Chapter 2: Fundamentals of Controlled Freezing for Ice‐templated Porous Materials – Ice Templating and Freeze-Drying for Porous Materials and Their Applications

2
Fundamentals of Controlled Freezing for Ice‐templated Porous Materials

2.1 Introduction

Freezing has been known to be an effective way for food storage and cell cryopreservation [1, 2]. Because molecules, particles, or impurities can be rejected and condensed between ice crystals during freezing, it has also been used as a drying technique or a method for cleaning water [3]. Owing to the phase separation that results from freezing, the removal of ice crystals by sublimation or washing after gelation of a concentrated solute phase can produce ice‐templated porous materials or fibrous structures. This was probably first demonstrated by the formation of silica fibres after Mahler et al. froze a silica sol [4]. More examples have been reported on the preparation of porous ceramics, biomaterials, and polymers [57]. Some early examples include the initial description of freeze casting by Maxwell et al. in 1954 for fabricating dense ceramics [8], porous polymers from agar gels in 1984 [9], porous ceramics in 2001 [10], and biomaterials from collagen [11].

The term ‘ice templating’ is referred to the process where ice crystals are used as template and a porous material can be generated after removing such a template. This process is also widely termed as freeze casting (particularly for preparation of porous ceramics) or simply freeze drying (although this name does not accurately describe the process) [57, 12]. The ice‐templating method has been widely used to produce a wide range of porous materials with a variety of morphologies. Because ice crystals are used as template, the ice‐templated structure and/or pore morphology can be controlled by the various ice morphologies that are formed during the freezing process. Therefore, the fundamental study on the formation and behaviour of ice crystals under controlled freezing conditions is critical for the production of advanced porous materials with desirable properties. Indeed, the investigation on freezing aqueous polymer and particulate systems is of wider scientific interests such as in frozen soils, freezing sea water/glaciers, and cryopreservation [12].

Although some organic solvents other than water have been used for ice‐templated porous materials, the fundamental study in controlled freezing in literature has been nearly exclusively focused on the systems using water. In this chapter, the basics of ice crystals is first described. In the freezing or solidification of aqueous systems, the majority of the articles in literature are based on aqueous particulate systems, either experimentally or theoretically. Different mathematic models describing the freezing phenomena have been developed, which are evidenced by experimental observation. We will then introduce the instruments or techniques used for observation or investigation of freezing phenomena and follow this by description/discussions of models/equations and observations in the freezing of aqueous particulate systems.

2.2 The Basics of Ice Crystals

Ice crystals may be formed when the temperature of the water is lowered below the equilibrium freezing point (i.e. 0 °C) and nucleation can be started. In practice, ice crystals do not form just below the equilibrium freezing point because a large energy barrier must be overcome so that water molecules may move closer and become ordered [1]. Supercooled water (e.g. as low as −40 °C) is usually required to initiate the nucleation and crystallization [13]. Very often, foreign particles (or impurities) and even the wall of the container act as nuclei to promote the growth of ice crystals.

There are about 10 crystalline phases available for ice crystals, as shown in Figure 2.1 [14]. Most of these phases are stable in a certain range of temperature and pressure. The structures of these crystalline phases, mostly investigated by the neutron‐diffraction crystallographic method, can be rationalized in terms of fully connected tetrahedral networks of water molecules, with each molecule donating hydrogen bonds to two neighbours and accepting two hydrogen bonds from two others. Although the oxygen atoms are related to an underlying lattice, the hydrogen atoms are in disorder [14]. The crystalline phases that are encountered daily, which are formed from atmospheric pressure and a temperature not lower than −200 °C, is the hexagonal ice (Ih). In the Ih, the O–O–O angles are close to the ideal tetrahedral angle of 109.47° [14]. The density of Ih is lower than that of water, which is essential for the lift on Earth. However, with the increase of pressure, other forms of ice crystals can be formed, with a higher density than water [15]. As the pressure is increased, water molecules have to rearrange themselves to occupy less volume. This can change the network structure (which can still remain tetrahedral) and increase the distortion of the O–O–O angles, e.g. the angles varying between 80° and 129° [14].

Figure 2.1 The solid–liquid phase diagram of ice (1 GPa = 10 000 bars).

Source: Bartels‐Rausch et al. 2012 [14]. Reprinted with permission from APS.

Ih shows a lattice of ‘puckered’ hexagonal crystal structure, with the basic ice crystal growth form being the hexagonal prism [16]. The overall morphology of natural ice crystals (e.g. snowflakes) or ice crystals formed under relatively ambient conditions can take different forms, e.g. they can be plate‐like, columnar, or dendritic, depending on the relative growth rate of ice crystals on the basal facet and prism facet. At ambient conditions, there exists another crystalline form, the cubic ice (Ic). Ic is metastable and is thought to exist in the coldest regions of the Earth. Ic is believed to be a benign form of ice crystals for cryopreservation. However, the observation of Ic is difficult because it can rapidly transform to the stable hexagonal phase Ih. The structures of both Ih and Ic consist of six‐membered puckered rings. However, the difference between them lies in the stacking of these layers [13]. It is also thought that Ic may be a poorly crystallized phase and may be placed between the crystalline and amorphous phases [14]. From X‐ray diffraction data and Monte Carlo simulations, Malkin et al. showed that ice crystals formed homogeneously from supercooled water (231.7 ± 1.0 K) were neither of Ih and Ic phases, but were composed of randomly stacked layers of cubic and hexagonal sequences [13]. Based on this study, they argued that almost all ice that had been identified as Ic was most likely stacking‐disordered ice with varying degrees of stacking disorder.

The morphologies of ice crystals have been mostly observed and investigated for snowflakes or snow crystals. Snow crystals grow from water vapour and they have been known for their beautiful and symmetrical patterns. When snow crystals are formed in air at atmospheric pressure near 1 bar, the morphologies vary, depending on the change of temperature and water vapour supersaturation [16]. Although the ice crystals are formed from a liquid phase in the ice‐templating technique, the morphologies of snow crystals grown from water vapour may be controlled by careful selection of freezing conditions when freezing aqueous solutions or suspensions. Indeed, most of the snow crystal morphologies have been observed in ice‐templated materials, as discussed by Deville in the review article [12].

In a study related to the ice templating or freeze‐casting technique, Shibkov et al. established the morphology diagram of ice crystal patterns growing freely in supercooled water [17]. The study classified eight non‐equilibrium macroscopic structures from heterogeneous nucleation of ice in pure water at atmospheric pressure and the temperature range of −0.1 to −30 °C. The morphology spectrum of non‐equilibrium crystal of ice crystals includes fragment of dense‐branching structure, dendrite, needle‐like crystals, fractal needled branch, compact needled branch, and the platelet, depending on the range of supercooling temperatures. Some of the structures co‐exist. Three different scenarios, tip‐splitting, dynamical tip oscillation, and selective application of noise, were thought to induce the initial perturbations and the formation of side‐branches. Based on the results of the real‐time tip‐velocity measurements, the morphology diagram of ice crystals (average tip velocity υt vs supercooling temperature ΔT) was constructed (Figure 2.2) [17]. It is clear that the ice crystal morphology could be tuned by varying the supercooling temperature/tip velocity. The morphology transition between different phases is achievable and can be investigated. For example, the transition between stable needle and platelet was found to be a first‐order kinetic transition because there was a υtT) jump by a factor of about 2 at about 7.5 °C.

Figure 2.2 Morphology diagram of non‐equilibrium growth patterns in υt–ΔT phase space: •, dendrite; ○, stable needle‐like crystal; solid square, fractal needled branch; solid triangle, compact needled branch; hollow triangle, platelet.

Source: Shibkov et al. 2003 [17]. Reprinted with permission from Elsevier.

2.3 Instruments and Techniques for Investigation of Freezing Aqueous Particulate Systems

When the ice‐templating technique or freeze casting is used to produce porous materials, the dissolution of polymers/small molecules and the suspended particles can considerably impact the growth of ice crystals. The morphologies of ice crystals and hence the ice‐templated pore structures can be directly related to the freezing system and the freezing condition. In order to investigate how the ice crystals grow, the impact of particles (the studies on impact of polymers are highly limited), and the interaction of particles and freezing front, a variety of instruments and techniques have been employed and are described further.

2.3.1 Optical Microscope Equipped with a Freezing Stage

The sizes of ice crystals or ice lenses during freezing of aqueous suspensions are normally in the micron range, which can be observed by a standard optical microscope. Of course, the macroscopic ice lenses rather than single ice crystals are observed. A CCD camera can be attached to the optical microscope to record the images or videos. The key element in the set‐up is the freezing stage that should allow directional solidification or freezing of aqueous suspensions. This is probably the most used technique discussed in literature to observe the initiation and growth of ice lenses [1821].

An example of such a freezing stage set‐up is illustrated in Figure 2.3 [18]. A similar system was used by our group [19] and a photo of such a set‐up is given in Figure 1.8. In order to investigate the directional freezing of aqueous colloidal silica suspensions (average diameter of ∼1.5 μm), a Linkam GS350 stage was employed with a home‐built Köhler lens microscope [18]. The temperature of the two metal plates, with a gap of 2 mm between them, is controlled by electrical resistance heaters and a pump‐controlled flow of liquid nitrogen. The temperature gradient is created by the difference in the temperatures of the two plates. Usually, the temperature of a cold plate (Tc) is lower than the freezing temperature Tf while the temperature of a warm plate (Tw) is higher than Tf. A computer‐controlled stepper motor holds the sample cell in contact with the plates and pulls along the temperature gradient (from the warm plate to the cold plate) at a defined moving velocity. The sample depth is small so that the ice lens can be clearly observed from the gap between the two plates. The sample cell is usually contained within a sealed box to prevent the condensation of water vapour, which could make the clear observation very difficult.

Figure 2.3 Schematics of a freezing stage set‐up used to directionally freeze the aqueous silica colloidal suspension. (a) The sample is contained within two glass walls and has depth h. (b) Initial and final positions of two thermocouples, represented by square and triangle symbols, placed inside the sample.

Source: Schollick et al. 2016 [18]. Reprinted with permission from American Chemical Society.

Different freezing stages may be used [1921]. To allow the observation of directional solidification of ice lens, they should always have two independently temperature‐controlled plates and a facility to allow the accurate moving velocity of samples across the plates.

2.3.2 Optical Interferometry

Optical interferometry is a good technique to study interfacial instability and dendritic ice growth because of its ability to measure small relative changes in refractive index in two dimensions. This method has been applied in the form of Michelson, Mach–Zehnder, wedge, and holographic interferometry. For example, Mach–Zehnder optical interferometry (a schematic diagram is shown in Figure 2.4) has been used to measure the concentration gradient at the ice–solution interface and the ice–solution interface morphology [21]. Like the optical microscope, a temperature gradient microscope stage (TGMS) is required to directly freeze the sucrose and pullulan solutions. Similarly, the TGMS consists of two plates kept at different temperatures with a gap of 5 mm. A drop of the solution is placed on a glass slide surrounded by a 75 μm spacer on the top of which a standard cover slip is placed. The slide is placed across the two plates and held by a movable sprung plate connected to a motor capable of moving the sample across the gag at speeds between 0.5 and 80 μm s−1. A constant stream of dry nitrogen gas is passed over the sample to prevent condensation from forming on the top of the coverslip.

Figure 2.4 Schematic diagram of the Mach–Zehnder optical interferometer.

Source: Butler 2000 [21]. Reprinted with permission from American Chemical Society.

In the Mach–Zehnder set‐up, an interference pattern, consisting of dark and light areas, is produced when the two beams are recombined, which can be analysed to obtain information on the refractive index. Usually, reference fringes are set up prior to introducing the samples into the sample path. The solution concentrations may be derived by analysing the optical difference from fringe positions, using the following equation [21]:

(2.1)

where C is the local solution concentration, C0 is the bulk concentration, λ is the wavelength, Δχ is the deviation of fringe position from that of the reference fringe, n(C) is the refractive index of the solution with concentration C, n0 is the refractive index of pure water (1.333), h is the sample thickness, and w is the separation of the reference fringes.

The ice–solution interface morphology can be observed directly from the interference images using the interferometer. The change of position of the interface as the planar interface develops instability is used to calculate the growth velocity at the point of instability.

2.3.3 Cryogenic Transmission Electron Microscopy (CryoTEM)

Optical microscopy cannot be used to investigate the nanoscale structure or the crystalline polymorphs of ice crystals due to its limited magnification power. Cryogenic transmission electron microscopy (CryoTEM) is ideal for direct experimental observation of hexagonal or cubic ice polymorphs. First of all, the aqueous solution or suspension must be frozen for the observation. To allow the electron beam to go through the sample, the thickness of the sample must be carefully controlled. However, because of the high vacuum in the CryoTEM system and also because the high current density of the electron beam irradiation can generate heat at the sample, the ice sample may be easily sublimed, thus causing great difficulty in observing the ice sample.

Dillon and co‐workers developed an in situ CryoTEM sample stage by joining an environmental cell with a copper cold finger stage [22]. This platform could access nanoscale information about ice crystallization. A gold nanoparticle (∼30 nm) suspension was pipetted (about 100 nl) onto a cleaned 50 nm thick SiNx window. A second SiNx window sealed the liquid within the cell. An average thickness of ∼300–500 nm was detected. A low beam current density (∼0.5–1.5 mA cm−2) was used to minimize the heating and charging effects. Both images and videos were recorded on the observation. The microstructures of ice crystallized at 220–260 K were imaged using this technique. It was found that the growth of nanotwinned cubic ice aligned along the plane with dendrite‐like morphology occurred between 220 and 245 K. Hexagonal ice was formed at 260 K growing along the plane with a planar ice/water interface [22].

A high‐resolution TEM was used by Kobayashi et al. to investigate the morphology and crystal structure of ice nanoparticles [23]. Both Ih and Ic crystals were found and assigned. The dynamic transition between Ih and Ic phases of individual ice nanoparticles was monitored by in situ transmission electron diffractometry. The preparation of ice nanoparticles was quite delicate. A TEM microgrid, wetted with distilled methanol, was held in a cryo‐holder. The grid was quenched by dipping the tip of the holder into liquid nitrogen, and subsequently cooled by supplying liquid nitrogen. The grid was then exposed to a moist atmosphere with a specific water vapour pressure of 1.2 kPa and a temperature of 298.9 K for 20 s. The specimen was then directly induced into a TEM column (vacuum of ∼10−5 Pa, keeping the low temperature). The TEM was operated at 120 kV with the point resolution better than 0.106 nm. The average size of the ice nanoparticles was around 390 nm and the larger ice nanocrystals tended to be faceted. To avoid instant evaporation of ice nanoparticles under the incident beam, the exposure time was minimized to 0.5 s, from a very thin region with a rapidly moving and evaporating frontier.

2.3.4 X‐ray Radiography and Tomography

Compared to the techniques mentioned earlier (and also the scanning electron microscopy (SEM) that can usually only image the surface of a freeze‐dried and/or sintered sample), X‐ray radiography and tomography provide obvious advantages: transparent materials are no longer required because opaque materials can be imaged by X‐ray absorption contrast; a high spatial resolution (μm) can be obtained on synchrotron instrument; and a three‐dimensional (3D) reconstruction of the particle arrangement and the ice crystals after complete freezing is possible [24, 25]. X‐rays are at the high energy end of the electromagnetic spectrum, with the energy level in the approximate range of less than 1 to over 100 keV. Although there is no clear classification, X‐rays with energy levels higher than 10 keV are termed as hard X‐rays while X‐rays with energy levels lower than 10 keV are termed as soft X‐rays. The wavelengths of soft X‐rays are in the order of 1 nm while the wavelengths are in the fraction of nanometres for hard X‐rays. Because of such short wavelengths (in the similar length region as atoms), the reflection is minimal when the X‐ray hits a material surface. The X‐ray can penetrate between the atoms and then interact with atoms and absorbed by the material. Different materials exhibit different attenuation coefficients, which are determined by the electron density of the material and the atomic weight of its chemical elements.

There are different modes of X‐ray imaging [26]. For X‐ray absorption imaging, as traditionally used in medical applications, there is almost no distance between the sample and the detector. However, for a homogeneous material with low attenuation coefficient or a heterogeneous material with a range of small attenuation coefficients, it is insufficient to produce absorption image with satisfactory contrast. Thus, X‐ray phase contrast imaging has been developed where there is a fixed distance between the sample and the detector. Phase contrast is possible because the beam is not only absorbed by the material but that the phase of the wave is also affected, depending on the material's refractive index. This technique requires a homogeneous and spatially coherent beam. Based on this, holotomography is introduced, which uses a combination of several distances as well as combines the phase shift information to produce phase maps that are subsequently used for the tomographic reconstruction. This technique can be employed for the materials that have very small variations in attenuation coefficients, which lead to insufficient imaging results even with phase contrast techniques. Holotomography is defined as quantitative phase tomography with micrometre resolution using coherent hard synchrotron radiation X‐rays [27]. The approach is based on the Fresnel diffraction approximation and is efficiently implemented using fast Fourier transform (FT) functions. The use of different distances reduces the challenge of finding a local optimum or a non‐unique solution. It should be mentioned that only two dimensional (2D) images (radiographs) can be produced by X‐ray imaging. However, once the phase maps are obtained via holographic reconstruction (disengaging the phase map from the image), it is then rather straightforward to bring many maps corresponding to different orientations of the sample together and produce the 3D tomographic reconstruction with quantitative information. For example, the quantitative phase maps of polystyrene foam from 700 angular positions were used to produce a 3D tomographic reconstruction, giving the 3D distribution of the refractive index decrement, or of the electron density [27].

The nucleation and growth of ice crystals during solidification of aqueous alumina slurries were investigated by direct in situ high‐resolution X‐ray radiography and tomography on the beamline ID19 of European Synchrotron Radiation Facility (ESRF) [25]. The energy was set to 20.5 keV. The distance between the sample and the detector was 20 mm. A set of 1200 projections were taken within 180°. Radiographs taken with different viewing angles of the sample provided a dynamic visualization of the solidification process. Using the HST program at the ESRF, the computing reconstruction provided a 3D map of the local absorption coefficients in each elementary volume of the sample from the set of absorption radiographs. An example of the radiographs showing global instabilities and the 3D local observation by X‐ray tomography of ice crystals are given in Figure 2.5. With this study, it was possible to establish global instabilities and the structure diagrams for the solidification of colloidal suspensions [25]. The dynamics of the freezing front during the solidification of aqueous alumina slurry was investigated by in situ X‐ray radiography, tomography, and modelling [28]. Freezing of alumina slurry was carried out by pouring it into a polypropylene cylindrical mould, invisible to X‐rays, placed at the tip of a copper cold finger inside a cryogenic cell. A set of 900 projections were taken within 180°. This work showed that the control of the morphology of packed alumina structures in the freezing direction could be achieved by choosing the suitable temperature profile at the base of the suspension. The effects of additives during directional freezing of aqueous titania suspensions were investigated by synchrotron X‐ray radiography on beamline 8.3.2 at the Advanced Light Source (Berkeley, CA) [29]. The freezing stage consisted of a cylindrical Teflon mould placed on a copper cold finger with the temperature controlled using liquid nitrogen via a thermocouple linked to a power unit and ring heater. The cooling rate was 2.5 °C min−1 starting from the ambient temperature. Samples were scanned with a 24 keV monochromatic beam. A field view of 3 mm was obtained using 5 × lens and a voxel size of 1.7 μm. It was concluded that adding binder could induce a transition from a disorientated structure to a lamellar structure. A depletion mechanism could be used to explain the particle redistribution at the ice/water interface.

Figure 2.5 Experimental observations of the development of global interface instabilities and their impact on the ice structure when directionally freezing aqueous alumina (0.2 μm particles) suspension. (a) The radiograph showing bright bands (less absorption of the beam, particles‐poor region) and interface displacement direction. (b) 3D local observation by X‐ray tomography of ice crystals, revealing lateral growth of ice crystals from localized (white arrows) and global (black arrows) instabilities.

Source: Deville et al. 2009 [25]. Reprinted with permission from Nature Publishing Group.

2.3.5 Small Angle X‐ray Scattering

In general, the X‐ray scattering techniques collect the information on variation of a sample's electron density to generate contrast. A spatial variation of electron density at the nanometre scale scatters an X‐ray beam to low angles while the high angle scattering results from the atomic scale [30]. Small angle X‐ray scattering (SAXS) is a technique used to characterize the samples based on a spatially averaged intensity distribution between the scattering angles of 0° and 5°. SAXS can provide detailed structural analysis and physical information for a variety of 1–100 nm and beyond particles or polymers, in various states (e.g. gaseous, liquid, solid) [30, 31]. SAXS uses small beams (submillimetre down to a micrometre). Thin samples with low to intermediate atomic number density for low absorption or intermediate to high atomic number density for scattering contrast can usually generate good quality data. This technique can benefit from the extreme brightness of X‐ray synchrotron source that provides excellent photon counting statistics [32]. The theoretical foundation for SAXS can be correlated with both the form factor and structure factor. The SAXS profile has three distinct regions [33, 34] that can be used to extract for information on the radius of gyration (the Guinier region), cross‐section structures (the Fourier region), and surface per volume (the Porod region) (Figure 2.6). In the Guinier region, the radius of gyration Rg may be obtained by fitting a line to the natural log of the intensity as a function of the square of the scattering vector q2. In the Fourier region, the pair distribution function may be determined by an indirect Fourier transformation of the experimental form factor, providing significant information regarding the particle shape P(q). In the Porod region, by determining the Porod invariant Q, it can provide surface information such as the surface to volume ratio and specific surface estimation for compact particles [33].

Figure 2.6 Diagram showing the regions of SAXS profile and the type of data that can be extracted from each region.

Source: Boldon et al. 2015 [33]. Reprinted with permission from Taylor & Francis.

During the process of freezing aqueous particulate suspension, the particles can be rejected from the freezing front and pack densely between the ice crystals. Understanding the packing at the particle scale is important both for the theoretical description and the potential for production of advanced ice‐templated materials. However, none of the above techniques described can provide information about the particle‐scale structure during freezing or for a completely frozen sample. CryoTEM would have the high resolution required to observe the particles. However, the densely packed particles region will have very low transparence and make it really difficult if not impossible to be observed by CryoTEM.

However, this structural information on the packing particles during freezing may be obtained by SAXS, via a Fourier‐space representation of the mass distribution within the samples on the scale of one to several times the particle radius. The directional solidification of aqueous silica colloidal suspensions (radii of silica colloids about 32 nm) was investigated by Spannuth et al. using SAXS [35]. The sample chamber within the cell was formed by sandwiching an approximately 400 μm thick aluminium washer between two copper blocks. The actual thickness of the chamber could be varied between 200 and 400 μm due to the flexibility of the windows combined with manual positioning of the cold finger abutting one window. The X‐ray scattering experiments were carried out at beam line 8‐ID of the Advanced Photon Sources at Argonne National Laboratory. It was first verified that the scattering pattern was isotropic and did not change significantly while acquiring a set of images. Thus, the images could be averaged azimuthally and over time to produce the intensity as a function of scattering vector I(q). The normalized intensity curve is given as in the following equation [35]:

(2.2)

where d is the cell thickness, Tr is the transmission coefficient, Vpart is the average particle volume, and Δρ is the electron density difference between silica and water or ice. The coefficients are grouped to the amplitude A which is q‐independent. P(q) is the particle form factor and S(q) is the structure factor.

One typical example of the SAXS profile (the scattering intensity I(q) vs scattering vector) is shown in Figure 2.7 [35]. The unfrozen data decrease smoothly as q increases, whereas the frozen profile has two features: a peak at high q and an upturn at low q. By fitting the intensities to a theoretical model for the unfrozen data and an empirical function for the frozen data, the structural information about the samples can be obtained. By comparing with the images acquired, the main peak from the SAXS profile from the frozen data can be attributed to the close packing of the particles in the densely packed particle region. The close packing of the silica colloids (so closely packed as to be touching) by freezing allows the short‐range attractive interparticle interaction to dominate, thereby forming long‐lived particle aggregates.

Figure 2.7 The SAXS profiles showing aqueous alumina slurry before freezing (circles) and when frozen (squares). The solid curve represents the fit of the unfrozen data to a polydisperse sphere form factor and monodisperse hard‐sphere structure factor. For comparison, the dotted line shows only the form factor with the same parameters but an arbitrary amplitude. For clarity, the unfrozen data have been offset from the frozen data.

Source: Spannuth et al. 2011 [35]. Reprinted with permission from APS.

2.4 The Interactions Between a Particle and the Freezing Front

2.4.1 Basic Models and Equations for the Critical Freezing Velocity

To obtain experimental evidence for theoretical studies, diluted particle suspensions are usually being directionally solidified. In order to develop theoretical models to describe the interactions between particles and the ice/water interface, the particles may be treated simply as hard spheres and the models may be described based on the interactions of a particle and the freezing front. This is, of course, not the real situation when ice templating is used to make porous materials. But it can help to develop the basic theoretical models and work towards the more complicated system.

During the freezing step, the advancing ice/water front pushes a particle ahead at a low freezing velocity. Figure 2.8 schematically depicts a spherical particle being pushed away by the freezing front [19]. The particle is not directly in contact with the freezing front. Instead, there is a flowing liquid film around the particle that is present in order to maintain the transport of molecules and growth of ice crystals around the particle. The particle experiences two counteracting forces: an attractive force resulting from viscous drag due to fluid flow that tends to trap the particle, and a repulsive force originating from van der Waals forces between the particle and freezing front. Theoretical models have been developed based on, to be more specific, non‐retarded van der Waals forces where the separation between the particle and the freezing front is small, compared to the particle radius.

Figure 2.8 Schematic representation for the movement of a particle in front of the freezing front.

Source: Zhang et al. 2005 [19]. Reprinted with permission from Nature Publishing Group.

The attractive drag force can be expressed as in the following equation, derived for the case of a flat interface [36, 37]:

(2.3)

where η means the viscosity of the liquid suspension medium, v is the velocity, r is the particle radius, and d is the vertical distance of the particle ahead of the freezing front.

By thermodynamics, a particle can be rejected from the freezing front if the interfacial free energy between the particle and the solid phase, σsp, is greater than the sum of surface free energies of solid–liquid, σsl, and liquid‐particle, σlp. That is:

(2.4)

If the above equation is not satisfied, the particle will be entrapped by the ice crystals to achieve the state of lowest energy.

For a spherical particle of radius r, with a thin film of liquid between the particle and the freezing point, the repulsive (disjoining) force can be expressed as:

(2.5)

where a0 is the average intermolecular distance in the liquid film, d is the vertical distance from the particle to the freezing front, and the exponent n ranges from 4 to 5 [36, 37]. However, there are also reports on n = 2 for small d, n = 3 for larger d, or n = 1–4 sometimes used [[37], and references therein].

The critical velocity, indicating the transition from particle rejection to encapsulation by increasing freezing velocity, can be derived by balancing the attractive force and the disjoining force (Fη = Fσ) [37]:

(2.6)

This equation is based on a flat freezing front. The critical velocity is inversely proportional to the particle radius. It is important to determine the values for d, n, and Δσ0.

Another equation can be used to calculate the critical velocity. Based on the numerical variation of some parameters affecting the shape of the solid/liquid interface, the critical velocity can be determined from the critical distance, the ratio of thermal conductivities, and a dimensionless parameter (X) including the solute concentration. It is assumed that the crystal and the melt have the same thermal conductivity and no gravitational action, and that the viscosity is independent of concentration and the surface free energies are independent of concentration and temperature. In the case of a pure melt, the critical velocity can be expressed as [38]:

where the critical distance is kept constant for all particle radii: dc = 2a0. When the thermal conductivities of the particles and the liquid are different, a parameter μ (the ratio of the thermal conductivities of the particles and the liquid) can be introduced into Equation (2.7) in the denominators.

2.4.2 Effects of Thermal Gradient, Particle Radius, and Viscosity on Critical Velocity

For large particles, both the gravitational force and buoyance force cannot be ignored. During a freezing process, the forces experienced by the particle include the attractive forces (gravity and viscous drag forces), which favour settling and entrapment, and the repulsive forces (buoyance and van der Waals forces), which facilitate the floating of the particles in the liquid and hence more likely the particles being pushed along. By balancing these forces and assuming that the velocities of the particles and the freezing front are the same, an equation to calculate the critical velocity is given as [5, 39, 40]:

(2.8)

where D is the particle diameter, η is the viscosity of the liquid phase, A* is the Hamaker constant (= −7.0 × 10−20 J for the ice–water–particle system) [41], d0 is the minimal distance between the particle and the freezing front, g is the gravitational constant, and ρs, ρl, and ρp are the densities of the solid frozen phase, the liquid phase, and the particle, respectively.

There are other forms of equations that take into account these factors. For a temperature gradient G applied across the sample system, the critical velocity can be expressed as [37]:

(2.9)

When the influence of surface curvature phenomena (Gibbs–Thomson effect) can be ignored and for a ‘large’ particle, the equation can be given in Equation (2.10), where B3 is an interaction constant [42]:

Equation (2.7) is based on a pure melt. When a solute is dissolved in the liquid phase, the critical velocity is influenced by the temperature gradient as:

(2.11)

The velocity can be independent from the particle radius if the solute concentration is very high:

(2.12)

When calculating the interaction energy between macroscopic bodies (the van der Waals force for two polarizable atoms can be expressed as W = −C/r6, where C is the coefficient in the atom–atom pair potential and r is the distance between the two atoms) it can be separated into two independent parts, the Hamaker constant (generally, A* = π21ρ2, where ρ1 and ρ2 are the number of atoms per unit volume in the two bodies; A is approximately 10−19 J), containing information about the materials involved, and a force equation taking into account the geometric parameters of the interactions. Equations (2.7) and (2.10) can be compared or linked by the interaction constant B3 and the Hamaker constant A* as given in the equation below [42]:

(2.13)

In a study on engulfment of latex particles (negatively charged polystyrene microspheres with the radii in the range of 1–10 μm), Lipp and Körber proposed an empirical function of particle radius r, temperature gradient G, and the viscosity η to fit the obtained data [42]:

The best fit of the data produced different values of x, y, z. When the value of x = 0.25, y = 1.0, and z = 1.0, Equation (2.14) corresponds to Equation (2.10).

The deformation of the ice–liquid interface and the presence of the liquid film (Figure 2.10) may be attributed to interfacial pre‐melting. When the particle is large enough, the effects of interface curvature may be minimal. However, the thermal conductivity and density of the particles can influence the freezing phenomena. It has been found that the critical velocity is less sensitive to the temperature gradient and the precise dependence changes with different interaction types. Particle buoyancy can enhance or reduce the tendency for the particle to be captured [43].

2.4.3 Geometry and Interfacial‐curvature Effects

When the particle is small, the effect of interfacial curvature is eminent. For a smooth spherical particle and freezing fronts of different geometry, the critical velocity equation can be generalized as [36, 37]:

(2.15)

where the exponent m varies between 1 and and k′ is a proportionality factor depending on the geometry and Δσ0.

The interface curvature can be considered specifically when developing theoretical models. When deviated from a planar freezing interface, the melting temperature of a pure substance is reduced by an amount proportional to the interface curvature and the surface energy σsl (Equation (2.16)). This phenomenon is often called as the Gibbs–Thomson effect. Intermolecular interactions can cause the formation of melted fluid between the particle and the freezing interface, even when the interface is planar. The interface temperature Ti can be expressed as [44]:

where Tm is the equilibrium melting temperature, ρs and qm are the solid density and the latent heat, respectively, and λ is a length scale characterizing the strength of the intermolecular interactions. The exponent v depends on the type of intermolecular forces that dominate; v = 2 for long‐range electrical interactions; v = 3 for non‐retarded van der Waals forces; and v = 4 for retarded van der Waals interactions [44, 45].

By comparing the relative importance of intermolecular interactions and interface curvature in controlling the interface temperature, the characteristic radius of the particle rc can be given as [44]:

(2.17)

Typically, rc is of order 10−4 m. When the particle radius r ≫ rc, the curvature effects may be neglected.

The calculated interface profile is used to evaluate the force balance and the particle behaviour. Both the non‐retarded van der Waals forces (controlling the thickness of the pre‐melted film) and the long‐range electrical interactions are considered. A generalized velocity scale ú, by balancing the counteracting forces, is expressed as [44]:

(2.18)

When long‐range electrical interactions dominate (v = 2), this indicates that vc ∝ r−1. When retarded van der Waals interactions dominate (v = 4), vc ∝ r−3/2. When non‐retarded van der Waals interactions dominate (v = 3, r ≪ rc), this will give vc ∝ r−4/3. It is noted that the velocity scale is independent of the temperature gradient G. The film thickness in the inner region is determined mainly by the balance between interfacial‐curvature effects and non‐retarded van der Waals interactions. Because the lubrication force (attractive force) and the thermomolecular force (disjoining force) are strongest here, the particle velocity is insensitive to G [44].

2.5 Morphology Instability at the Freezing Front

During the freezing of particle suspension, the freezing front rejects and pushes the particles at low freezing rate. With the increase of freezing rates (usually by applying a higher temperature gradient across the liquid sample), the planar interface is disturbed and broken into dendritic or cellular ice crystals while the particles are closely packed between the ice dendrites. This is the key point that indicates that the ice‐templating technique via directional freezing can be used to produce porous materials with different pore morphologies [7, 12, 19, 46, 47].

The interface instability is a result of constitutional supercooling in the liquid phase just in front of the freezing front. This indicates a lower freezing temperature in the local zone than the equilibrium temperature. The reason for supercooling in a colloidal system is the higher concentration of the particles close to the freezing front. In the constitutionally supercooled region, a small protrusion of ice crystals experience a greater driving force than the planar front and hence grow faster. With the protrusions becoming larger, a cellular or dendritic morphology is formed [21].

2.5.1 Mullins–Sekerka Equation

A theoretical equation was developed by Mullins and Sekerka by calculating the time dependence of the amplitude of a sinusoidal perturbation of infinitesimal initial amplitude introduced into the planar shape. The critical mathematical simplification was the use of steady‐state values for the thermal and diffusion fields [48]. The equation was developed to investigate the stability of a planar interface during solidification of a dilute binary alloy. But it has also been applied to the freezing of aqueous colloidal suspensions. The equation is given below [19, 21, 48, 49]:

(2.19)
(2.20)

where δ is the size of the perturbation,V is the growth velocity, ω is the frequency, TM is the equilibrium melting point, Γ is the ratio of the surface energy and the latent heat of fusion, D is the diffusion coefficient, k is the partition coefficient, G is the temperature gradient, Gc is the solute concentration gradient at the interface, m is the slope of the liquidus line on the phase diagram, and ks and kl are the thermal conductivities of the ice and solution, respectively.

When using this equation to predict instability, there is always a sinusoidal perturbation of certain wavelength λ0, above which waves grow, below which waves decay, and at which the wave must have constant amplitude.

Usually, the actual primary spacing of the formed ice dendrites is greater than the instability wavelength, which is likely the result from some degree of coarsening of the structure during growth. Different models have been developed to predict the primary spacing, based on different assumptions. However, the expression for the primary spacing (λd) can be generally given as [21]:

where A is dependent on variables such as the diffusion coefficient, partition coefficient, and liquidus slope; G is the temperature gradient and V is the ice crystal growth velocity.

In a directional solidification study of ice crystallization in sucrose and pullulan solutions, it was found that the Mullins–Sekerka equation could provide a reasonable prediction of the instability wavelength over a range of velocities. The primary spacing found is in agreement with Equation (2.21). However, it was not good for the pullulan solution. The discrepancy was attributed to the long polymer chains and hence the hindered diffusion mechanism [21].

From the investigation of freezing colloidal suspensions or polymer/colloidal suspensions [19, 46], the primary spacing or the lamellae thickness is inversely proportional to the freezing velocity. That is, the faster the freezing velocity, the narrower the primary spacing. An empirical dependence can be correlated between the primary spacing λ (or wavelength) and the freezing velocity ν in parallel to the temperature gradient:

(2.22)

where n may be in the range of 1–4 [6, 50].

2.5.2 Linear Stability Analysis

It is well known that dissolution of ionic compounds can significantly reduce the freezing temperature. The degree of freezing temperature depression is directly related to the concentration of the particles. It has been noticed that a colloidal suspension can become constitutionally supercooled as well. Peppin et al. developed a mathematical model for the solidification of a suspension of hard‐sphere colloids [51]. It showed that the highly non‐linear functional dependence of the diffusion coefficient on the volume fraction gave rise to a range of behaviours. For small particles where Brownian diffusion dominates, constitutional supercooling occurs at the interface. For larger particles where Brownian diffusion is weak, the particles form a porous layer above the interface.

In a further study, following the hard‐sphere equation of state, a linear stability analysis of a planar freezing interface has been described [52]. After solving steady‐state configuration, the steady profile is perturbed via normal modes. The governing equation in the perturbed quantities is linearized and the resulting ordinary differential equation is solved to determine conditions under which the interface is stable (perturbations decay in time) or unstable (perturbations grow in time). Considering the dilute limit (the concentration C → 0 so that the diffusion coefficient D → 0), the characteristic equation can be obtained as [51]:

(2.23)

where σ is the growth rate of the disturbance and α is the wavenumber of the normal modes along the interface. The growth rate σ determines whether or not a particular perturbation will grow in time. ks is the segregation coefficient that depends on the interfacial concentration and the solidification velocity. M is the morphological number while Γ is a surface energy parameter (detailed definition can be found in Ref. [52]). This equation is similar to the Mullins–Sekerka equation developed from dilute alloys, with an important distinction that here the coefficients depend on the particle radius. It is found that the interfacial stability depends strongly on the size and concentration of the particles. A stabilizing effect is observed when increasing the particle sizes whilst increasing the concentration can destabilize the interface.

Experimental evidence has been obtained to verify the morphological instability [20]. The compressibility (osmotic pressure) is measured to predict the freezing point depression as a function of particle volume fraction (bentonite particles). The freezing point may decrease to −8 °C at a volume fraction of about 0.5. Measurement of permeability is used to predict the concentration‐dependent diffusivity. By considering these parameters, for a given temperature gradient, the critical conditions for the onset of constitutional supercooling can be obtained from the following equation [20]:

(2.24)

where dimensionless temperature gradient and diffusivity are used. Pe is the Peclet number and is the particle volume fraction. This theoretical threshold can be determined experimentally, demonstrating that the colloidal suspensions can be treated analogously to atomic or molecular alloys [20].

2.5.3 Morphology Zones and Stability Diagram

From the initial freezing of a colloidal suspension to the steady‐state freezing, the morphologies in different zones can vary significantly. This may be shown by imaging the freeze‐dried and sintered samples (Figure 2.9) [47]. When freezing aqueous alumina slurry by contacting cold finger on two sides, the initial freezing rate is fast and the freezing fronts encapsulate all the particles. This results into a dense layer of the materials. With the increasing distance from the cold finger, the freezing velocity decreases gradually and the microstructures of the resulting material change from columnar, lamellar, lamellar/dendritic, to a more steady‐state of lamellar/dendritic ice front with homogeneous lamellar thickness (Figure 2.9). Two possibilities can be considered for the ice morphology transition from planar to columnar [47]. The first possibility is the constitutional supercooling as described in the Mullins–Sekerka theory. The second possibility for the breakdown of the planar interface may be triggered by the presence of particles in the liquid. The breakdown may occur when the velocity is below the threshold for the onset of constitutional supercooling [20].

Figure 2.9 Scanning electron microscopic (SEM) images of a sintered freeze‐cast alumina sample, showing the change of ice front morphology with varying distance from the cold finger. The schematic on the right represents how the particles are encapsulated in or rejected from the ice front.

Source: Deville et al. 2007 [47]. Reprinted with permission from Elsevier.

The morphological change of ice front during the freezing of aqueous alumina slurry has also been directly observed by in situ X‐ray radiography and tomography [53, 54]. The morphology and evolution of the ice crystals are classified based on the presence and percentage of crystal orientations. Two types of ice crystals are described: z‐crystals where the crystals are orientated to the freezing direction (z‐axis) and are growing faster in this direction, and r‐crystals (r means random) where the crystals are more or less in the xy plane (radial direction). In the zone very close to the cold surface, because nucleation is spatially homogeneous, the populations of z‐crystals and r‐crystals are spatially homogenously distributed. In the next zone, the crystals grow larger and the packing of the particles become more efficient, the percentage of z‐crystals slowly increase. In the zones further away from the cold surface, the r‐crystals stop growing and then only z‐crystals grow from the suspension [53]. This leads to a steady‐state freezing stage where the particles are rejected from z‐crystals and particle redistribution occurs in the xy‐plane. Particle redistribution depends on the interface velocity. From moderate to slow velocities (1 μm s−1), the particles migrate by Brownian diffusion, leading to a built‐up layer of concentrated particles ahead of the interface. However, for the conditions normally used in the freezing of colloidal suspensions for freeze‐cast materials, the Brownian diffusion may be neglected, as observed in this study [54]. The concentrated particle layers observed in the radiographs may be due to the larger particles approaching the crystal separation distance that cannot be accommodated between the primary dendrites.

Theoretical models are often developed based on single particle in dilute colloidal suspensions [3652]. For real situations where freezing of colloidal suspensions is used for the preparation of porous materials, those models cannot predict the complex behaviour. Direct observation of such a freezing system and the corresponding explanation by multi‐particle models, where the possible interactions between the particles are considered, are required in this research field. Global and localized interface instabilities have been observed by in situ X‐ray radiographs (Figure 2.5) [25]. The instability can be regarded as inducing growth of ice crystals in the plane perpendicular to the freezing direction. The localized instabilities are transverse crystals between the ice crystals. A stability diagram can be built based on this study (Figure 2.10), including unstable domains and metastable domains. The metastable domains are defined as those where no instabilities are observed and homogeneous crystals are obtained at high interface velocity and/or large particles. The instability is attributed to partial diffusion of the particles ahead of the freezing interface. Thus, the resulting higher concentration of the suspension in front of the freezing front leads to the depressing of the freezing point [20], hence a constitutional supercooling situation. For high freezing velocity, the diffusion is very limited and the freezing zone (in a constitutional supercooling state) is within the growing crystal tips. The zone ahead of the crystal tips is in a stable state, which does not incur instability and yields homogenous materials. This situation is referred to be in the metastable domains. At lower freezing velocities, the diffusion is enhanced. The particle redistribution leads to the formation of a diffusion layer (still in a constitutional supercooling state) appearing beyond the growing crystal tips. The nucleation of the crystals occurs in the supercooled zone and hence the instability [25].

Figure 2.10 The stability diagrams constructed from freezing aqueous alumina slurry as observed by in situ X‐ray radiography. (a) The ice stability diagram. The hatched area is an estimated domain where the metastable to unstable regime transition can occur. (b) General stability and the structural diagram. The hatched area indicates the conditions investigated. The diamond and triangle symbols indicate the data points that are measured.

Source: Deville et al. 2009 [25]. Reprinted with permission from Nature Publishing Group.

The traverse ice crystals caused by the localized instability can lead to porosities in the wall of the freeze‐casting material. This may explain why many of the materials are prepared under similar freezing conditions although the compression stabilities can vary considerably among different reports [55]. The traverse growth of ice crystals can be regarded as ice lenses, which is often observed in geophysics. Ice lenses are the ice crystals growing in the supercooled zone in the direction perpendicular to the freezing direction or the temperature gradient. The formation of the supercooled zone has been attributed to the partial diffusion of the particles. During the freezing of alumina dispersions with varying concentrations of ionic additives, different zones ahead of the crystals tips are observed [55]. At a low concentration of the ionic additive (but in the optical range to obtain the strongest repulsion), accumulated particle layer (APS), particle‐depleted region (PDR), and then the liquid suspension are in front of the crystals tips in turn. With the increasing concentration of the ionic additive, the PDR becomes thinner and then disappears. The formation of the PDR is quite unusual. This is explained by the concentration of free ions rejected from the freezing front, increased ionic strength compressing the repulsion charged layer of the alumina particles, and the flocculation and aggregation of the particles. The settlement of large aggregates leads to the formation of a PDR [55]. As such, the supercooled zone should originate from the particle flocculation rather than the partial particle diffusion. The PDR is initially formed between the ice crystals but can move above the crystals tips when the suspension contains low concentration of ionic dispersants, favoured by low viscosity and high zeta potential. The ice lenses formed in the PDR can explain the defects formed in the freeze‐casting materials. The growth of ice lenses in the high velocity region, sometimes termed as ‘banding’, may question the credibility of the assumption of local equilibrium at the freezing interface. In order to model this non‐equilibrium segregation of ice crystals, the Boltzmann velocity distribution of a particle has been used to derive an expression for the segregation coefficient at the interface as a functional of the freezing velocity and particle radius [56]. A phase diagram can be constructed and used to explain the formation of band‐like defects.

Time‐lapse, 3D in situ imaging shows that the thickness of the ice crystals and the horizontal growth of ice crystals (ice lenses) increase with freezing time [57]. The thickness of the horizontal crystals reaches a plateau of about 25 μm after about 60 s. A close‐up view of the 3D reconstruction of the ice crystals shows the tilted growth of ice crystals from the imposed freezing direction and also the dendrites on one side of the ice crystals. This is also observed by SEM imaging of the sintered freeze‐casting alumina [47]. As schematically shown in Figure 2.11, tilting of the ice crystals is the result of the direction imposed by the temperature gradient and the favoured direction of anisotropic crystal growth [57]. Tilting of the ice crystals becomes greater when the freezing rate (usually corresponding to the temperature gradient) decreases. A threshold below which ice crystals tilt exists in the 3–3.5 μm s−1 range [57]. The dendrites grow on one side with respect to the temperature gradient direction. Dendrite growing on the other side is energetically unfavourable [47, 57].

Figure 2.11 The schematic representation shows the tilted growth of ice crystals, resulting from the relationship between the imposed temperature gradient and the preferred growth direction.

Source: Deville et al. 2013 [57]. Reprinted with permission from Elsevier.

Ice banding is a pattern of ice segregation that results from freezing concentrated colloidal suspensions, which is a familiar phenomenon seen in geophysics such as frost heave. Directional solidification of the concentrated alumina slurry (mean particle diameter ∼0.32 μm, 60 wt% or 27 vol%) was performed to investigate the ice segregation patterns [58]. As shown in Figure 2.12, at the low freezing rate (0.5–2 μm s−1), the segregated ice lenses are formed in the frozen solid with a partially frozen layer and a compacted layer above the freezing interface [58]. When the freezing velocity is increased (3–4 μm s−1), like the breakdown of planar interface in the dilute system, the ice lenses become disordered with the compact layer diminishing. In the high velocity region (5–10 μm s−1), with no ice lenses or compacted layer, the ice banding is highly modulated, giving a wavy and jagged appearance. The compacted layer is randomly closely packed, with particle fraction Φp ≈ 0.64. The compaction of the particles occurs as a sol–gel transition because they cannot be re‐dispersed in the solvent. It indicates an irreversible aggregation, overcoming the maximal repulsion between the particles. According to the Derjaguin–Landau–Verwey–Overbeek (DLVO) theory, there is a peak of repulsion force at a critical distance between the particles. Passing that critical distance, the cohesive force is dominating and the particles are aggregated. The cryostatic suction during ice segregation can easily produce compressive stresses that are much larger than the typical repulsive force value (≤10 kPa), ensuring the particles bond together [59]. This is also evidenced by a SAXS study demonstrating the particles in touching distance [35].

Figure 2.12 Images show the quasi‐steady ice segregation patterns from directional solidification of concentrated alumina slurries at different pulling speed. The segregated ice appears black while the colloid appears grey. (a) At low velocity, the ice bands are crack‐like in the frozen region, illustrating the formation of ice lenses, with compacted layers (dark grey) and partially frozen layer (light grey). (b) At medium velocity, ice lenses become disordered, with diminishing compaction layer. (c) At high velocity, periodic ice banding with a wavy appearance forms. Vertical ice‐filled cracks can also be observed.

Source: Anderson and Worster 2012 [58]. Reprinted with permission from American Chemical Society.

Quite importantly, the partial frozen layer can be regarded as a frozen fringe, which is an interconnected liquid network maintained in the frozen fringe due to the existence of unfrozen water near the particle surface and in the pendular region near the pore throat. This supercooled water is the result of curvature near the throat of the pore and interfacial pre‐melting. The thermomolecularly generated pressure gradient pushes the particles towards the warmer region and drives the liquid moving towards the colder region to sustain ice lens growth. When a new ice lens is nucleated within the frozen fringe, it can intercept the water supply of its predecessor that will stop growing. The repeating of this process can eventually produce a regular pattern of ice lenses and compaction layer [59].

Rigid‐ice models have been developed for this ice segregation phenomenon (ice lenses and compaction). It assumes that the pore ice and ice lens form a rigid interconnected body of ice and the forces generated do not deform the pore‐ice network [60]. A related theoretical framework in the context of frost heave is developed, where the forces are derived directly from the thermo‐physical interactions between the constituents of partially frozen suspension [61]. This model is extended to allow for an evolving compaction front, where particles aggregate and form a cohesive matrix. The developed model is able to elucidate the role of compaction layer in ice segregation [59].

2.6 Effect of Parameters on the Structure and Morphology of Ice‐templated Materials

There are not many studies on how the freeze‐drying process impacts the structure of ice‐templated materials, except the evaluation of freeze‐dried cakes in the pharmaceutical and food industries. The parameters that have been investigated extensively can be classified into two categories: formulation of solution/suspensions and control of the freezing conditions.

2.6.1 Formulation of Solution/Suspensions

2.6.1.1 Solutes or Particles

A solute can be dissolved in a solvent (usually water) to form a solution, which is then frozen and freeze‐dried to produce an ice‐templated material. Because small molecular compounds would normally form powders after freeze‐drying, polymers are usually added to the solution. The entanglement of long polymer chains can lead to the formation of stable porous structures. The main parameters for polymers include molecular weight, hydrophobicity, and charges. Stronger interaction between the polymer chains, e.g. H bonding, π–π interaction, and electrostatic interaction, can have big impacts on the stability and pore morphology of the resulting structures. Small molecules may be included in polymer solutions as additives to introduce additional functions or properties into the ice‐templated materials.

Both microparticles and nanoparticles have been extensively used for freeze‐casting materials. Different solvents have been used but water is always the most preferred one because of its low cost and green credentials. The main parameters for particles include: particle size, nature of particle material (e.g. organic, metal, or metal oxide), and particle shape. As particulate suspensions are the most commonly investigated systems both for materials preparation and fundamental studies, the following discussion of parameters is mostly relevant to particles.

2.6.1.2 Particle Size and Shape

For large particles, depending on their density (particularly the density difference compared to the density of the solvent), the gravitational force/buoyance force can play an important role on rejection/encapsulation of the particles by the freezing front, resulting in a significant impact on the morphology and stability of the resulting materials. Although most models are based on spheres, the shapes of microparticles (which are usually prepared by grinding or precipitation) used in freeze casting may not be spherical. The irregular shape and larger size of the particles can result in low packing efficiency and slow moving during freezing, which may have significant effects on porosity, morphology, and stability of the prepared materials [5, 6, 62]. Nanoparticle suspensions are usually quite stable. They may be readily self‐assembled into microwire or nanowire structures by the ice‐templating process [7, 12].

The size of the polymers can be varied by molecular weight. But the difference in the structures can be quite subtle, providing that ice‐templated materials can be formed.

2.6.1.3 Concentration of Polymers or Particles

The concentration of a polymer solution can have a distinct effect on the structure of the ice‐templated polymeric materials. At medium or high concentrations (e.g. 5 wt% or higher), aligned porous or randomly porous polymers can be produced [7, 19]. However, at very low concentrations (e.g. 0.5 wt% or lower), polymeric nanofibres may be formed [63, 64]. This effect of polymer concentration depends on the type and molecular weight of the polymers, and also the type of the solvent used.

Similarly, the concentration of particulate suspensions may result in wire structures or porous materials [7, 65]. Most of the reports have employed suitable concentrations in a view to produce porous ceramics with controlled morphology and mechanical stability [5, 6, 12]. However, the concentrated particulate suspensions may generate porous ceramics with periodic ice lens‐templated structures rather than the usual layered or unidirectional microchannel structure  [55, 58].

2.6.1.4 Additive or Binder

Additives and/or binders are required when processing particulate suspensions [5, 6]. First, it helps to stabilize a homogeneous suspension which is crucial for the freeze‐casting process. Common surfactants (e.g. ionic surfactants) or polymers with stabilizing effects, including commercial polymers with trade names, poly(vinyl alcohol) and poly(vinyl pyrrolidone), can be added for this purpose. The binder is to hold the particles together and maintain a stable structure after freeze‐drying. A polymer additive may act both as a stabilizer for the suspension and as a binder for the freeze‐dried material. If the target is to produce porous ceramics, the amount of additives used is usually kept to optimum and minimum. This is to reduce the processing costs and also to reduce the potential shrinkage or defects after sintering the freeze‐dried bodies.

This approach can be further utilized to produce porous composite materials. Depending on what is required, the main component can be polymer or particles. The distribution of particles in the polymer and the interaction between polymer and particles are two important parameters. Usually, it is required to have a homogenous distribution of particles and strong interaction between the polymer and particles to generate porous composites with desired pore structure and mechanical stability.

2.6.1.5 Solvent

In addition to the cost and environmental consideration, the selection of solvents mainly depends on their suitability for freeze‐drying. In this regard, high melting point and high vapour pressure are the criteria to be considered. There is no doubt that water is the most used solvent. But other solvents have been used as well. For example, camphene, tert‐butylalcohol, naphthalene‐camphor have been used as solvents for particle suspensions to produce freeze‐casting materials [5, 6, 12]. Dichloroethane, o‐xylene, dimethyl sulfoxide (DMSO), cyclohexane, and compressed CO2 have been used for solutions or emulsions [7, 19, 6669].

2.6.1.6 Emulsion

In addition to solution and suspensions, freezing and freeze‐drying of emulsions have also been used to prepare porous polymers and composite materials. Emulsion is a mixture of two immiscible liquids (usually water and an immiscible organic solvent), with one liquid as droplet phase dispersed in the other continuous liquid phase. Surfactants are usually required to stabilize the droplets in the emulsion. The volume percentage of an emulsion can be varied in a wide range. This can be very powerful when emulsion templating is used to prepare porous materials [70]. The emulsion structure is usually locked in by polymerization. Freezing has provided an effective and alternative way of locking in the emulsion structure. Usually, a polymer can be dissolved in the continuous phase with the droplets only consisting of solvent. After the removal of the solvents from both the continuous phase and the droplet phase by freezing‐drying, porous materials can be produced. For an oil‐in‐water emulsion (where oil phase is the droplet phase), hydrophobic compounds may be dissolved in the droplet phase. After freeze‐drying, hydrophobic nanoparticles in hydrophilic porous polymer matrix can be formed [68, 69]. This nanocomposite can be readily re‐dissolved in water, producing stable aqueous nanoparticle dispersions, which have important applications for nanomedicine [71]. While particles can be suspended in the continuous phase for preparation of composite materials, colloids or nanoparticles can also be used as surfactants to form Pickering emulsions to make either inorganic or composite materials. By varying the percentage of the oil droplet phase in an emulsion, combining emulsion templating and ice templating, a highly efficient route can be established for producing porous materials with systematically tuned pore morphology and porosity. This approach can also be extended to double emulsions [72].

2.6.2 Control of the Freezing Conditions

In order to produce the ice‐templated materials with the desired pore structure and formats, the freezing process may be carried out in different ways, as detailed in Chapter 1. The key point in controlled freezing is the control of freezing velocity. Indeed, the theoretical models described above are always related to the freezing velocity. Qualitatively, the freezing velocity can be varied to tune the pore sizes [6, 19, 50]. In general, fast freezing leads to small ice crystals and hence small pores in the resulting freeze‐dried materials while the freezing rate should be reduced if a material with larger pore sizes is required.

However, it is very difficult to accurately control the freezing velocity when preparing porous materials via the ice‐templating approach. It is generally accepted that the freezing velocity may be determined by the temperature between the freezing front and the cooling plate and the thermal conductivities of solid phases involved. For the fundamental studies in freezing, quite often, a sample cell is placed across two separately temperature‐controlled metal plates and is pulled at certain speeds towards the cold plate. A temperature gradient is created between the two plates (the temperature difference divided by the gap between them), which is the driving force for the directional solidification of solutions or colloidal suspensions. However, the same temperature gradient can be obtained from different temperatures of the plates, as long as the temperature difference is the same. This may have a significant impact on the removal of the heat generated by the freezing (solidification) of the solvent.

When a layer of liquid freezes at its thermal transition temperature TE, the solidification enthalpy hE is given out as heat and has to be conducted away through the frozen solid to the cooling plate. The velocity at a given time t where the solidification front progresses can be described as [6]:

where s is the thickness of the liquid layer, λS is the thermal conductivity of the frozen solid, k is the heat–transfer resistance, ρL is the liquid density, and T0 is the temperature of the cold plate (or the temperature of the cold liquid that cools the plate). It is clear that to maintain a constant velocity, T0 should vary according to Equation (2.25).

When freezing a slurry, in order to calculate the thermal conductivity of the solidified particles, the packed particles between the ice lamellae crystals can be treated as a bed of closed particles. The thermal conductivity of such a packed bed can be predicted using the Krischer–Kröll model, from the amount of the serial component, the volume fraction of the continuous ice phase, and the thermal conductivities of the particles and ice. Assuming that the frozen slurry has alternate ice lamellae and packed particle layers and the lamellae to be parallel to the heat flow direction, the total thermal conductivity of the frozen slurry can be calculated by the following equation [6]:

(2.26)

where Ψ is the volume fraction of the ice phase and λI is the thermal conductivity of pure ice, while λB is the predicted thermal conductivity of the packed particles between ice lamellae. Once λS is obtained, one can use Equation (2.25) to calculate the freeze velocity.

2.7 Summary

In this chapter, the fundamental aspects of controlled freezing for ice‐templated materials have been described. The basics of ice crystals, the instruments used for the fundamental study, and the different theoretical models to predict particle fate and ice morphology and dimensions when freezing aqueous colloidal suspensions are covered. The important processing parameters, with the aim to fabricate the ice‐templated materials of desired structures and properties, are outlined.

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