efficient tool to deal with the long-ranged contributions. RISM has advantages over other theoretical models in structure description of sophisticated systems, where the effects of intramolecular structure and complicated interactions including hydrogen bonding can be fully taken into account. The main disadvantage of RISM lies in its inaccurate free-energy expression, especially for inhomogeneous system. Very recently, this drawback has been partly overcome by integrating more reliable force field or bridge function.35 Two typical successes of RISM for quantitative calculation are the hydration free-energy36, 37 and surface wetting.38 In this work, we try to propose a new theoretical procedure to analyze the thermodynamics of ice crystallization by combining the merits of RISM and the classical DFT. During the theoretical calculations, the crystal–liquid phase equilibria need to be predicted in advance. As such, we firstly use RISM to calculate direct and total correlation function to illuminate the effect of hydrogen-bond association. With the reasonable direct and total correlation function, the free-energy can be calculated. Then we calculate the vapor−liquid and liquid−solid equilibria using a sample grand canonical ensemble method. Under the condition of liquid-solid equilibrium, the local density distribution in hexagonal ice crystal and the interfacial density distribution of water−ice are determined by a three-dimensional (3D) DFT approach, which is constructed by the direct correlation function obtained from the RISM. The combined theory shares the merits of the two theories in detailed structure description and reliable freeenergy calculation that often require careful calibration of the molecular model. Finally, we consider the nucleation of ice crystal in undercooled water. The free-energy barrier and critical cluster radius are calculated and compared with the corresponding simulation values. This work is expected to provide a theoretical insight to ice nucleation from the molecular level. ACS Paragon Plus Environment
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THEORY AND EQUATIONS The essential work is to study the properties of ice nucleation. To address this issue, the critical nucleus radius and free-energy barrier are the two crucial parameters, which are coherently related to the interfacial structure and free-energy. The details of structure and energy at the liquid–solid interface can be calculated by a density functional method, in which the density at liquid–solid equilibrium acts as the input. Since liquid–solid equilibrium is correlated to vapor-liquid equilibrium, we also present the theoretical model for the vapor–liquid and liquid−solid phase equilibrium calculation. The density functional approach is given to determine the interfacial structure and free-energy. DFT is promising in the description of the interfacial properties.39, 40 For inhomogeneous system, DFT is applied to obtain the density distribution in the equilibrium state by minimizing the grand potential Ω [ ρ (r )] Ω [ ρ (r ) ] = F [ ρ (r )] + ∫ dr ρ (r ) (Vext (r ) − µ )
(1)
where F [ ρ (r ) ] accounts for the Helmholtz free-energy functional, ρ (r ) stands for the density distribution with configuration r , Vext (r ) presents the external potential, and µ indicates the chemical potential of the bulk fluid. For the homogenous system, Vext (r ) = 0 . The functional can conveniently be split into ideal (id) and excess (ex) contributions, i.e.
F ρ ( r ) = F id ρ ( r ) + F ex ρ ( r ) = kBT ∫ drρ ( r ) ln ( ρ ( r ) Λ 3 ) − 1 + F ex ρ ( r )
(2)
where Λ is the thermal de Broglie wavelength with Λ = 2π mk BT h 2 . Formally, F ex ρ ( r ) can be expressed relatively to that of the bulk liquid at the same temperature and chemical potential39, 41
F ex ρ ( r ) = F ex [ ρ 0 ] + µ0ex ∫ dr∆ρ ( r ) −
1 2 drdr ' ∆ρ ( r ) c ( ) ( r , r ', ρ 0 ) ∆ρ ( r ' ) ∫∫ 2
(3)
where ∆ρ ( r ) = ρ ( r ) − ρ0 , ρ0 denotes bulk water density, µ0ex implies the excess chemical potential of the reference liquid, and the function c ( 2) ( r , r ', ρ 0 ) switches to the two-body direct correlation function of oxygen−oxygen. ACS Paragon Plus Environment
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The liquid−solid equilibrium densities are necessary factors in the free-energy calculation. It is generally believed that grand canonical ensemble is fundamental to describe the phase behavior of system and it is better than canonical ensemble by accounting for density fluctuations. Thus, to make the transform much easier in practical calculation of phase equilibrium, we use the grand canonical ensemble method,42 which can be simply represented by
F ( ρ + ϕ ) + Fm ( ρ − ϕ ) Fm +1 ( ρ ) = min m 0
