Ice and Its Formation

Abstract

Ice formation is ubiquitous in nature, and plays a pivotal role in natural processes such as cloud formation and precipitation. Its occurrence in biological organisms is, however, usually problematic, and the main function of antifreeze proteins (AFPs) is to prevent freezing of organisms that inhabit at subzero temperatures. Understanding the physical processes that impact ice formation is, therefore, a prerequisite to elucidating how AFPs function in biological cells and organisms. The primary focus of this chapter is to provide a basic overview of ice and its formation. In particular, the structure of ice and its polymorphism, the thermodynamics and kinetics of ice formation from pure water and aqueous solutions, and the processes that result in the formation of amorphous glassy water will be discussed in detail.

1 Introduction

Freezing of water into ice is a ubiquitous phenomenon that has fascinated mankind for millennia. In addition to biological cells (Padayachee et al. 2009), ice formation can occur in a wide variety of other environments, including atmospheric clouds (Baker 1997; Carslaw et al. 2002), soil (Chamberlain and Gow 1979), freshwater resources (Beltaos and Prowse 2009), aircraft (Potapczuk 2013), and aviation and transportation infrastructure (Ye et al. 2013). The ubiquity and relative ease of freezing water have, since antiquity, been used in applications such as traditional medicine (Yang and Mochizuki 2003), transportation (Li et al. 2013a), mummification (Cockburn et al. 1998), and food preservation (Varshney and Singh 2015) and processing (Arbuckle 1986). Ice formation also plays a pivotal role in many natural processes, such as cloud microphysics (Fowler et al. 1996; Herbert et al. 2015). For instance, freezing of supercooled microdroplets of water in atmospheric clouds triggers a cascade of events that eventually result in precipitation. Predicting the spatiotemporal distribution of ice droplets within a cloud is, therefore, an important—and yet challenging—component of climate modeling (Wilson and Ballard 1999).

Freezing is usually disruptive in biological systems, and can result in cell death. Not only can ice particles damage cell membranes (Muldrew and McGann 1990, 1994), but they can also result in cold denaturation of proteins (Privalov 1990). A large portion of this book is dedicated to understanding how antifreeze proteins (AFPs) prevent and/or control ice formation within cells and organisms. In order to understand how AFPs function, however, it is first necessary to have a fundamental understanding of the microscopic structure of ice, and the thermodynamics and kinetics of its formation.

Considering the ubiquity of freezing and its important ramifications in many disciplines, the question of ice formation has been extensively studied, and this current chapter offers only a brief introduction. In this context, the term “ice” refers to any condensed phase of water that exhibits mechanical properties of a solid. This notion encompasses both crystalline ices, i.e., structures in which the oxygen atoms of constituent water molecules are arranged into periodic lattices, and amorphous ices, or kinetically arrested glassy states of liquid water. Like almost everything else about water, crystalline and glassy ices exhibit various interesting anomalies some of which will be discussed in this chapter.

This chapter is organized as follows. Section 3.2 is dedicated to a brief discussion of the thermodynamics of crystallization. Section 3.3 discusses ice anomalies, including water’s complicated phase diagram and its ability to form open-network crystals. Section 3.4 is dedicated to ice I, the thermodynamically stable crystalline phase of water under ambient conditions, and discusses the thermodynamics and kinetics of its formation in pure water and aqueous solutions. The chapter is concluded with a discussion of amorphous ices in Sect. 3.5.

2 Physics of Crystallization

Before discussing ice, its anomalies, and the thermodynamics and kinetics of its formation, it is useful to provide a quick precept of some basic thermodynamic concepts that will be frequently referred to in this chapter. According to classical thermodynamics, a system will reach thermodynamic equilibrium by minimizing its free energy. The precise notion of what constitutes free energy, i.e., the particular thermodynamic function that needs to be minimized at equilibrium, depends on the thermodynamic ensemble or the set of thermodynamic variables controlled during an experiment. In most experiments, it is the temperature and pressure that are kept constant, and therefore the quantity that needs to minimize at equilibrium is the Gibbs free energy, G, which is related to internal energy E, volume V, and entropy S by:

$$ G=E+ PV- TS $$

(3.1)

Here, P and T are pressure and temperature, respectively. Therefore, under isothermal isobaric conditions, a crystal becomes more stable when its molar Gibbs free energy—also known as chemical potential, μ—is smaller than that of its competitors.

Crystallization is a first-order phase transition in the sense that it involves discontinuities in first derivatives of Gibbs free energy, such as enthalpy, H, and volume. For instance, a latent heat of fusion is released during crystallization, which is due to the fact that the solid has a lower enthalpy than the liquid. Similarly, the crystal and the liquid of the same substance usually have different densities. Despite this discontinuity in first derivatives, a crystal and a liquid can coexist under conditions at which their chemical potentials are equal. The loci of temperatures and pressures at which this can happen are called the coexistence line, with its slope given by the Clausius–Clapeyron equation:

$$ \frac{dP}{dT}=\frac{\Delta {h}_{\mathrm{m}}}{T\Delta {v}_{\mathrm{m}}} $$

(3.2)

Here, Δh_m is the latent enthalpy of fusion and Δv_m is the difference between specific volumes of liquid and the solid. Each point on a solid–liquid coexistence line is technically a melting point, but the term usually refers to the coexistence temperature at ambient pressure, i.e., at 1 bar. The temperature and pressure at which a solid–liquid coexistence line intersects the solid–gas and liquid–gas coexistence lines are called a triple point. Also, coexistence lines that separate two disordered phases, such as gas and liquid, usually end in a critical point at which the transition between the two phases becomes second order and the macroscopic distinction between them, e.g., in terms of density and enthalpy difference, disappears. Beyond the critical point, a transformation between the two phases will become continuous, i.e., without any discontinuities in thermodynamic functions.

3 Anomalies of Crystalline Ices

Water is a material with numerous anomalous properties, the origins of which are far from fully understood (Debenedetti 2003; Gallo et al. 2016). In order to understand anomalies associated with crystalline ices, it is first necessary to discuss what we expect from the crystallization of a typical single-component liquid. For most materials, liquid–solid transition will result in the formation of a single thermodynamically stable crystalline solid, which will always be denser than the liquid at coexistence. Therefore, according to Eq. (3.2), the coexistence line will have a positive slope. From a molecular perspective, this behavior emanates from the fact that intermolecular interactions between most small molecules are nonspecific and nondirectional in nature. Therefore, only the densely packed crystalline structures can be energetically favored, and since there are not very many such structures (Hales 2006; Damasceno et al. 2012), the phase diagram will be simple, with one—or a handful of—crystalline phases only.

Water is a material that deviates from this generic picture as it not only can exist in several distinct crystalline forms, but it can also form crystals that are less dense than the liquid. These two anomalies will be discussed at length in Sects. 3.3.1 and 3.3.2, respectively. A simple conceptual framework to understand them, however, is to note the importance of hydrogen bonding in water. As strong and highly directional specific interactions, hydrogen bonds contribute so strongly to the internal energy of condensed phases of water that such phases can only be stable if their constituent molecules form as many hydrogen bonds with their neighbors as possible. As will be discussed later, this constraint can be satisfied by a wide variety of permitted—including some open network—crystalline structures. This is unlike simple liquids, which, as discussed above, mostly form densely packed crystals.

3.1 Polymorphism in Water

In materials science, polymorphism refers to a material’s ability to form distinct crystalline structures, i.e., structures with distinct sets of point symmetry operations. Each such structure is typically called a polymorph. As explained above, most single-component systems only form one thermodynamically stable polymorph, and therefore have simple phase diagrams with one solid/liquid/gas triple point and one liquid/gas critical point. Water is a notable exception, as evidenced by its very complex phase diagram (Fig. 3.1), and has 18 experimentally observed crystalline polymorphs, denoted by Roman numerals according to their chronological order of discovery. Some of these polymorphs, including five thermodynamically stable ones, can directly form within—and coexist with—liquid water, while the remaining ones are only accessible through solid–solid transformations. Due to this rich polymorphism, the phase diagram of water has five solid–solid–liquid (Bridgman 1912; Schwager et al. 2004; Goncharov et al. 2005, 2009), eight solid–solid–solid (Bridgman 1912; Kauzmann and Eisenberg 1969; Fletcher 1970; Mercury et al. 2001; Song et al. 2003; Salzmann et al. 2009; Yen and Chi 2015), and one gas–solid–solid (Petrenko and Whitworth 1999) triple points, in addition to the generic gas–liquid–solid triple point commonly observed for other materials. Despite their structural differences, all these crystalline ices are similar in the sense that every water molecule in them forms exactly four hydrogen bonds with its neighbors, two as a proton donor and two as a proton acceptor (Zheligovskaya and Malenkov 2006). These constraints are typically referred to as Bernal Fowler rules or ice rules (Bernal and Fowler 1933).

Fig. 3.1

Equilibrium phase diagram of water. Note that ice can exist in many different polymorphs. Reproduced from Zhang et al. (2015)

Ice I is the most common polymorph of ice, and forms at ambient pressures. It has two polymorphs that are stacking variants of one another: the thermodynamically stable hexagonal ice (I_h) and the thermodynamically metastable cubic ice (I_c). The structure of ice I and the thermodynamics and kinetics of its formation will be discussed in detail in Sect. 3.4. Other ice polymorphs can only form under non-ambient conditions, such as very high pressures, and/or very low temperatures. Historically, the first non-ambient polymorph of ice is ice II, which was discovered in the beginning of the twentieth century, and has a rhombohedral unit cell composed of 12 water molecules (Kamb 1964). Ice II can be obtained from ice I at 198 K and 300 MPa (Fuentes-Landete et al. 2015). Since then, 16 more ice polymorphs have been discovered. The most recent polymorphs, ice XVII and XVIII, were only discovered in 2016 (del Rosso et al. 2016a, b) and 2019 (Millot et al. 2019), respectively. Ice polymorphs span a wide range of densities (from 0.85 g cm⁻³ in the case of ice XVII (del Rosso et al. 2016a, b) to 2.785 g cm⁻³ in the case of ice X (Hemley et al. 1987) and ~3 g cm⁻³ in the case of ice XVIII (Millot et al. 2019)) and have unit cells with as few as two molecules—e.g., in the case of ice VII (Kamb and Davis 1964) and ice X (Hemley et al. 1987)—to 136 molecules—e.g., in the case of ice XVI (Falenty et al. 2014).

Polymorphism in water is very rich and unprecedented even in comparison to other tetrahedral liquids such as silicon (Jones and Stevanovíc 2017), which also have multiple polymorphs. One important factor that contributes to this richness is the asymmetric nature of a hydrogen bond. In other words, a hydrogen atom that bridges two oxygen atoms is covalently bonded—and is, therefore, closer—to one of those oxygens. This asymmetry allows for degeneracy in assigning protons to oxygen that are located on a periodic lattice, with the number of all possible assignments scaling exponentially with the number of water molecules in the lattice.¹ There are usually small energetic differences between these distinct arrangements, but such differences are usually too small for any particular arrangement to be favored unless at very low temperatures. Therefore, ice polymorphs at higher temperatures are usually proton-disordered, and the entropy associated with such proton disorder lowers their free energy and enhances their thermodynamic stability. When temperatures are sufficiently low, however, the entropic gain associated with proton disorder will become smaller, and eventually a transition into a proton-ordered phase will take place. Without such proton disorder–order transitions, far fewer ice polymorphs would have existed, as twelve of the eighteen known polymorphs come in pairs that only differ in their proton order (Fuentes-Landete et al. 2015). For instance, ice I transforms into its proton-ordered counterpart, ice XI, at 72 K (Fukazawa et al. 1998). Ice X is the only polymorph in which this hydrogen bonding asymmetry is broken, and each proton is equally spaced between the two oxygens that it bridges (Hirsch and Holzapfel 1984).

In addition to these experimentally observed polymorphs, there are many more polymorphs whose existence has been predicted using computational methods such as density functional theory (Gross and Dreizler 2013) or classical force fields (Guillot 2002; Vega and Abascal 2011). Such polymorphs are typically known as computer ices, and are usually difficult to isolate in experiments. One important class of computer ices constitutes those that are predicted to form at experimentally inaccessible pressures. The most notable example is metallic ice, which is predicted to form at pressures above 4.8 TPa. Metallic ice has the crystallographic space group C₂/m, and its electrons are delocalized like a metal (Hermann et al. 2012). [A possible experimental candidate for metallic ice has only been observed recently, though its electronic properties vis-a-vis the diffusion of protons are yet to be confirmed (Millot et al. 2019).]

Not all computer ices are expected to form at high pressures. For instance, ice 0, which was recently proposed as a precursor for the formation of ice I, has an open tetragonal structure with a unit cell composed of 12 water molecules (Russo et al. 2014). In principle, any crystalline structure that is tetrahedral, satisfies ice rules, and is mechanically stable, can at least be thermodynamically metastable. As to which of these computer ices can form in reality is an open question that can only be addressed experimentally.

The polymorphs discussed so far can form—or be stable—in the bulk. In ultra-confined geometries, such as nanotubes and nanopores, however, water’s hydrogen bond network can be disrupted, which can, in turn, culminate in the formation of novel ice polymorphs. Such polymorphs are typically called two-dimensional ices and can only exist under the geometrical constraints induced as a result of confinement. Historically, such two-dimensional ices were originally discovered in molecular simulations, but the existence of some of them has been recently confirmed in experiments (Zhao et al. 2014b). 2D ices can exist in many different forms, including monolayers (Zangi and Mark 2003; Kumar et al. 2005; Bai et al. 2010; Kaneko et al. 2013; Zhao et al. 2014a; Chen et al. 2016; Corsetti et al. 2016; Barati Farimani and Aluru 2016), bilayers (Koga et al. 1997; Kastelowitz et al. 2010; Johnston et al. 2010; Algara-Siller et al. 2015; Zhu et al. 2016; Chen et al. 2017), and ice nanotubes (Koga et al. 2001; Bai et al. 2006; Takaiwa et al. 2008). Unconventional 2D ice polymorphs can, in theory, form in biological systems due to the presence of ultra-confined geometries within biological cells. Such possibilities have, however, not been thoroughly investigated.

3.2 Formation of Open-Network Crystals

Another interesting consequence of hydrogen bonding in water is its ability to form open-network crystalline structures, or phases that are less dense than the liquid. This rather striking feature is only shared by a handful of other small molecules, e.g., Group IV elements such as carbon and germanium (Brygoo et al. 2007; Jayaraman et al. 1963). Open-network ices typically reside in the low-P region of the phase diagram. Among thermodynamically stable ices, only ice I has an open structure and is ≈8% less dense than liquid water at coexistence. The remaining four ice polymorphs that can coexist with liquid water, i.e., ices III, V, VI, and VII, are all denser than the liquid.

The fact that ice I is less dense than liquid water has important physiological consequences. First of all, one of the reasons that makes ice formation so disruptive to biological cells is the volumetric expansion of liquid water upon freezing. Therefore, ice particles that form intracellularly or extracellularly will occupy a larger volume than the liquid and can, therefore, exert mechanical stress on cell membranes (Meryman 1956). Ecologically, however, this density anomaly is critical to the sustenance of life in lakes and rivers during cold winters, since such freshwater resources freeze from the top and the marine life can still survive within the liquid entrapped below the ice cap.

Another important consequence of this density anomaly is what is known as pressure melting. Indeed, for a liquid that is denser than its crystal, such as liquid water and ice I, Δv_m < 0 and according to Eq. (3.2), the solid–liquid coexistence line will have a negative slope. This will imply that ice I can become unstable upon both heating and pressurization. At temperatures above 251 K, ice will eventually melt at some pressure below ~210 MPa (Fuentes-Landete et al. 2015). Pressure–melting of ice is believed to play a pivotal role in glacier mobility (Weertman 1957).

Ice I is not the only open-network crystal of pure water. In recent years, several other such polymorphs have been discovered. These ices are metastable, can usually form at negative pressures only, and are obtained by exposing two-component crystals of water and small gaseous molecules to vacuum. For instance, ice XVI is obtained by vacuum pumping a clathrate hydrate of neon (Falenty et al. 2014). Gas hydrates are crystalline solids in which water molecules form cages that encompass small guest molecules such as methane, carbon dioxide, hydrogen, and neon (Maslin et al. 2010; Knott et al. 2012), and will be discussed in further detail in Chap. 12 of Vol. 2. At ambient pressures, such cages will collapse in the absence of guest molecules. Applying negative pressure to such hydrates, however, can generate a driving force for the removal of the guest molecules and the subsequent formation of guest free clathrates. Ice XVI is the first experimentally observed guest-free clathrate of water, even though guest-free clathrates had been previously observed in other systems such as germanium (Guloy et al. 2006). Not all open-network ice polymorphs are obtained from applying negative pressures to clathrate hydrates. For instance, ice XVII (del Rosso et al. 2016a, b) is the guest-free variant of a two-component crystal composed of water and hydrogen, which has a quartz-like structure (Strobel et al. 2016).

4 Ice I: The Ambient and Biologically Relevant Form of Ice

As mentioned above, ice I is the only ice polymorph that forms at ambient pressures, and is, therefore, the only form of crystalline ice relevant to biological systems. This section is dedicated to a discussion of its structure and the thermodynamics and kinetics of its formation in pure water and aqueous solutions.

4.1 Structure of Ice I

As mentioned in Sect. 3.3.1, every water molecule in a crystalline ice forms hydrogen bonds with four of its neighbors in accordance with Bernard–Fowler rules. In ice I, each molecule has exactly four neighbors in its first hydration shell, and forms hydrogen bonds with all of them. These nearest neighbors are all arranged tetrahedrally around the central molecule with an average O–O distance of 0.276 nm. As will be discussed in Sect. 3.5, this local tetrahedral arrangement is not unique to ice I and can exist in amorphous ices as well. What distinguishes ice I, however, is a hierarchy of structures that culminate in a periodic lattice with long-range translational order. There are different ways of describing this hierarchy, two of which will be discussed here.

The first approach is based on viewing ice I as a layered structure with water molecules within each layer organizing into hexagons. The two-dimensional projection of each such layer will thus be the honeycomb lattice (Fig. 3.2a, e). Due to the tetrahedral nature of hydrogen bonding, however, these layers are not flat, and their constituent hexagons adopt chair-like configurations. In Fig. 3.2b, f, the oxygens that are above and below the average plane of a hexagon are depicted in purple and green, respectively. Among the four hydrogen bonds that each molecule forms, three are with molecules that are within the same layer, while the fourth is with a molecule in an adjacent layer. This constraint is, however, not sufficient for uniquely determining the arrangement of adjacent layers, and ice I can, therefore, exist in different stacking variants, also known as polytopes.

Fig. 3.2

Polymorphs of ice I. (a–d) Hexagonal ice, I_h, in which adjacent layers are reflected with respect to one another, with their up (square) and down (circular) oxygens sitting on top of one another (a, b). This culminates in a layered structure (d) composed of hexagonal cages (c). (e–h) Cubic ice, I_c, in which adjacent layers have the same orientation, but are shifted with respect to one another by $4\sqrt{2}{R}_{OO}/3$. (e, f) The corresponding layered structure (h) is composed of double-diamond cages (g). (i) Stacking disordered ice, with hexagonal and cubic layers depicted in red and blue, respectively. Stacking faults are shown in light yellow

The most commonly known ice I polytope is hexagonal ice, which is denoted by I_h and has the space group P6₃/mmc (Fig. 3.2a–d). In hexagonal ice, adjacent layers are mirror images of one another, and each upward oxygen (purple) is adjacent to a downward oxygen (green) in the layer above it (Fig. 3.2b). Ice I_h, therefore, has a stacking sequence of ABAB ^…, with A and B corresponding to layers related to one another via reflection. The second idealized polytope of ice I is cubic ice (Fig. 3.2e–h), which is denoted by I_c and has the space group Fd3m (König 1943). In I_c, adjacent layers are not mirror images of one another but are instead related to one another via a translation by a distance $4\sqrt{2}{r}_{OO}/3$. Since each layer will be recovered after three such translations, I_c has a stacking sequence of ABCABC ^…. Both cubic and hexagonal ices are proton disordered, with a zero-point entropy of 3.41 J mol⁻¹ K⁻¹ (Kolafa 2014).

These two polytopes can also be understood in terms of their constituent topological building blocks (Haji-Akbari and Debenedetti 2015), as hexagonal and cubic ices are composed of hexagonal and double-diamond cages, respectively. Hexagonal cages (HCs) (Fig. 3.2c) are anisotropic motifs composed of five hexagons. Two of these hexagons participate in stacking layers, have chair like conformations, and are known as basal faces, while the remaining three have boat-like conformations and are referred to as prismatic faces. Double-diamond cages (DDCs) (Fig. 3.2g) are, however, structurally compact and are composed of six identical chair-like hexagons surrounding an internal chair-like hexagon that participates in the stacking layer. A DDC can, therefore, be viewed as two tetrahedra sitting on top of one another, and rotated with respect to each other by 60°.

Since these two polytopes differ in their stacking sequence only, their lattice energies are almost identical. Hexagonal ice is, however, thermodynamically more stable (Handa et al. 1986), a fact that has recently been attributed to its lower anharmonic nuclear vibrational energy (Engel et al. 2015). Cubic ice, which is thermodynamically metastable, has never been isolated in its single crystalline form. It is instead an idealized representation of ices that form at very low temperatures (Dowell and Rinfret 1960; Kohl et al. 2000; Murray et al. 2005; Shilling et al. 2006) and/or in ultra-confined geometries such as nano-droplets (Johari 2005) and nano-pores (Steytler et al. 1983; Morishige and Uematsu 2005; Morishige et al. 2009). Such metastable ices are usually a mixture of cubic and hexagonal stacks with a high (≈60–70%) fraction of cubic stacks, and are collectively referred to as stacking disordered ice I_sd (Fig. 3.2i). Geometrically, I_sd can be constructed in a layer-by-layer fashion by randomly applying reflection or translation to any new layer that is added on top of the existing layers. The importance of cubic ice and stacking disorder in ice nucleation will be discussed in detail in Sect. 3.4.2. The kinetics of transformation from I_sd into the thermodynamically stable I_h is, however, very slow at low temperatures (Murray et al. 2005; Gainaru et al. 2018). Further information about stacking disordered ice can be found in a review by Malkin et al. (2015).

4.1.1 Crystallography of Hexagonal Ice

Considering the importance of I_h in biological systems, it is worthwhile to discuss its crystallography in further detail. I_h belongs to the hexagonal family of crystals, and has two crystallographic axes (Fig. 3.3a). Its first major axis is denoted by c and is perpendicular to the basal faces of HCs. Its second major axis—denoted by a—lies on a plane perpendicular to c and points toward the vertices of individual hexagons. Note that there are three equivalent directions for a, denoted by a₁, a₂, and a₃ in Fig. 3.3a. These three degenerate vectors, along with c, uniquely determine the unit cell of hexagonal ice. Using such a mathematically overdetermined basis set might seem strange at first, but has important mathematical advantages explained in Allen (2010).

Fig. 3.3

Major crystallographic planes of hexagonal ice. (a) Major crystallographic axes overlaid on the structure of I_h. Note that there are three equivalent a axes. (b) Basal, (c) primary prismatic and (d) secondary prismatic planes

In principle, it is possible to define countably infinite number of distinct crystallographic planes of any crystal by constructing integer valued linear combinations of its basis set. Most such planes will be experimentally uninteresting, either due to their high surface energy, or because of their similarity to simpler crystallographic planes with smaller integer indices. Hexagonal ice has three major crystallographic planes. The basal plane (or the 0001 plane) is perpendicular to the c-axis and correspond to basal faces of HCs (Fig. 3.3b). The basal plane is the lowest energy plane of I_h and has a surface enthalpy of 5.57 μJ cm⁻² (Shultz et al. 2015). The primary prismatic plane—or the 01$\overline{1}$0 plane—is perpendicular to the a-axis (Fig. 3.3c), and has a surface enthalpy of 5.95 μJ cm⁻² (Shultz et al. 2015), while the secondary prismatic plane—or the 11$\overline{2}$0 plane—is perpendicular to a prismatic face and has a surface enthalpy of 6.90 μJ cm⁻² (Shultz et al. 2015). These crystallographic planes are important in understanding the function of antifreeze proteins, as different AFPs can preferentially attach to different crystallographic planes of ice, and affect the mechanism of ice formation—or lack thereof—accordingly (Olijve et al. 2016; Haji-Akbari 2016).

4.2 Thermodynamics and Kinetics of Ice Formation in Pure Water

4.2.1 Thermodynamics of Ice Formation

In Sect. 3.2, a brief overview is provided on the basics of the thermodynamics of crystallization, and concepts such as melting point and triple point are introduced. Table 3.1 summarizes the thermodynamic properties of ice I and its corresponding liquid–solid transition. Ice I has a melting temperature of T_m = 273.15 K, and has an enthalpy of fusion of 1.436 kcal mol⁻¹ at T_m. The gas–liquid–solid triple point of water is at 273.16 K and 61 Pa. Note that the triple point temperature is slightly higher than T_m due to the negative slope of the ice I–liquid coexistence line. Also, water has an unusually high melting point in comparison to similar molecules, a fact that has been attributed to the dominant role that hydrogen bonding plays in condensed phases of water.

Table 3.1 Thermodynamic and structural properties of hexagonal ice (Petrenko 1993)

4.2.2 Kinetics of Ice Formation: Nucleation and Growth

According to classical thermodynamics, a supercooled liquid, i.e., a liquid below its equilibrium melting temperature, is thermodynamically unstable and will eventually freeze into its respective crystal. But the fact that such a transformation is thermodynamically favored does not imply that it will occur within experimentally accessible timescales. Indeed, supercooled water—or any other supercooled liquid for that matter—can remain liquid, sometimes for very long times. Such a delay arises from the fact that crystallization—like any other first-order phase transition—proceeds through a mechanism known as nucleation and growth (Fig. 3.4). Nucleation is an activated process and involves the formation of a sufficiently large crystalline nucleus within the supercooled liquid, while growth refers to the subsequent enlargement of that nucleus. The activated nature of nucleation arises from the free energetic penalty associated with the formation of a solid–liquid interface. This penalty, which exists due to structural mismatch between an ordered crystal and a disordered liquid, scales linearly with the surface area of the crystalline nucleus, and is eventually overcome by the thermodynamic driving force for crystallization, which scales linearly with the volume of the nucleus. For small nuclei, however, this surface contribution is dominant and makes nuclei that are smaller than a critical size thermodynamically unstable. This makes the formation of a critical nucleus a stochastic activated process unless its free energy of formation is comparable to thermal fluctuations in the system.

Fig. 3.4

(a) Ice can nucleate either homogeneously (e.g., in pure water in the absence of a facilitating surface) or heterogeneously in the vicinity of an interface. (b) Schematic free energy of the system as a function of nucleus size for both homogeneous and heterogeneous nucleation

The qualitative argument provided above is made more rigorous in classical nucleation theory (CNT). Proposed in the early twentieth century (Volmer and Flood 1934; Becker and Döring 1935), CNT is the most widely used theoretical framework for crystal nucleation (Oxtoby 1992). In the simplest variant of CNT, it is assumed that crystalline nucleus is spherical in shape, and the properties of the solid and the liquid are bulk-like and are not affected by the presence of an interface. The free energy of formation of a crystalline nucleus composed of N molecules will then be given by (Debenedetti 1996):

$$ G(N)=\kern0.37em -\frac{N\left|\Delta \mu \right|}{N_{\mathrm{A}}}+{\left(\frac{36\;\pi }{\rho_{\mathrm{s}}^2}\right)}^{1/3}{\sigma}_{\mathrm{ls}}{N}^{2/3} $$

(3.3)

Here, Δμ is the chemical potential difference between the liquid and the solid, N_A is the Avogadro number, and ρ_s and σ_ls are the solid number density and the solid–liquid surface tension, respectively. Equation (3.3) has a maximum at:

$$ {N}_{\mathrm{c}}=\frac{\sigma_{\mathrm{ls}}}{\left|\Delta \mu \right|}\ {\left(\frac{32\pi }{3{\rho}_{\mathrm{s}}^2}\right)}^{2/3} $$

(3.4)

The free energy of formation for a nucleus with N_c molecules is called the nucleation barrier and is given by:

$$ \Delta {G}_{\mathrm{nuc}}=G\left({N}_{\mathrm{c}}\right)=\frac{N_{\mathrm{c}}}{2{N}_{\mathrm{A}}}\left|\Delta \mu \right|=\frac{16{\pi \sigma}_{\mathrm{ls}}^3{N}_{\mathrm{A}}^2}{3{\rho}_{\mathrm{s}}^2{\left|\Delta \mu \right|}^2} $$

(3.5)

Due to this non-monotonicity in G(N), forming a critical nucleus is an uphill battle thermodynamically. Assuming that a steady-state nucleus size distribution is established at timescales much shorter than the nucleation time, it can be demonstrated that a fluctuation resulting in barrier crossing will have an occurrence probability proportional to exp[−ΔG_nuc/kT] with k the Boltzmann constant. The nucleation rate, or the expected number of nucleation events per unit time per unit volume, will then be given by:

$$ R=\frac{24{\rho}_{\mathrm{liq}}{DN}_{\mathrm{c}}^{3/2}}{l^2}\kern0.24em {\left(\frac{\left|\Delta \mu \right|}{6\pi { kT N}_{\mathrm{c}}}\right)}^{\frac{1}{2}}\ \exp \left[-\frac{\Delta {G}_{\mathrm{nuc}}}{kT}\right] $$

(3.6)

with ρ_liq and D the density and the self-diffusivity of the liquid, and l the atomic jump distance in the liquid. Note that the stochastic nature of nucleation will imply that the nucleation time τ will be a stochastic random variable and will have a mean given by 〈τ〉 = 1/(RV), with V the volume of the supercooled liquid. It is generally assumed that nucleation is a Poisson process² and nucleation time is exponentially distributed (Koop 2004). Upon the formation of a critical nucleus, however, there is no longer any thermodynamic resistance for its further growth, and the growth timescale is governed by the pace of structural relaxation in the system.

Depending on the magnitude of the nucleation barrier, either nucleation or growth can be the rate-limiting step in crystallization. When the nucleation barrier is very large, i.e., when ΔG_nuc ≫ kT, crystallization will be nucleation limited and nucleation times will be orders of magnitude larger than growth times. For ΔG_nuc ≈ kT, however, crystallization will be growth limited as a large number of nucleation events will occur simultaneously in the system, and the crystallization timescale will be determined by the intrinsic dynamics of the system, most importantly by its self-diffusivity.

In order to identify the conditions under which crystallization is nucleation limited or growth limited, it is first necessary to examine how |Δμ| and σ_ls change with thermodynamic variables such as temperature. In general, solid–liquid surface tension is extremely difficult to measure in the deeply supercooled regime, and even though it is known that σ_ls is an increasing function of temperature, its precise dependence on T is not known due to technical difficulties of measuring it far from equilibrium (Gŕańasy et al. 2002). |Δμ|, however, can be rigorously calculated from integrating the equations of state of the supercooled liquid and the crystal:

$$ \left|\Delta \mu \right|=T{\int}_T^{T_{\mathrm{m}}}\frac{\left|{h}_{I_{\mathrm{h}}}-{h}_{\mathrm{liq}}\right|}{{\overline{T}}^2}d\overline{T} $$

(3.7)

with ${h}_{I_{\mathrm{h}}}$ and h_liq the molar enthalpies of ice I_h and the supercooled liquid, respectively. Assuming that molar enthalpies are not strong functions of temperature, it can be easily shown that:

$$ \Delta \mu =\Delta {h}_{\mathrm{m}}\left[1-\frac{T}{T_{\mathrm{m}}}\right] $$

(3.8)

where Δh_m is the enthalpy of fusion at T_m. Combining Eqs. (3.5) and (3.8) yields:

$$ \frac{\Delta {G}_{\mathrm{nuc}}}{kT}=\frac{16{\pi \sigma}_{\mathrm{ls}}^3\;{N}_{\mathrm{A}}^2\;{T}_{\mathrm{m}}^2}{3 kT\;{\rho}_{\mathrm{s}}^2\;\Delta\;{h}_{\mathrm{m}}^2{\left({T}_{\mathrm{m}}-T\right)}^2}=\frac{B}{kT{\left(T-{T}_{\mathrm{m}}\right)}^2} $$

(3.9)

Therefore, the nucleation barrier depends on ΔT = T_m − T, which is typically referred to as the degree of supercooling.³ Equation (3.9) predicts that ΔG_nuc is a decreasing function of ΔT, an assertion that is generally true even if Eq. (3.8) is not valid. Therefore, nucleation rate should increase exponentially upon decreasing temperature. Such an increase in rate is, however, eventually offset by the slowdown of intrinsic dynamics at low temperatures. Assuming that transport properties such as self-diffusivity have an Arrhenius-type dependence⁴ on temperature, i.e., D ~ exp[–E_a/kT], the overall crystallization rate will be given by:

$$ R=A\kern0.24em \exp \left\{-\frac{1}{kT}\left[\underset{\mathrm{diffusion}}{\underbrace{\ {E}_{\mathrm{a}}\ }}+\frac{B}{\underset{\mathrm{nucleation}}{\underbrace{\ \Delta {T}^2}}}\right]\right\} $$

(3.10)

Therefore, nucleation-limited crystallization will occur at small supercoolings where the rate of increase in exp[–ΔG_nuc/kT] is much faster than the slowdown in D. Growth-limited crystallization, however, will take place under conditions at which Eq. (3.10) is dominated by the E_a/kT term, i.e., at large supercoolings. In the middle, there will always be a temperature at which the nucleation rate will be the largest. Figure 3.5 depicts the schematic dependence of nucleation rate on ΔT, with the nucleation- and growth-limited regimes depicted in green and blue, respectively.

Fig. 3.5

Schematic representation of the dependence of the rate of ice nucleation on temperature. Nucleation- and growth-limited regimes are depicted in green and blue, respectively

The discussion presented so far only applies to homogeneous nucleation in which a critical nucleus forms in the absence of an external surface. In the case of water, barriers for homogeneous nucleation are so large that it can only occur at temperatures below ≈235 K, and is practically impossible for temperatures above –20 °C (Sanz et al. 2013). With extra care, water microdroplets can be supercooled for temperatures as low as 227 K (Sellberg et al. 2014). All our daily experience of freezing, therefore, involve heterogeneous nucleation in which an external surface facilitates freezing by decreasing nucleation barriers. There are a wide variety of materials that can heterogeneously nucleate ice, including mineral dust (Chen et al. 2008; Zimmermann et al. 2008; Hoose et al. 2010; Atkinson et al. 2013; Kiselev et al. 2017), volcanic ash (Schumann et al. 2011), soot (Jensen and Toon 1997; Chen et al. 2008; Hoose et al. 2010), organic materials (Wilson et al. 2015), ammonium sulfate particles (Hoose and Möhler 2012), and biological materials such as pollen and bacteria (Chen et al. 2008; Hoose et al. 2010; Joly et al. 2013; Wilson et al. 2015; O’Sullivan et al. 2015) and diatoms (Knopf et al. 2011). Interestingly biological materials are among the most potent ice-nucleating agents (Murray et al. 2012).

An extension of CNT for heterogeneous nucleation was formulated by Turnbull (1950), in which it is assumed that the crystalline nucleus is a spherical cap and its shape does not change during nucleation. According to CNT for heterogeneous nucleation, the nucleation barrier will be given by $\Delta {G}_{\mathrm{nuc}}^{\mathrm{het}}=f\left({\theta}_c\right)\Delta {G}_{\mathrm{nuc}}^{\mathrm{homo}}$. Here, θ_c is the three-phase contact angle between the crystal, the supercooled liquid and the external surface, and $f\left({\theta}_{\mathrm{c}}\right)=\frac{1}{4}{\left(1-\cos\;{\theta}_{\mathrm{c}}\right)}^2\left(2+\cos\;{\theta}_{\mathrm{c}}\right)$ is called the potency factor. Note that whenever θ_c < π, i.e., when the external surface has a higher propensity for the crystal than for the supercooled liquid, the barrier for heterogeneous nucleation will always be smaller than the barrier for homogeneous nucleation at the same temperature. Therefore, the temperature at which the nucleation rate becomes maximum increases in heterogeneous nucleation, but the overall qualitative picture depicted in Fig. 3.5 remains unchanged. Also, the nucleation rate in heterogeneous nucleation should be expressed as the number of nucleation events per time per area, and nucleation time will have a mean given by 〈τ〉 = 1/(RS) with S the total surface area of all ice-nucleating particles in the system. It is, however, usually possible to obtain an effective volumetric nucleation rate by accounting for the volumetric concentration of ice-nucleating particles, and the surface area of each such particle.

Another remarkable feature about heterogeneous nucleation is that it does not occur on solid-like surfaces only, but can also take place at a fluid–fluid interface. For instance, it has been demonstrated that amphiphilic alcohol monolayers (Gavish et al. 1990; Popovitz-Biro et al. 1994; Seeley and Seidler 2001) and terpenes (Rosinski 1980; Rosinski and Lecinski 1981; Rosinski et al. 1990) can facilitate ice nucleation. A more interesting possibility, which has been studied extensively both in experiments (Djikaev et al. 2002; Tabazadeh et al. 2002; Kuhn et al. 2011) and simulations (Gianetti et al. 2016; Vrbka and Jungwirth 2006; Pluhárová et al. 2010; Li et al. 2013b; Haji-Akbari et al. 2014; Haji-Akbari and Debenedetti 2017a), is that water–vapor interfaces can facilitate ice nucleation at their vicinity. This latter scenario is called surface freezing and has been predicted to become dominant in submicron microdroplets (Kuhn et al. 2011). Due to the difficulty of generating monodispersed submicron droplets, however, whether surface freezing occurs or not is still controversial (Sigurbjörnsson and Signorell 2008) and is regarded as one of the ten biggest open questions about ice and snow (Bartels-Rausch 2013). A detailed discussion of surface freezing can be found in an earlier review (Haji-Akbari and Debenedetti 2017b).

4.2.3 Ice Nucleation and Ice I Polytopes

As mentioned earlier, hexagonal ice is the thermodynamically stable form of ice I under ambient conditions, and the type of ice that forms at temperatures close to T_m. However, this does not necessarily imply that the crystalline nuclei that emerge during nucleation are exclusively hexagonal. Indeed, the ice that nucleates homogeneously at low temperatures (Dowell and Rinfret 1960; Mayer and Hallbrucker 1987; Kohl et al. 2000; Murray et al. 2005; Shilling et al. 2006) or in ultra-confined geometries (Johari 2005; Steytler et al. 1983; Morishige and Uematsu 2005; Morishige et al. 2009) is stacking disordered, with a considerable fraction of cubic stacks (Malkin et al. 2012; Amaya et al. 2017). If temperatures are sufficiently high, stacking disordered ice will eventually anneal to the thermodynamically stable hexagonal ice (Murray et al. 2005; Gainaru et al. 2018).

The fact that a liquid would nucleate into a thermodynamically metastable crystal is not surprising and has been observed in other systems (ten Wolde and Frenkel 1999). It has indeed been argued that if that metastable crystal has a lower solid–liquid surface tension than the thermodynamically stable crystal, it will be the first to nucleate in the liquid. The emerging nucleus will then nucleate the thermodynamically stable phase from within when it becomes sufficiently large (ten Wolde et al. 1996). This is typically called Ostwald’s step rule, and stipulates that crystallization of a liquid into the thermodynamically stable crystal will usually occur in stages (Van Santen 1984). Ostwald’s step rule has been invoked to explain the formation of I_sd during ice nucleation (Takahashi 1982). It has, however, not been established whether I_c and I_sd have lower solid–liquid surface tensions than the thermodynamically stable hexagonal ice. Furthermore, the validity of Ostwald’s step rule at a molecular level cannot be confirmed using existing experimental techniques due to their insufficient spatiotemporal resolution.

In recent decades, molecular simulations have shed more light on the molecular origin of stacking disorder in ice nucleation. Molecular simulations utilizing different water models and simulation techniques have indeed revealed the formation of stacking disordered ice during nucleation (Brukhno et al. 2008; Moore and Molinero 2010, 2011a, b; Johnston and Molinero 2012; Reinhardt and Doye 2012; Reinhardt et al. 2012; Haji-Akbari and Debenedetti 2015, 2017a; Lupi et al. 2017). In unbiased molecular dynamics simulations of the monoatomic water (mW) potential (Molinero and Moore 2009), a coarse-grained model of water, it has been observed that homogeneous nucleation in the deeply supercooled regime results in the formation of nuclei that are stacking disordered, and their fraction of cubic stacks increases when they become larger, eventually reaching a plateau of ∼60% (Moore and Molinero 2010, 2011a, b). The formation of I_sd is, however, not confined to mW and has been observed for atomistic models of water as well (Brukhno et al. 2008; Reinhardt et al. 2012; Haji-Akbari and Debenedetti 2015, 2017a).

So far, two distinct explanations have been proposed for the emergence of stacking disorder during ice nucleation. The first explanation is kinetic in nature, and was provided by Haji-Akbari and Debenedetti (2015), who utilized atomistic simulations and forward-flux sampling (FFS) (Allen et al. 2006) to compute the rate and explore the mechanism of homogeneous ice nucleation for TIP4P/Ice (Abascal et al. 2005), one of the most accurate atomistic models of water. They monitored the presence of HCs and DDCs, the topological building blocks of I_h and I_c, and observed that DDC-rich nuclei have smaller surface-to-volume ratios and are therefore more likely to grow and become post-critical. HC-rich nuclei, however, emerge mostly as a result of the attachment of new HCs to prismatic faces of existing HCs. The dominance of prismatic growth culminates in HC-rich nuclei that are composed of chains of hexagonal cages, and that have large surface-to-volume ratios. The fact that new HCs are added prismatically has a kinetic origin, and results in disfavoring HC-rich crystallites and favoring the formation of stacking disordered ice with a high fraction of cubic stacks. This mechanism has been confirmed in a later study utilizing a more accurate variant of the FFS technique (Haji-Akbari 2018). Computational studies of bulk ice also confirm that ice growth is the fastest at prismatic planes, presumably due to their higher surface energies (Seo et al. 2012; Rozmanov and Kusalik 2012).

The second explanation is thermodynamic in nature and was proposed by Lupi et al. (2017), who used molecular dynamics simulations and the aimless shooting method (Mullen et al. 2015) to compute the free energy landscape for homogeneous nucleation in the mW system, as a function of nucleus size and cubicity. They concluded that the entropy associated with mixing cubic and hexagonal layers makes stacking disordered nuclei more stable than their pure cubic and hexagonal counterparts. This entropic contribution is proportional to the number of stacks, and scales as N ^1/3, with N the number of water molecules in the nucleus. In bulk ice, i.e., for N → ∞, this entropic contribution will be negligible in comparison to the free energy difference between I_c and I_h, which will scale as $\mathcal{O}(N)$. For small nuclei with a few hundred water molecules, however, this entropic contribution will be significant and will decrease the nucleation barrier by enhancing the stability of stacking disordered nuclei. It is necessary to note that these explanations are not mutually exclusive, and the preference for stacking disordered ice is likely a consequence of both thermodynamic and kinetic effects.

4.2.4 Melting of Ice I: Gibbs–Thomson Equation and Recrystallization

Similar to freezing, melting of crystals, such as ice, is a first-order transition, proceeds through nucleation and growth, and can be delayed at temperatures higher than T_m for kinetic reasons. In other words, similar to liquid water that can be supercooled and remain liquid at temperatures below T_m, ice can also be superheated and remain crystalline at temperatures above T_m (Iglev et al. 2006). What is different about melting, however, is that every crystal has an intrinsic stability limit, i.e., a temperature above which it becomes mechanically unstable due to high-amplitude thermal vibrations of its constituent atoms and/or molecules (Lindemann 1910). Homogeneous nucleation of the liquid can, however, occur before this stability limit is reached, as confirmed in several experimental (Wang et al. 2012, 2015) and computational (Mochizuki et al. 2013; Samanta et al. 2014) studies. In reality, most crystals melt heterogeneously at temperatures that are only slightly higher than T_m and are thus much lower than this stability limit. Heterogeneous melting can be induced at external surfaces, or at internal defects such as grain boundaries. What is remarkable about heterogeneous melting is the ability of most external surfaces to destabilize crystals even at temperatures below T_m, resulting in the formation of a thin pre-melted quasi-liquid layer in their vicinity. This phenomenon—also known as premelting—has been extensively studied for ice (Dash et al. 1995; Bluhm et al. 2002; Sánchez et al. 2017) and other materials (Ronchi and Hiernaut 1996; Kristensen and Cotterill 1977) in recent decades. A crystal that is in contact with a surface that induces premelting cannot be superheated, as at any T > T_m, the pre-melted layer will immediately destabilize the bulk crystal at its vicinity and will result in its melting without a need to cross a nucleation barrier.

Considering the preponderance of surface premelting, superheating is not very common, and in order for a crystal to be superheated, it needs to be engulfed by surfaces that do not induce premelting. This can, for instance, be achieved by embedding nanocrystals of the target crystal within a matrix of a second crystal with a higher melting point, a method that has been used to superheat metals (Banhart et al. 2003) and Group IV elements such as germanium (Xu et al. 2006). Crystals that have been frozen within solutions with kinetic inhibitors such as antifreeze proteins can also be superheated, as such inhibitors can bind to crystalline particles and provide the engulfment discussed above in an irregular manner. Therefore, the ice that forms in the presence of antifreeze proteins can be superheated (Knight and DeVries 1989; Celik et al. 2010; Cziko et al. 2014), a process that will be discussed in further detail in other chapters.

It is necessary to emphasize that confinement does not only affect the kinetics of melting, but can also impact its thermodynamics. For instance, the equilibrium melting temperature can change considerably for water entrapped in nanopores (Findenegg et al. 2008; Sliwinska-Bartkowiak et al. 2008; Moore et al. 2012). One of the most notable examples of how confinement can affect the thermodynamics of melting is the depression of equilibrium melting point for small crystalline particles. In order to understand the origin of this depression, consider a spherical ice particle of radius r, which, due to its surface curvature, will experience a heightened internal pressure. The pressure difference between the crystal interior and the surrounding liquid is called the Laplace pressure and can be estimated using the Young–Laplace equation (Chen et al. 2006):

$$ \Delta P=\frac{2{\sigma}_{\mathrm{ls}}}{r} $$

(3.11)

This ΔP will increase the chemical potential of the crystal by Δμ = v_iceΔP = 2σ_ls/ρ_icer and will make it thermodynamically unstable at the bulk melting temperature T_m. Instead, thermodynamic equilibrium will be achieved at a lower temperature T_m(r) where this curvature-induced penalty is offset by the bulk thermodynamic driving force given by Eq. (3.8). It can be shown using simple thermodynamic reasoning that T_m(r) will be given by:

$$ {T}_{\mathrm{m}}(r)={T}_{\mathrm{m},\mathrm{bulk}}-\frac{2{\sigma}_{\mathrm{ls}}\;{T}_{\mathrm{m},\mathrm{bulk}}}{\rho_{\mathrm{s}}\;r\;\Delta {h}_{\mathrm{m}}} $$

(3.12)

Equation (3.12) is known as the Gibbs–Thomson equation (Jackson and McKenna 1990; Johari 1997). A similar argument can be made to obtain the depressed T_m for arbitrarily shaped surfaces (Johnson 1965; Jones 1974), including a spherical cap of radius r sitting on top of an ice-nucleating surface. A closely related equation is the Kelvin equation, which describes the dependence of the equilibrium vapor pressure of a droplet as a function of its size (Thomson 1872):

$$ P(r)=P\left(\infty \right)\kern0.24em \exp\ \left[\frac{2{\sigma}_{\mathrm{l}\mathrm{v}}}{\rho_{\mathrm{l}} rkT}\right] $$

(3.13)

with P (r) and P (∞) the equilibrium vapor pressure of a droplet of radius r, and bulk liquid, respectively, σ_lv, the liquid–vapor surface tension, and ρ_l the number density of the liquid.

An important consequence of this size-dependent stability is a phenomenon known as recrystallization or Ostwald ripening (Voorhees 1985), which refers to the growth of larger crystalline particles at the expense of smaller ones. This occurs when the nucleation barrier is so small that a large number of post-critical nuclei emerge within the liquid. However, not all such nuclei will be identical in shape and size. Such polydispersity will generate a thermodynamic driving force for the growth of larger nuclei at the expense of smaller ones, which, as explained above, are less stable. One of the important functions of AFPs is to inhibit recrystallization, i.e., to prevent small crystalline nuclei—that are still larger than their surrounding nuclei—from growing any larger. This is important from a biological perspective since smaller crystallites are less harmful to biological cells (Knight et al. 1984). The ability of AFPs to prevent recrystallization will be discussed in detail in Chap. 7 of Vol. 2.

4.2.5 Thermal Hysteresis

Nucleation is, in essence, an out-of-equilibrium process in which a thermodynamically metastable system, such as a supercooled liquid, turns into a thermodynamically stable structure, such as a crystal. Therefore, when the coexistence line is crossed, e.g., by cooling a liquid, freezing does not occur immediately. Instead, the system remains in the metastable supercooled state until nucleation occurs. However, when nucleation is complete, the system reaches thermodynamic equilibrium, and remains there until the coexistence line is crossed in the opposite direction. Therefore, the behavior of a system that crosses a coexistence line depends on the direction of such crossing. This is known as hysteresis and is an important feature of all first-order phase transitions, including crystallization. As explained in Sect. 3.4.2, hysteresis can occur for both ice formation and ice melting, but is less common in the case of melting. An important manifestation of this is thermal hysteresis in biological systems, which refers to a situation in which a biological solution can exist as a liquid at temperatures below T_m (Finger and Bischof 2018) or as a solid at temperatures above T_m (Knight and DeVries 1989). In other words, at finite timescales, temperature is not sufficient for determining whether a system is frozen or not, and the system might behave differently based on its initial state and processing history. One measure of an AFP’s potency is its ability to induce the largest possible thermal hysteresis, which will enable cells and/or organisms to survive freezing at lower temperatures.

4.3 Ice Formation in Solutions

So far, only the thermodynamics and kinetics of ice formation in pure water has been discussed. Biological water, however, is never pure, and in order to explore freezing in biological systems, it is important to understand how soluble impurities affect ice formation. The conceptual approach for analyzing freezing in aqueous solutions will depend on whether any of the solutes participates in the thermodynamic stable form of the aqueous crystal. Most solutes are excluded from the crystal, and therefore the freezing of the corresponding solution results in the formation of pure ice I. As mentioned in Sect. 3.3.2, however, there are notable exceptions to this rule including aqueous solutions of small gaseous molecules that can crystallize into gas hydrates (Maslin et al. 2010; Knott et al. 2012). Here, the focus will be on the former scenario, i.e., on solutes that do not participate in the ice structure, and on their impact on the thermodynamics and kinetics of freezing.

4.3.1 Thermodynamics of Ice Formation in Solutions

The effect of soluble entities on the thermodynamics of ice formation is well understood, and is discussed in detail in all standard thermodynamics textbooks. In summary, adding a solute to water results in a decrease in its chemical potential given by:

$$ {\mu}_{\mathrm{w},\mathrm{liq}}\left(T,P,{x}_{\mathrm{w},\mathrm{liq}}\right)={\mu}_{\mathrm{w},\mathrm{liq}}^0\left(T,P\right)+ RT\ln {a}_{\mathrm{w},\mathrm{liq}} $$

(3.14)

Here, x_w,liq is the mole fraction of water and a_w,liq < 1 is a dimensionless quantity known as activity. As mentioned earlier, thermodynamic coexistence is achieved when the chemical potentials of water in the liquid and the crystal become identical. Assuming that the solute does not participate in the thermodynamically stable crystal structure, the condition for thermodynamic coexistence between the solution and ice I will be given by:

$$ \ln {a}_{\mathrm{w},\mathrm{liq}}\left({T}_{\mathrm{m}}\right)=\frac{\mu_{\mathrm{w},\mathrm{ice}}^0\left({T}_{\mathrm{m}},P\right)-{\mu}_{\mathrm{w},\mathrm{liq}}^0\left({T}_{\mathrm{m}},P\right)}{RT_{\mathrm{m}}}=-\frac{\left|\Delta \mu \right|}{RT_{\mathrm{m}}} $$

(3.15)

Since a_w,liq < 1, Eq. (3.15) can only be satisfied if ${T}_{\mathrm{m}}<{T}_{\mathrm{m}}^{\mathrm{pure}}$. This decrease in equilibrium freezing temperature is typically referred to as freezing point depression, which is defined as $\Delta {T}_{\mathrm{m}}:= {T}_{\mathrm{m}}^{\mathrm{pure}}-{T}_{\mathrm{m}}\left({x}_{\mathrm{w}}\right)$. Assuming the availability of |Δμ(T)|, which can be obtained by integrating the equation of state, and the activity data versus temperature, which can, for instance, be readily obtained from vapor pressure—or fugacity—measurements, the depressed freezing point can be accurately calculated from Eq. (3.15). A simpler approach can, however, be employed for dilute solutions, i.e., solutions in which x_S, the total mole fraction of all solutes in the system, is much smaller than unity, i.e., when x_S ≪ 1. Under such circumstances, water activity will approach its mole fraction, and its chemical potential can be expressed as:

$$ {\mu}_{\mathrm{w},\mathrm{liq}}\left(T,P,{x}_{\mathrm{w},\mathrm{liq}}\right)={\mu}_{\mathrm{w},\mathrm{liq}}^0\left(T,P\right)+ RT\ln \left(1-{x}_{\mathrm{S}}\right) $$

$$ \overset{x_{\mathrm{S}}\ll 1}{\approx }{\mu}_{\mathrm{w},\mathrm{liq}}^0\left(T,P\right)-{x}_{\mathrm{S}} RT $$

(3.16)

Combining Eqs. (3.8) with Eqs. (3.15 and 3.16) yields:

$$ {T}_{\mathrm{m}}\left({x}_{\mathrm{S}}\right)=\frac{T_{\mathrm{m}}^{\mathrm{pure}}}{1+\frac{x_{\mathrm{S}}{RT}_{\mathrm{m}}^{\mathrm{pure}}}{\Delta {h}_{\mathrm{m}}}}\approx {T}_{\mathrm{m}}^{\mathrm{pure}}\left[1-\frac{x_{\mathrm{S}}{RT}_{\mathrm{m}}^{\mathrm{pure}}}{\Delta {h}_{\mathrm{m}}}\right] $$

(3.17)

Therefore, freezing point depression in dilute solutions is a colligative property as it only depends on the overall molal concentration of the solutes and not on their chemical identities. Other notable colligative properties include vapor pressure depression, boiling point elevation, and osmotic pressure. Due to their relatively large molar masses, aqueous solutions of most biomolecules can be usually considered dilute, and Eq. (3.17) can usually be used for computing ΔT_m.

4.3.2 Kinetics of Ice Formation in Solutions

The presence of solutes in liquid water will alter its molecular structure, such as its hydrogen bond network. Such changes will inevitably affect the kinetics—and possibly the mechanism—of ice nucleation, and considering the out-of-equilibrium nature of nucleation and its strong sensitivity to even modest changes in thermodynamic properties, it is not a priori clear how the nucleation kinetics will be affected by the identities and concentrations of the corresponding solutes. Even if we assume the validity of the classical nucleation theory, it is not trivial to determine how properties such as solid–liquid surface tension will depend on solute concentration. The situation is even more hopeless in the case of heterogeneous nucleation, since an ice-nucleating surface can interact differently with water molecules and the solutes, and such differences can induce compositional anisotropies that can, in turn, affect the kinetics and mechanism of ice nucleation in nontrivial ways.

Despite these complexities, there seems to be overwhelming experimental evidence for the colligative nature of nucleation kinetics. In 1977, Kanno and Angel discovered that the amount of decrease in the kinetic freezing temperature⁵ of an aqueous alkali halide solution is a colligative property and depends on cation concentration only (Kanno and Angell 1977). This idea was later generalized in the pioneering work of Koop et al. (2000) who studied sixteen different aqueous solutions, and demonstrated that water activity is sufficient for predicting the nucleation kinetics, irrespective of the identity of the solute. The qualitative argument for this empirical observation is that water activity is a proxy for how much its hydrogen bond network is perturbed in the presence of solute(s). This empirical framework is known as the activity shift model and works remarkably well for most simple solutes. Small deviations from it, however, have been observed in several systems (Swanson 2009; Knopf and Rigg 2011). Antifreeze proteins are among the most remarkable example of solutes that deviate from this colligative picture, resulting in kinetic suppressions considerably larger than what would otherwise be expected from their concentrations (Dolev et al. 2016; Olijve et al. 2016; Haji-Akbari 2016). The activity shift model was originally proposed for homogeneous nucleation. Interestingly, the kinetics of heterogeneous nucleation has also been shown to follow the activity shift model (Zobrist et al. 2008; Wilson and Haymet 2009; Knopf and Alpert 2013; Zeng et al. 2015) with minor deviations also reported in some systems (Cantrell and Robinson 2006).

Despite its success in predicting the kinetics of ice nucleation in aqueous solutions, the activity shift model does not provide any information about how the nucleation mechanism is affected by a solute. For instance, the presence of soluble impurities has been shown to affect polymorph selection (i.e., the ratio of cubic to hexagonal ice) in deeply supercooled water (Murray and Bertram 2007; Murray 2008). In addition, there is a wide gap in our understanding of when and why the activity shift model is violated, and studying such violations can be a potent area of research.

4.3.3 Solute Exclusion and Its Consequences

As mentioned above, most aqueous solutions crystallize into ice I, and their constituent solutes do not participate in the crystal structure. This results in a phenomenon known as solute exclusion or solute rejection in which the freezing of an aqueous solution results in the formation of ice crystals that coexist with regions composed of more concentrated—and possibly saturated—aqueous solutions. Solute rejection has very important consequences in biological systems. Not only such concentrated regions can be toxic simply due to their high solute concentrations, but there will also be a steep osmotic gradient between them and less concentrated solutions, e.g., in the cytoplasm. Such a gradient will induce a strong water flow, which can, in turn, exert mechanical damage to cell membranes. This osmotic shock scenario is thought to be an important mechanism of ice toxicity. Interestingly, osmosis-induced ice toxicity usually occurs when freezing is extracellular, in which case water flow will be from the cytoplasm to the extracellular region (Muldrew and McGann 1990, 1994).

5 Amorphous Ices

The focus of Sect. 3.4 was to discuss water’s equilibrium behavior at ambient pressures and subfreezing temperatures. This section, however, will deal with out-of-equilibrium glassy states of water that share the molecular structure of liquid water but, due to their slow dynamics, exhibit mechanical properties of a solid. Like most other liquids, water can be turned into a glass via a number of means including—but not limited to—rapidly quenching the liquid. Such kinetically arrested states are known as amorphous ices and the process employed for making them is called vitrification. Amorphous ices can avoid crystallization within experimentally accessible timescales as long as they are kept at temperatures below the glass transition temperature T_g, which, for liquid water, is ∼136 K at ambient pressures (Fuentes-Landete et al. 2015). Note that crystallization can be delayed for relatively long times even at temperatures slightly above T_g, since in order for the viscous water to crystallize, a large number of molecular rearrangements will still need to occur. At ambient pressures, for instance, crystallization will become sufficiently fast to be considered instantaneous at or above T_x ∼ 150 K (Fuentes-Landete et al. 2015). In the interstellar space, temperatures are usually much lower than T_g. Consequently, amorphous ice is the predominant form of ice in the outer space (Mitchell et al. 2017). This is in contrast to earth where subfreezing temperatures are much higher than T_x, and therefore it is the crystalline ice that is prevalent.

Amorphous ices are out of equilibrium in nature, and their properties depend heavily on their processing history. Consequently, classifying amorphous ices based on their structural and physical properties is not trivial, and there are a large number of names and acronyms currently in use in the literature to describe different types of amorphous ices. In terms of processing history, amorphous ices are broadly classified into five categories as depicted in Fig. 3.6a. Historically, glassy water was first obtained through a process in which water vapor was physically deposited onto the surface of a cold copper rod at temperatures below 130 K (Burton and Oliver 1935). Vapor-deposited amorphous ice is typically referred to as amorphous solid water (ASW), is usually very porous, and can have densities as low as 0.3 g cm⁻³. Upon annealing, however, ASW can undergo pore collapse and its specific surface area can decrease by as much as 2–3 orders of magnitude (Baragiola 2003). Vapor deposition is the primary mechanism of vitrification in comets and in the interstellar space (Mayer and Pletzer 1986).

Fig. 3.6

Metastable phase diagram of liquid water. (a) Procedures for preparing different types of amorphous ices, adapted and simplified from Fig. 8 of Fuentes-Landete et al. (2015). (b) O–O radial distribution functions for LDA, HDA, and VHDA adapted and replotted from Finney et al. (2006). (c) Metastable phase diagram of liquid water (Reproduced from Amann-Winkel et al. 2016)

Hyperquenched glassy water (HGW) is a type of glassy water obtained via rapid quenching of water microdroplets, which are then deposited onto the surface of a cold “cryoplate” (Mayer 1985). In order for HGW to form, the quenching rate must be ~10⁶ K/s or higher. High-density amorphous ice (HDA) is another type of amorphous ice, obtained via pressure melting of hexagonal ice, e.g., at 77 K and 1.0 GPa (Mishima et al. 1984). Annealing HDA at low and high pressures results in the formation of low-density amorphous ice (LDA) (Mishima et al. 1985) and very high-density amorphous ice (VHDA) (Loerting et al. 2001), respectively.

Three of these five classes, i.e., ASW, HGW, and LDA, are low-density amorphous ices, and share the same molecular structure. The local environment of each water molecule within LDA (and other related low-density glasses) is remarkably similar to that of ice I in the sense that each molecule has four tetrahedrally arranged nearest neighbors and there is a clear separation between the first and second hydration shells, as manifested in the oxygen–oxygen radial distribution functions depicted in Fig. 3.6b. Unlike ice I, however, LDA lacks long-range translational order. The local tetrahedrality of the first hydration shell is disrupted in both HDA and VHDA in which the first hydration shell is penetrated by one and two water molecules from the second shell, respectively, as evident from the emergence of a weak second peak at ≈0.4 nm in HDA (Finney et al. 2002b) and ≈0.35 nm in VHDA (Finney et al. 2002a). Therefore, from a structural perspective, amorphous ices are classified into three categories: LDA, HDA, and VHDA.

Another remarkable feature of these three structurally distinct amorphous ices is that they can be reversibly (Winkel et al. 2008) transformed into one another without crystallizing. The transition from LDA to the other two types is particularly sharp and is believed to be first order in nature (Finney et al. 2002a; Lemke et al. 2017). The notion that a material can exist in multiple amorphous phases that can reversibly transform into one another via first-order transitions is called polyamorphism. Studying glassy water polyamorphism is particularly interesting in the context of low-temperature water anomalies (Ball 2008; Brovchenko and Oleinikova 2008). There are a large number of such anomalies in the literature, including density maximum at 4 °C, isothermal compressibility minimum 46 °C and isobaric heat capacity minimum at and 35 °C. The origin of such anomalies are, however, yet to be fully understood, and four distinct scenarios have been proposed to explain them (Stanley and Teixeira 1980; Stanley et al. 1981; Speedy 1982; Poole et al. 1992; Sastry et al. 1996; Angell 2008; Stokely et al. 2009). Among these scenarios, the liquid–liquid transition hypothesis (Poole et al. 1992) is inspired by amorphous ice polyamorphism and attributes water anomalies to the existence of two distinct liquids in the supercooled regime, namely the low-density liquid (LDL) and the high-density liquid (HDL). According to this hypothesis, HDL and LDL are separated by a coexistence line that culminates in a liquid–liquid critical point (LLCP), and are the liquid counterparts of LDA and HDA, respectively. As can be seen in the metastable phase diagram of supercooled water in Fig. 3.6c (Amann-Winkel et al. 2016), the predicted liquid–liquid critical point is located below the homogeneous nucleation line and is therefore not accessible to existing experimental techniques. In recent years, there have been numerous experimental efforts to access this LLCP (Sellberg et al. 2014; Laksmono et al. 2015). Also, molecular simulations have been extensively utilized to understand liquid–liquid transition in water (Poole et al. 1992; Errington and Debenedetti 2001; Sastry et al. 1996; Palmer et al. 2014, 2018a) and other tetrahedral liquids (Saika-Voivod et al. 2000; Sastry and Angell 2003; Xu et al. 2005; Smallenburg et al. 2014). The topic of liquid–liquid transition in supercooled water is, however, far from resolved and is an active area of research (Palmer et al. 2018b).

Due to their metastable nature, amorphous ices should, in principle, crystallize at sufficiently long times. Crystallization, however, becomes practically implausible below T_g, and only becomes instantaneous at temperatures considerably higher than T_g. The process in which an amorphous solid turns into a crystal is called devitrification or recrystallization. Note that the general meaning of the term “recrystallization” is already introduced in Sect. 3.4.2. Crystallization within a glass is a growth-limited process resulting in the formation of a large number of nuclei, and therefore falls into the general framework of recrystallization introduced in Sect. 3.4.2.

Different amorphous ices can freeze into different crystalline phases. LDA crystallizes within experimentally accessible timescales at around 150 K and transforms into stacking disordered ice with a large fraction of cubic stacks (Johari et al. 1996; Johari 1998). High-density amorphous ices, however, crystallize into other ice polymorphs depending on pressure and processing history. For instance, HDA has been shown to crystallize into ice VII and ice VIII (Hemley et al. 1989), and ice IX (Seidl et al. 2013). VHDA, however, can transform into a wide variety of ice polymorphs at different pressures (Klotz et al. 2005). Amorphous ices can also be directly transformed into liquid water if they are heated so quickly that crystallization is avoided. The heating rates needed for this to occur are, however, around two orders of magnitude larger than the cooling rates needed for vitrification.

Similar to pure water, rapid quenching of an aqueous solution can result in the formation of an amorphous solid. As outlined in Sect. 3.4.3, the presence of solutes results in melting point depression, and a considerable decline in the nucleation rate. Meanwhile, aqueous solutions usually have higher viscosities in comparison to pure liquid, and their viscosities are stronger functions of temperature (Angell 2002). This deceleration of dynamics can become very pronounced in concentrated aqueous solutions to the extent that their T_g can considerably exceed that of pure water. For instance, concentrated divalent electrolytic solutions such as Mg(NO₃)₂ and Ca(NO₃)₂ can have T_g’s that are tens of degrees Kelvin higher than that of pure water (Angell and Sare 1970), and aqueous solutions of sugars such as trehalose can have T_g’s at or above room temperature (Green and Angell 1989). For most solutes, however, the extent of increase in T_g is not that large, but it can be generally stated that aqueous solutions have lower T_m’s and higher T_g’s. Consequently, homogeneous ice nucleation can occur over a narrower range of temperatures. This will imply that vitrifying an aqueous solution can be achieved at cooling rates considerably lower than what would be needed for vitrifying pure water. Similarly, it will be easier to bypass crystallization during the thawing of such glasses, as the required heating rates will also be considerably lower.

5.1 Biological Applications of Vitrification

Vitrification is, in essence, a process in which the liquid is kinetically arrested without considerable change to its microscopic structure. Therefore, whenever a biological cell is vitrified, its structure and the spatial distribution of its composition remain unchanged, unlike freezing that can damage it through a number of means, such as volumetric expansion, solute exclusion, and osmotic shock. Vitrification is therefore of considerable interest in cryopreservation applications (Fahy et al. 1984). Vitrified cells and tissues can, in principle, be preserved for years, if not decades, and considering that biological cells are composed of concentrated aqueous solutions of electrolytes, carbohydrates, and other biomolecules, they can be vitrified (and thawed) at considerably lower cooling (heating) rates in comparison to what will be needed for pure water. Yet, it is still difficult to vitrify entire tissues and organs due to possible heat transfer limitations. In other words, when a tissue or organ is rapidly cooled, its interior will be at a higher temperature than its surface at all times, and such temperature differences can result in the freezing of the interior while the surface is being vitrified. A similar problem can arise during thawing, and the tissue or organ can freeze from within due to inadequate heat transfer. Therefore, vitrification is generally used for preserving individual cells, such as sperms (Isachenko et al. 2004), oocytes (Kuwayama et al. 2005; Cobo et al. 2008) and embryos (Rall and Fahy 1985; Rall 1987), or small tissues, such as cryo-electron microscopy specimen (Dubochet et al. 1988).

6 Conclusions

Water is a material that exhibits numerous anomalies, and as discussed in detail in this chapter, its crystallization is also anomalous in many different ways. In particular, water can form a large number of crystalline polymorphs, some of which are less dense than the liquid water at coexistence. In all these ice polymorphs, each water molecule is hydrogen bonded to four of its neighbors, with two as a proton donor, and two as a proton acceptor. The polymorph of ice that forms under ambient pressures is ice I, which is a layered structure in which each water molecule has four neighbors within its first hydration shell. Ice I has two stacking variants—or polytopes—known as hexagonal (I_h) and cubic (I_c) ice. Hexagonal ice is the thermodynamically stable polytope, and is the form that emerges upon freezing under physiologically relevant conditions. Structurally, I_h is anisotropic and has three major crystallographic planes with differing surface energies. Cubic ice, which is structurally less anisotropic, is metastable and only becomes relevant during crystallization at very low temperatures.

As a first-order phase transition, ice formation proceeds through a nucleation and growth mechanism, and can occur both homogeneously and heterogeneously. At temperatures close to T_m, the thermodynamic driving force for crystallization is small, and freezing is nucleation-limited. At deep supercoolings, however, nucleation becomes almost instantaneous and crystallization becomes growth-limited. What is particularly interesting about ice I is that its homogeneous nucleation can be delayed for temperatures as low as 235 K. Freezing of liquid water into ice is therefore primarily heterogeneous under physiologically relevant conditions.

Freezing is generally slower in aqueous solutions, in comparison to pure water. Not only the thermodynamic freezing temperature is depressed in the presence of soluble impurities, but also the nucleation rate decreases considerably. For most solutions, the deceleration of nucleation kinetics can be predicted using the activity shift model, which is based on the empirical observation that solutions with identical water activities will freeze at the same rate at any given temperature.

Finally, water can be transformed into an amorphous solid via a number of means, including rapid quenching of the liquid, or depositing water vapor onto the surface of a cold substrate. Such glassy states are known as amorphous ices, and exhibit the mechanical properties of a solid despite being structurally liquid-like. Amorphous ices form during vitrification, which is widely used in cryopreservation applications.