Physico-chemical Processes that Occur During Freezing
Disclaimer: This dissertation has been submitted by a student. This is not an example of the work written by our professional dissertation writers. You can view samples of our professional work here.
Any opinions, findings, conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of UK Essays.
Lyophilization respectively freeze-drying is an important and well-established process to improve the long-term stability of labile drugs, especially therapeutic proteins. About 50% of the currently marketed biopharmaceuticals are lyophilized, representing the most common formulation strategy. In the freeze-dried solid state chemical or physical degradation reactions are inhibited or sufficiently decelerated, resulting in an improved long-term stability. Besides the advantage of better stability, lyophilized formulations also provide easy handling during shipping and storage. 
A traditional lyophilization cycle consists of three steps; freezing, primary drying and secondary drying. During the freezing step, the liquid formulation is cooled until ice starts to nucleate, which is followed by ice growth, resulting in a separation of most of the water into ice crystals from a matrix of glassy and/or crystalline solutes.[4-5] During primary drying, the crystalline ice formed during freezing is removed by sublimation. Therefore, the chamber pressure is reduced well below the vapor pressure of ice and the shelf temperature is raised to supply the heat removed by ice sublimation. At the completion of primary drying, the product can still contain approximately 15% to 20% of unfrozen water, which is desorbed during the secondary drying stage, usually at elevated temperature and low pressure, to finally achieve the desired low moisture content.
In general, lyophilization is a very time- and energy-intensive drying process. Typically, freezing is over within a few hours while drying often requires days. Within the drying phase, secondary drying is short (~hours) compared to primary drying (~days).[1, 4] Therefore, lyophilization cycle development has typically focused on optimizing the primary drying step, i.e., shortening the primary drying time by adjusting the shelf temperature and/or chamber pressure without influencing product quality.[5, 9] Although, freezing is one of the most critical stages during lyophilization, the importance of the freezing process has rather been neglected in the past.
The freezing step is of paramount importance. At first, freezing itself is the major desiccation step in lyophilization  as solvent water is removed from the liquid formulation in the form of a pure solid ice phase, leading to a dramatic concentration of the solutes.[11-12] Moreover, the kinetics of ice nucleation and crystal growth determine the physical state and morphology of the frozen cake and consequently the final properties of the freeze-dried product.[11-13] Ice morphology is directly correlated with the rate of sublimation in primary and secondary drying. In addition, freezing is a critical step with regard to the biological activity and stability of the active pharmaceutical ingredients (API), especially pharmaceutical proteins.
While simple in concept, the freezing process is presumably the most complex but also the most important step in the lyophilization process. To meet this challenge, a thorough understanding of the physico-chemical processes, which occur during freezing, is required. Moreover, in order to optimize the freeze drying process and product quality, it is vital to control the freezing step, which is challenging because of the random nature of ice nucleation. However, several approaches have been developed to trigger ice nucleation during freezing.
The purpose of this review is to provide the reader with an awareness of the importance but also complexity of the physico-chemical processes that occur during freezing. In addition, currently available freezing techniques are summarized and an attempt is made to address the consequences of the freezing procedure on process performance and product quality. A special focus is set on the critical factors that influence protein stability. Understanding and controlling the freezing step in lyophilization will lead to optimized, more efficient lyophilization cycles and products with an improved stability.
2. Physico-chemical fundamentals of freezing
The freezing process first involves the cooling of the solution until ice nucleation occurs. Then ice crystals begin to grow at a certain rate, resulting in freeze concentration of the solution, a process that can result in both crystalline and amorphous solids, or in mixtures. In general, freezing is defined as the process of ice crystallization from supercooled water. The following section summarizes the physico-chemical fundamentals of freezing.
At first, the distinction between cooling rate and freezing rate should be emphasized. The cooling rate is defined as the rate at which a solution is cooled, whereas the freezing rate is referred to as the rate of postnucleation ice crystal growth, which is largely determined by the amount of supercooling prior to nucleation.[16-17] Thus, the freezing rate of a formulation is not necessarily related to its cooling rate.
2.1 Freezing phenomena: supercooling, ice nucleation and ice crystal formation
In order to review the physico-chemical processes that occur during freezing of pure water, the relationship between time and temperature during freezing is displayed in figure 1. When pure water is cooled at atmospheric pressure, it does not freeze spontaneously at its equilibrium freezing point (0°C). This retention of the liquid state below the equilibrium freezing point of the solution is termed as “supercooling”. Supercooling (represented by line A) always occurs during freezing and is often in the range of 10 to 15°C or more.[12, 18] The degree of supercooling is defined as the difference between the equilibrium ice formation temperature and the actual temperature at which ice crystals first form and depends on the solution properties and process conditions.[1, 6, 11, 20] As discussed later, it is necessary to distinguish between “global supercooling”, in which the entire liquid volume exhibits a similar level of supercooling, and “local supercooling”, in which only a small volume of the liquid is supercooled. Supercooling is a non-equilibrium, meta-stable state, which is similar to an activation energy necessary for the nucleation process. Due to density fluctuations from Brownian motion in the supercooled liquid water, water molecules form clusters with relatively long-living hydrogen bonds  almost with the same molecular arrangement as in ice crystals.[11, 15] As this process is energetically unfavorable, these clusters break up rapidly. The probability for these nuclei to grow in both number and size is more pronounced at lowered temperature. Once the critical mass of nuclei is reached, ice crystallization occurs rapidly in the entire system (point B).[15, 21-22] The limiting nucleation temperature of water appears to be at about -40°C, referred to as the “homogeneous nucleation temperature”, at which the pure water sample will contain at least one spontaneously formed active water nucleus, capable of initiating ice crystal growth. However, in all pharmaceutical solutions and even in sterile-filtered water for injection, the nucleation observed is “heterogeneous nucleation”, meaning that ice-like clusters are formed via adsorption of layers of water on “foreign impurities”.[6, 11] Such “foreign impurities” may be the surface of the container, particulate contaminants present in the water, or even sites on large molecules such as proteins.[23-24] Primary nucleation is defined as the initial, heterogeneous ice nucleation event and it is rapidly followed by secondary nucleation, which moves with a front velocity on the order of mm/s through the solution. [14, 25] Often secondary nucleation is simply referred to as ice crystallization, and the front velocity is sometime referred to as the crystallization linear velocity.
Once stable ice crystals are formed, ice crystal growth proceeds by the addition of molecules to the interface. However, only a fraction of the freezable water freezes immediately, as the supercooled water can absorb only 15cal/g of the 79cal/g of heat given off by the exothermic ice formation.[12, 22] Therefore, once crystallization begins, the product temperature rises rapidly to near the equilibrium freezing point.[12, 26] After the initial ice network has formed (point C), additional heat is removed from the solution by further cooling and the remaining water freezes when the previously formed ice crystals grow. The ice crystal growth is controlled by the latent heat release and the cooling rate, to which the sample is exposed to. The freezing time is defined as the time from the completed ice nucleation to the removal of latent heat (from point C to point D). The temperature drops when the freezing of the sample is completed (point E).
The number of ice nuclei formed, the rate of ice growth and thus the ice crystals` size depend on the degree of supercooling.[14, 20] The higher the degree of supercooling, the higher is the nucleation rate and the faster is the effective rate of freezing, resulting in a high number of small ice crystals. In contrast, at a low degree of supercooling, one observes a low number of large ice crystals.[14, 19] The rate of ice crystal growth can be expressed as a function of the degree of supercooling. For example for water for injection, showing a degree of supercooling of 10°C +/- 3°C, an ice crystal growth rate of about 5.2cm/s results. In general, a slower cooling rate leads to a faster freezing rate and vice versa. Thus, in case of cooling rate versus freezing rate it has to be kept in mind “slow is fast and fast is slow”.
Nevertheless, one has to distinguish between the two basic freezing mechanisms. When global supercooling occurs, which is typically the case for shelf-ramped freezing, the entire liquid volume achieves a similar level of supercooling and solidification progresses through the already nucleated volume.[12, 14] In contrast, directional solidification occurs when a small volume is supercooled, which is the case for high cooling rates, e.g. with nitrogen immersion. Here, the nucleation and solidification front are in close proximity in space and time and move further into non-nucleated solution. In this case, a faster cooling rate will lead to a faster freezing rate.[12, 14]
Moreover, as ice nucleation is a stochastically event [6, 18], ice nucleation and in consequence ice crystal size distribution will differ from vial to vial resulting in a huge sample heterogeneity within one batch.[6, 14, 27] In addition, during freezing the growth of ice crystals within one vial can also be heterogeneous, influencing intra-vial uniformity.
Up to now, 10 polymorphic forms of ice are described. However, at temperatures and pressures typical for lyophilization, the stable crystal structure of ice is limited to the hexagonal type, in which each oxygen atom is tetrahedrally surrounded by four other oxygen atoms. The fact that the ice crystal morphology is a unique function of the nucleation temperature was first reported by Tammann in 1925. He found that frozen samples appeared dendritic at low supercoolings and like “crystal filaments” at high supercooling. In general, three different types of growth of ice crystals around nuclei can be observed in solution: i) if the water molecules are given sufficient time, they arrange themselves regularly into hexagonal crystals, called dendrites; ii) if the water molecules are incorporated randomly into the crystal at a fast rate, “irregular dendrites” or axial columns that originate from the center of crystallization are formed; iii) at higher cooling rates, many ice spears originate from the center of crystallization without side branches, referred to as spherulites. However, the ice morphology depends not only on the degree of supercooling but also on the freezing mechanism. It is reported that “global solidification” creates spherulitic ice crystals, whereas “directional solidification” results in directional lamellar morphologies with connected pores.[12, 14] While some solutes will have almost no effect on ice structure, other solutes can affect not only the ice structure but also its physical properties. Especially at high concentrations, the presence of solutes will result in a depression of the freezing point of the solution based on Raoults`s Law and in a faster ice nucleation because of the promotion of heterogeneous nucleation, leading to a enormously lowered degree of supercooling.
2.2 Crystallization and vitrification of solutes
The hexagonal structure of ice is of paramount importance in lyophilization of pharmaceutical formulations, because most solutes cannot fit in the dense structure of the hexagonal ice, when ice forms. Consequently, the concentration of the solute constituents of the formulation is increased in the interstitial region between the growing ice crystals, which is referred to as “cryoconcentration”.[11-12] If this separation would not take place, a solid solution would be formed, with a greatly reduced vapor pressure and the formulation cannot be lyophilized. The total solute concentration increases rapidly and is only a function of the temperature and independent of the initial concentration. For example, for an isotonic saline solution a 20-fold concentration increase is reported when cooled to -10°C and all other components in a mixture will show similar concentration increases. Upon further cooling the solution will increase to a critical concentration, above which the concentrated solution will either undergo eutectic freezing or vitrification.
A simple behavior is crystallization of solutes from cryoconcentrated solution to form an eutectic mixture. For example, mannitol, glycine, sodium chloride and phosphate buffers are known to crystallize upon freezing, if present as the major component. When such a solution is cooled, pure ice crystals will form first. Two phases are present, ice and freeze-concentrated solution. The composition is determined via the equilibrium freezing curve of water in the presence of the solute (figure 2). The system will then follow the specific equilibrium freezing curve, as the solute content increases because more pure water is removed via ice formation. At a certain temperature, the eutectic melting temperature (Teu), and at a certain solute concentration (Ceu), the freezing curve will meet the solubility curve. Here, the freeze concentrate is saturated and eutectic freezing, which means solute crystallization, will occur.[7, 19] Only below Teu, which is defined as the lowest temperature at which the solute remains a liquid the system is completely solidified. The Teu and Ceu are independent of the initial concentration of the solution. In general, the lower the solubility of a given solute in water, the higher is the Teu. For multicomponent systems, a general rule is that the crystallization of any component is influenced, i.e. retarded, by other components. In practice, analogous to the supercooling of water, only a few solutes will spontaneously crystallize at Teu. Such delayed crystallization of solutes from a freezing solution is termed supersaturation and can lead to an even more extreme freeze concentration. Moreover, supersaturation can inhibit complete crystallization leading to a meta-stable glass formation, e.g. of mannitol.[12, 23] In addition, it is also possible that crystalline states exist in a mixture of different polymorphs or as hydrates. For example, mannitol can exist in the form of several polymorphs (a, b and d) und under certain processing conditions, it can crystallize as a monohydrate.
The phase behavior is totally different for polyhydroxy compounds like sucrose, which do not crystallize at all from a freezing solution in real time. The fact that sucrose does not crystallize during freeze-concentration is an indication of its extremely complex crystal structure. The interactions between sugar -OH groups and those between sugar -OH groups and water molecules are closely similar in energy and configuration, resulting in very low nucleation probabilities. In this case, water continues to freeze beyond the eutectic melting temperature and the solution becomes increasingly supersaturated and viscous. The increasing viscosity slows down ice crystallization, until at some characteristic temperature no further freezing occurs. This is called glassification or vitrification. The temperature at which the maximal freeze-concentration (Cg`) occurs is referred to as the glass transition temperature Tg`.[11, 29] This point is at the intersection of the freezing point depression curve and the glass transition or isoviscosity curve, described in the “supplemented phase diagram”  or “state diagram” (figure 2). Tg´ is the point on the glass transition curve, representing a reversible change between viscous, rubber-like liquid and rigid, glass system. In the region of the glass transition, the viscosity of the freeze concentrate changes about four orders of magnitude over a temperature range of a few degrees. Tg` depends on the composition of the solution, but is independent of the initial concentration.[4, 11, 27] For example, for the maximally freeze concentration of sucrose a concentration of 72-73% is reported. In addition to Tg` the collapse temperature (Tc) of a product is used to define more precisely the temperature at which a structural loss of the product will occur. In general Tc is several degrees higher than Tg`, as the high viscosity of the sample close to Tg` will prevent . The glassy state is a solid solution of concentrated solutes and unfrozen, amorphous water. It is thermodynamically unstable with respect to the crystal form, but the viscosity is high enough, in the order of 1014 Pa*s, that any motion is in the order of mm/year.[4, 11, 29]
The important difference between eutectic crystallization and vitrification is that for crystalline material, the interstitial between the ice crystal matrix consists of an intimate mixture of small crystals of ice and solute, whereas for amorphous solutes, the interstitial region consists of solid solution and unfrozen, amorphous water.[19, 23] Thus, for crystalline material nearly all water is frozen and can easily be removed during primary drying without requiring secondary drying. However, for amorphous solutes, about 20% of unfrozen water is associated in the solid solution, which must be removed by a diffusion process during secondary drying. Moreover, the Teu for crystalline material or the Tg` respectively Tc for amorphous material define the maximal allowable product temperature during primary drying. Eutectic melting temperatures are relatively high compared to glass transition temperatures, allowing a higher product temperature during primary drying, which results in more efficient drying processes. If the product temperature exceeds this critical temperature crystalline melting or amorphous collapse will occur, resulting in a loss of structure in the freeze-dried product, which is termed “cake collapse”.[11, 19]
2.3 Phase separation and other types of freezing behavior
A characteristic property of multicomponent aqueous solutions, especially when at least one component is a polymer, is the occurrence of a liquid-liquid phase separation during freezing into two liquid equilibrium phases, which are enriched in one component.[11, 19] This phase separation behavior has been reported for aqueous solutions of polymers such as PEG/dextran or PVP/dextran but is also reported for proteins and excipients.[32-33] When a critical concentration of the solutes is reached, the enthalpically unfavorable interactions between the solutes exceed the favorable entropy of a solution with complete miscibility. Another proposed explanation is that solutes have different effects on the structure of water, leading to phase separation.
Besides the separation into two amorphous phases, two other types of phase separation are stated in literature; crystallization of amorphous solids and amorphization from crystalline solids. Crystallization of amorphous solids often occurs when metastable glasses are formed during freezing. In this case, e.g. upon extremely fast cooling, a compound that normally would crystallize during slower freezing is entrapped as an amorphous, metastable glass in the freeze-concentrate.[12, 23] However, with subsequent heating above Tg`, it will undergo crystallization, which is the basis for annealing during freeze-drying (see 3.3). Without annealing, the metastable glass can crystallize spontaneously out of the amorphous phase during drying or storage. Amorphization from crystalline solids, that can be buffer components or stabilizers, predominantly occurs during the drying step and not during the freezing step.[18, 36]
Additionally, lyotropic liquid crystals, which have the degree of order between amorphous and crystalline, are reported to form as a result of freeze-concentration. However, their influence on critical quality attributes of the lyophilized product are not clarified. Moreover, clathrates, also termed gas hydrates, are known to form, especially in the presence of non-aqueous co-solvents, when the solute alters the structure of the water.
3. Modifications of the freezing step
As aforementioned, the ice nucleation temperature defines the size, number and morphology of the ice crystals formed during freezing. Therefore, the statistical nature of ice nucleation poses a major challenge for process control during lyophilization. This highlights the importance of a controlled, reproducible and homogeneous freezing process. Several methods have been developed in order to control and optimize the freezing step. Some of them only intend to influence ice nucleation by modifying the cooling rate. Others just statistically increase the mean nucleation temperature, while a few allow a true control of the nucleation at the desired nucleation temperature.
3.1 Shelf-ramped freezing
Shelf-ramped freezing is the most often employed, conventional freezing condition in lyophilization. Here, at first, the filled vials are placed on the shelves of the lyophilizer and the shelf temperature is then decreased linearly (0.1°C/min up to 5°C/min, depending on the capacity of the lyophilizer) with time.[37-38] As both water and ice have low thermal conductivities and large heat capacities and as the thermal conductivity between vials and shelf is limited, the shelf-ramped cooling rate is by nature slow. In order to ensure the complete solidification of the samples, the samples must be cooled below Tg` for amorphous material respectively below Teu for crystalline material. Traditionally, many lyophilization cycles use a final shelf temperature of -50°C or lower, as this was the maximal cooling temperature of the freeze-drier. Nowadays, it is suggested to use a final shelf temperature of -40°C if the Tg` or Teu is higher than -38°C or to use a temperature of 2°C less than Tg` and Teu. Moreover, complete solidification requires significant time. In general, the time for complete solidification depends on the fill volume; the larger the fill volume the more time is required for complete solidification. Tang et al. suggest that the final shelf temperature should be held for 1 h for samples with a fill depth of less than or equal to 1 cm or 2 h for samples with a fill depth of greater than 1 cm. Moreover, fill depth of greater than 2 cm should be avoided, but if required, the holding time should be increased proportionately.
In order to obtain a more homogeneous freezing, often the vials are equilibrated for about 15 to 30 min at a lowered shelf temperature (5°C - 10°C) before the shelf temperature is linearly decreased. Here, either the vials are directly loaded on the cooled shelves or the vials are loaded at ambient temperature and the shelf temperature is decreased to the hold temperature. [1, 5, 9] Another modification of the shelf-ramped freezing is the two-step freezing, where a “supercooling holding” is applied.(7) Here, the shelf temperature is decreased from room temperature or from a preset lowered shelf temperature to about -5 to -10°C for 30 to 60min hold. This leads to a more homogenous supercooling state across the total fill volume.[1, 5] When the shelf temperature is then further decreased, relatively homogeneous ice formation is observed.
In general, shelf-ramped frozen samples show a high degree of supercooling but when the nucleation temperature is reached, ice crystal growth proceeds extremely fast, resulting in many small ice crystals.[9, 39] However, the ice nucleation cannot be directly controlled when shelf-ramped freezing is applied and is therefore quite random. Thus, one drawback of shelf-ramped freezing is that different vials may become subject to different degrees of supercooling, typically about +/- 3°C about the mean. This results in a great variability in product quality and process performance. Moreover, with the shelf-ramped freezing method it is not practical to manipulate the ice nucleation temperature as the cooling rates are limited inside the lyophilizer and the degree of supercooling might not change within such a small range.[1, 14]
3.2 Pre-cooled shelf method
When applying the pre-cooled shelf method, the vials are placed on the lyophilizer shelf which is already cooled to the desired final shelf temperature, e.g. -40°C or -45°C.[1, 13-14] It is reported that the placement of samples on a pre-cooled shelf results in higher nucleation temperatures (-9,5°C) compared to the conventional shelf-ramped freezing (-13.4°C). Moreover, with this lowered degree of supercooling and more limited time for thermal equilibration throughout the fill volume, the freezing rate after ice nucleation is actually slower compared to shelf-ramped freezing. In addition, a large heterogeneity in supercooling between vials is observed for this method. A distinct influence of the loading shelf temperature on the nucleation temperature is described in literature.[13-14] Searles et al. found that the nucleation temperatures for samples placed on a shelf at -44°C were several degrees higher than for samples placed on a -40°C shelf. Thus, when using this method the shelf temperature should be chosen with care.
Annealing is defined as a hold step at a temperature above the glass transition temperature. In general, annealing is performed to allow for complete crystallization of crystalline compounds and to improve inter-vial heterogeneity and drying rates.[1, 19] Tang et al. proposed the following annealing protocol: when the final shelf temperature is reached after the freezing step, the product temperature is increased to 10 to 20°C above Tg` but well below Teu and held for several hours. Afterwards the shelf temperature is decreased to and held at the final shelf temperature. Annealing has a rigorous effect on the ice crystal size distribution [17, 41] and can delete the interdependence between the ice nucleation temperature and ice crystal size and morphology. If the sample temperature exceeds Tg`, the system pursues the equilibrium freezing curve and some of the ice melts.[12, 41] The raised water content and the increased temperature enhance the mobility of the amorphous phase and all species in that phase. This increased mobility of the amorphous phase enables the relaxation into physical states of lower free energy. According to the Kelvin equation ice crystals with smaller radii of curvature will melt preferentially due to their higher free energy compared to larger ice crystals.[12, 37, 41] Ostwald ripening (recrystallization), which results in the growth of dispersed crystals larger than a critical size at the expense of smaller ones, is a consequence of these chemical potential driving forces.[12, 41] Upon refreezing of the annealed samples small ice crystals do not reform as the large ice crystals present serve as nucleation sites for addition crystallization. The mean ice crystal radius rises with time1/3 during annealing.[37, 41] A consequence of that time dependency is that the inter-vial heterogeneity in ice crystal size distribution is reduced with increasing annealing time, as vials comprising smaller ice crystals “catch up” with the vials that started annealing containing larger ice crystals.[12, 17, 37, 41] Searles et al. found that due to annealing multiple sheets of lamellar ice crystals with a high surface area merged to form pseudo-cylindrical shapes with a lower interfacial area. In addition to the increase in ice crystal size, they observed that annealing opened up holes on the surface of the lyophilized cake. The hole formation is explained by the diffusion of water from melted ice crystals through the frozen matrix at the increased annealing temperature. Moreover, in the case of meta-stable glass formation of crystalline compounds, annealing facilitates complete crystallization. Above Tg` the meta-stable glass is re-liquefied and crystallization occurs when enough time is provided. Furthermore, annealing can promote the completion of freeze concentration (devitrification) as it allows amorphous water to crystallize. This is of importance when samples were frozen too fast and water capable of crystallization was entrapped as amorphous water in the glassy matrix. In addition, the phenomenon of annealing also becomes relevant when samples are optimal frozen but are then kept at suboptimal conditions in the lyophilizer or in a freezer before lyophilization is performed.
3.4 Quench freezing
During quench freezing, also referred to as vial immersion, the vials are immersed into either liquid nitrogen or liquid propane (ca. -200°C) or a dry ice/ acetone or dry ice/ ethanol bath (ca. -80°C) long enough for complete solidification and then placed on a pre-cooled shelf.[9, 16] In this case the heat-transfer media is in contact with both the vial bottom and the vial wall , leading to a ice crystal formation that starts at the vial wall and bottom. This freezing method results in a lowered degree of supercooling but also a high freezing rate as the sample temperature is decreased very fast, resulting in small ice crystals. Liquid nitrogen immersion has been described to induce less supercooling than slower methods [9, 37, 39] , but more precise this faster cooling method induces supercooling only in a small sample volume before nucleation starts and freezes by directional solidification.[12, 14] While it is reported that external quench freezing might be advantageous for some applications , this uncontrolled freezing method promotes heterogeneous ice crystal formation and is not applicable in large scale manufacturing.
3.5 Directional freezing
In order to generate straight, vertical ice crystallization, directional respectively vertical freezing can be performed. Here, ice nucleation is induced at the bottom of the vial by contact with dry ice and slow freezing on a pre-cooled shelf is followed. In this case, the ice propagation is vertically and lamellar ice crystals are formed.
A similar approach, called unidirectional solidification, was described by Schoof et al. . Here each sample was solidified in a gradient freezing stage, based on the Power-Down principle, with a temperature gradient between the upper and the lower cooling stage of 50 K/cm, resulting in homogenous ice-crystal morphology.
3.6 Ice-fog technique
In 1990, Rowe  described an ice-fog technique for the controlled ice nucleation during freezing. After the vials are cooled on the lyophilizer shelf to the desired nucleation temperature, a flow of cold nitrogen is led into the chamber. The high humidity of the chamber generates an ice fog, a vapor suspension of small ice particles. The ice fog penetrates into the vials, where it initiates ice nucleation at the solution surface. Rambathla et al.  successfully implemented this technique for temperature-controlled nucleation, in the range of -1°C to -11°C, in laboratory scale lyophilizers. One challenge in this study was to induce ice nucleation in all vials at the same time because the small ice particle of the ice fog will not reach all the vials simultaneously, which resulted in inter-vial heterogeneity.  Therefore, Patel et al. introduced a variation of the ice fog method, in which a reduced pressure in the chamber was applied to enable a faster and more uniform freezing. With this modification a rapid ice nucleation within 1 minute and a uniform ice crystal structure in all vials was observed. Although it is a promising technique to control ice nucleation inside the lyophilizers, the technique is not implemented yet in large-scale, commercial lyophilization.
Another method to control ice nucleation is electrofreezing (EF). Here, a high voltage pulse is applied to generate an ice nucleus at a platinum electrode, that initiates ice crystallization.[46-47] The capability of high voltage to induce ice nucleation in supercooled water, was first reported by Rau  in 1951. However, there are still discussion about the basic mechanism of EF including the influence on molecular dynamics , bubble formation and breakdown  and electrolytic formation of hydrated metal-ion complexes . For this external freezing method, samples are first cooled to the desired temperature, ice nucleation is then induced by EF and samples are further cooled. For direct EF, a simple and disposable electrode setup made of a gold wire, allowing the ice nucleation of many samples at the same time, can be used. However, the presence of high amounts of excipients, especially salts, inhibited ice nucleation.[46-47] Consequently, special electrode caps were developed to achieve ice nuclei generation independent form the sample composition, called indirect EF. In these caps, the ice nucleus is formed at the platinum electrode in a separate small volume of water and grows through a narrow cannula and PTFE-tube into the sample. This method only allowed the parallel freezing of eight samples under identical conditions. The ice nucleation temperature could be induced at the desired temperature by electofreezing (-1.5 to -8.5°C for indirect EF and -4.5 to -8.5°C for direct EF). Small spherical ice crystals grew when the ice nucleation temperature was low and large plate-like ice crystals formed at higher ice nucleation temperatures. Up to date, this freezing method was only applied in modified cryotubes. Moreover, the need for individual electrodes for each sample diminishes the applicability in manufacturing.
3.8 Ultrasound controlled ice nucleation
In addition, nucleation can be induced by ultrasonic vibration, which was first applied in the food science field, e.g. for manufacturing of ice cream. Inda et al.  reported that the phase change from supercooled water to ice by ultrasonic vibration can be actively controlled at the desired freezing temperature. The mechanisms of sono-nucleation are still discussed [51-53]. Acoustic cavititation, which results in the formation of air bubbles in the liquid, is one key factor. In addition, at the final stage of collapse of a cavitating bubble, the equilibrium freezing temperature of water is increased due to very high pressures. This results in an increased supercooling level which is the driving force for ice nucleation.
Nakagawa et al. introduced ultrasound controlled nucleation for lyophilization of pharmaceutical proteins. Here, an ultrasound transducer, which is connected to an ultrasound generator, is attached to an aluminium plate, which is combined with a cooling stage to cool the vials (figure 4). When vials have reached a desired temperature, ice nucleation is triggered with an ultrasound wave. Subsequently, samples were continually cooled down to the final temperature to allow for complete solidification. Larger and directional ice crystals of the dendrite type were found when the sample nucleated at higher temperatures, while smaller und heterogeneous ice crystals were formed at lower nucleation temperatures. Moreover, it was observed that ice crystal initiation by ultrasound started at the bottom of the vial and progressed to the top, resulting in the possible formation of a cryoconcentrated solution layer at the top of the sample. Compared to samples nucleated at the same nucleation temperature but without ultrasound no significant differences in ice morphology were observed, indicating that the ice morphology only depends on the nucleation temperature and not on the mode of nucleation. In a follow on-study Hottot et al.  investigated the effect of ultrasound-controlled nucleation on structural and morphological properties of freeze-dried mannitol solutions. They found that a compromise between nucleation temperature level and ultrasound pulse power is necessary to get the most stable mannitol polymorph with a highly permeable cake structure. Saclier et al. [53, 56] found in a theoretical model and also experimentally that the size and circularity of the ice crystals depends on both supercooling and acoustic power used. In the aforementioned studies, the controlled ice nucleation during freezing was always performed externally. Passot et al.  used a prototype freeze-dryer, in which one of the shelves is equipped with the ultrasound technology. In accordance, they found that the controlled nucleation by ultrasound was possible at a nucleation temperature close to the equilibrium freezing point and that the homogeneity of the whole batch (100 vials) could be improved.
By applying ultrasound induced ice nucleation a significant intra-vial heterogeneity of ice crystal distribution with smaller ice crystals at the vial bottom compared to larger ice crystals at the top was observed. Thus, Nakagawa et al. applied an additional annealing step to reduce the intra-vial heterogeneity. Another challenge for successful implementation is a good mechanical and thermal contact between the more or less curved vial bottoms and the plate surface. An advantage of ultrasound controlled nucleation is that it can be applied without the need for a direct contact with the product and is thus chemically non-invasive.
3.9 Vacuum-induced surface freezing
At low pressure, evaporation of water is favored. The associated enthalpy of evaporation reduces the local temperature in the water surface, which is known as self-cooling, such that the water surface freezes and a thin film of ice is formed.[58-59] Based on this concept, Kramer et al. introduced a “vacuum-induced surface freezing” technique. The vials were placed onto the precooled shelves (+10°C) of the freeze-drier and the pressure was reduced to 1mbar (equals 750 mtorr). After 5 min under these conditions, a 1-3 mm thick layer of ice formed on the surface of the sample. Then, in order to prevent further water loss by boiling and to inhibit melting of the ice film on the surface, the chamber pressure was released to atmospheric pressure as fast as possible and the shelf temperature was decreased to 3-4°C below the eutectic melting temperature of the formulation simultaneously. This temperature was held for 1 h and subsequenlty the shelf temperature was decreased to -40°C. Due to the release of the vacuum and the reduction of the shelf temperature, growth of dendritic ice crystals occurs, resulting in the formation of long, chimney-like, extremely large ice crystals. It is unclear wether this technique can be scaled up for commercial applications. Moreover, the major drawback of this method is the high risk of uncontrolled boiling, which can result in a “puff off” , when the unfrozen portion in the vial boils and blows up the frozen surface, which can influence the concentration of the sample  and which might also influence the API. Therefore, Liu et al.  modified this method: The temperature was held at -10°C for equilibration before pulling a vacuum to 600 mTorr to induce freezing, in parallel the shelf temperature was rapidly (>1°C/min) decreased to -45°C. They showed that the lowered initial shelf temperature was necessary, because when vacuum induced freezing was performed at higher temperatures, ice is only formed at the top of the vial, as the heat uptake by evaporation is not enough to lower the temperature of the whole fill volume, and therefore results in a two-layer solidification (the first from vacuum-induced freezing, the second from shelf cooling).
3.10 High-pressure shift freezing or depressurization technique
“High-pressure assisted freezing” was the fist freezing method wherein increased pressures were applied to promote freezing.[15, 60] Under high pressure the freezing point of water is lowered and a large number of small ice crystals is formed. It also has been shown that this method generates smaller ice crystals compared to other conventional rapid freezing methods such as liquid nitrogen immersion. Up to now this method was only applied in food science.
Later, researchers from the same group investigated a “high-pressure shift freezing” method.[63-64] Here, the pressure is released slowly or quickly and the phase transition from liquid to solid occurs as a result of a pressure change and instantaneous ice formation is promoted. In 2007, Gasteyer et al.  introduced an analogous method, referred to as “depressurization method”, for the controlled ice nucleation in samples intended for lyophilization, wherein the samples are initially cooled to the equilibrium freezing temperature in a pressurized gas atmosphere, which is subsequently de-pressurized to induce ice nucleation. The detailed construction of a freeze-dryer, that is applicable for that purpose, is described in more detail in a follow-on patent by Rampersad et al. . The system consists of a freeze-drying chamber, a gas circuit to pressurize the freeze-drying chamber and a separate circuit for depressurization. This freezing method involves several steps. The freeze-dryer is loaded and air within the chamber is purged with a pressurization gas, e.g. argon or nitrogen. The chamber is pressurized up to less than 50 psig (about 2600 torr or 3.5 bar) and then the samples are cooled to and equilibrated at the desired nucleation temperature. The samples are then nucleated by depressurizing the freeze-drying chamber and after the nucleation, the samples are further cooled to the final shelf temperature. Bursac et al.  demonstrated that this freezing method could be applied in both laboratory and small commercial-scale freeze-dryers for a wide range of formulations and containers. They showed that the nucleation of aqueous samples could be well controlled within 1°C of the equilibrium freezing point. One main advantage of this technique is that the samples are only contacted by inert gas, which will be removed from the vials during lyophilization. The technique can basically be implemented with minor equipment additions on freeze-dryers which are designed to withstand pressures e.g. during steam sterilization. [66-67] However, e.g. in the presence of a capacitance manometer or if the product chamber is not intended and approved for such high pressures, the adaptation of this technique will be very cost intensive. This required equipment is now also integrated in commercially available freeze-dryers as the “ControLyoTM nucleation on demand technology”.
3.11 Addition of ice nucleating agents
In general, all insoluble impurities have the potential to serve as ice nucleating agent (INA). INAs promote a heterogeneous ice nucleation process that occurs at higher temperatures compared to samples that do not contain INAs. The most studied non-biogenic INA is silver iodine (AgI), that is also used for cloud seeding and snow-making.[15, 69] AgI enhances ice nucleation as the crystal structure shows a similarity to ice , but also an electric field mechanism is discussed . Among the biogenic INAs, six different species of ice nucleation bacteria have been studied in food science, of which Pseudomonas syringae is the most widely used.[15, 69] These biogenic INAs favor ice nucleation, as their structure is similar to that of the ice crystal lattice, lacks surface charge and is of high hydrophobicity.[15, 72-73] For instance, Searles et al.  used P. syringae (0.001% w/v) and AgI (1mg/ml) to alter the ice nucleation temperature during freezing of a 10% HES solution. The addition of P. syringae reduced supercooling (nucleation at -1.8°C compared to -13.4°C for control samples) and direction solidification with completely lamellar ice crystal structure was observed. Samples seeded with AgI nucleated in a termperature range between -5° to -7.5°C in the center of the meniscus where most of the AgI concentrated and a mixture between spheroidal and lamellar ice crystal structures formed. In contrast, Liu et al. found that also in the case of high fill depth, AgI (0.1 mg per vial) limited supercooling (about -2°C) but that ice grew from bottom to the top of the vial and observed dendritic ice crystals. For P. syringae it is known that the nucleation efficiency depends on the INA concentration  and this also seems to be the case for AgI. Searles et al.  also showed that vials with high particulate contamination, which was achieved by drying the open vials after washing in an uncontrolled laboratory environment, also slightly decreased the degree of supercooling to -11.4°C compared to -13.4°C for conventional shelf-ramped freezing. Overall, the presence of INAs increases the average nucleation temperature but does not allow controlled nucleation and individual vials may show a great heterogeneity. Moreover, the addition of ice nucleating agents is not of practical use for a FDA-regulated and approved pharmaceutical product.
3.12 Non-aqueous co-solvents
A comprehensive review on the potential of non-aqueous co-solvents in lyophilization has been published by Teagarden and Baker. The use of non-aqueous co-solvents has both, advantages and disadvantages. The advantages include: increased sublimation rate leading to a decreased drying time, potentially increased wettability, improved reconstitution characteristics, increased solubility and stability of some drugs in solution and enhanced sterility assurance. Disadvantages might be: operator safety concerns due to high degree of flammability or explosion potential, toxicity and regulatory issues because of residual solvent levels. The most extensively used co-solvent used in lyophilization is tertiary butyl alcohol, tert-butanol (TBA), as it is 100% miscible in water, shows a high freezing point (24°C) and a high vapor pressure (26.8 mmHg at 20°C). Other co-solvents which do not freeze completely in commercial freeze-driers are very difficult to use and often result in unacceptable freeze dried cakes. In general, the specific effect of TBA is related to the modification of the ice crystal habit, leading to the growth of needle shaped crystals. However, the water-TBA mixture shows an extremely complex series of eutectic, peritectic and hydration phenomena. Kasraian et al.  suggested a phase diagram for TBA-water systems, which can be described as side by side placement of two simple eutectic phase diagrams, one for water-TBA hydrate with Teu at 20% TBA and one for TBA hydrate-TBA with Teu at 90% TBA. Depending on the concentration of TBA used, ice, solid TBA hydrate or solid TBA will separate upon cooling. At concentrations lower than 20%, pure ice will form at first leading to an increasingly concentrated TBA solution, wherein TBA crystallizes out as TBA hydrate when the concentration is increased to 20% TBA.
The size and morphology of the ice crystals depends on the amount of TBA present. In the presence of 1% TBA, the ice crystal morphology does not differ from pure water.  At a concentration of 3% larger, dendritic ice crystals form and above this concentration but below the eutectic concentration of 20% needle-shaped ice crystals form. In accordance, Liu et al. found that in the presence of 5% TBA, large needle-like ice crystals were formed that grew faster than the ice crystals in the control. In addition, it is reported that the freezing rate also influences the size of the ice crystals in the presence of TBA, with smaller ice crystals for fast freezing and bigger ice crystals for slow freezing. Moreover, the collapse temperature of e.g. sucrose is not influenced by TBA addition (3-10% w/v).
The level of residual solvent in the final product can be critical when TBA is added to the samples. It is influenced by the initial TBA concentration, the freezing rate and the physical state of the solutes. In general, crystalline samples contain very low levels of residual TBA. In the presence of an amorphous solid the removal of TBA is hindered when used at low initial concentrations (<2%), as TBA does not crystallize but is dispersed in the amorphous phase. Additionally, fast-freezing results in high residual TBA levels, as the complete crystallization is inhibited. Annealing, which promotes TBA crystallization, lowers residual TBA levels. An additional critical point is the time between filling the highly volatile solvent and freezing of the solution. This time span should kept as short as possible to avoid reduced cake heights and to avoid dry powder spots near the neck of the vial after drying, as some dissolved substances can be carried along the evaporating stream and recondense near the top of the vial.
Vial pretreatment by scoring, scratching or roughening was also applied to lower the degree of supercooling during freezing. Searles et al. used scored vials, which were scratched at the bottom interior surface with a metal scribe. The produced surface defects or scraped glass particles were supposed to catalyze ice nucleation. However, only a marginally increase in ice nucleation temperature to -13.1°C compared to -13.4°C for shelf-ramped frozen samples was observed. With regard to particle contamination this method is undesirable for pharmaceuticals. Randolph et al.  suggested the incorporating of ice nucleation chemistry into the vial interior. But up to now, no successful implementation was demonstrated.
4. Consequences of the freezing step on product quality and process performance
As aforementioned, the freezing process directly influences number, size and shape of ice crystal formation. The ice crystal properties are set early in the freezing process by the ice nucleation temperature, but also by the freezing rate and the time required for complete solidification and will directly impact several quality attributes of the product like morphology, product uniformity, physical state, residual moisture content or reconstitution time and also primary and secondary drying performance, as summarized in table 1. However, it should also be emphasized that the product quality and process performance is not only influenced by the process conditions but also by other factors like formulation composition, fill volume and fill depth or properties of the glass vials.
4.1 Intra-vial and inter-vial uniformity
During freezing the growth of ice crystals as well as the distribution of solutes across the vial can be heterogeneous, which is both reflected by intra-vial uniformity. Intra-vial heterogeneity results in unpredictable changes in sublimation rate during drying and in the most extreme case unacceptable product quality. In general, the lower the temperature equilibration in the sample, the more heterogeneous will ice crystals form across the vial. For example, ice crystal distribution is more homogeneous for the two-step shelf-ramped frozen samples compared to shelf ramped freezing without a holding step. Moreover, the slower the freezing rate, the more time is available for the solute to concentrate ahead of the advancing freezing front. For shelf-ramped freezing high heterogeneity in solute distribution across the vial is reported, especially at high fill volumes. Patapoff et al.  concluded from a first fast and later slower increase in dry layer resistance that the structure of the dried cake varied in the vertical direction when the sample was frozen by shelf-ramping. Liu et al.  determined the vertical heterogeneity by three-section weight analysis. For the shelf-ramped freezing a highly concentrated core in the middle was found. In this study, the best intra-vial uniformity resulted from shelf-ramped two-step freezing or by addition of TBA. Vacuum induced freezing often results in a two-layer solidification, one from vacuum-induced freezing and one from shelf cooling, if the controlling parameters are not sufficiently adjusted. Annealing is suggested to improve intra-vial and also inter-vial heterogeneity. [20, 41, 54] During annealing, larger ice crystals grow at the expense of smaller ones, leading to a large and more uniform ice crystal size that is no longer dependent on ice nucleation temperature.
Inter-vial or batch uniformity is a consequence of the stochastic phenomenon of nucleation. Thus, all the vials in a batch do not have the same nucleation temperature and will not behave equally during drying. Passot et al.  found that the ultrasound and precooled shelf method allowed a significant increase in inter-vial uniformity when compared the shelf-ramped freezing. The best batch uniformity was obtained by the addition of a nucleating agent. As aforementioned annealing has a inter-vial homogenization effect, as well. Webb et al.  demonstrated that that the variations in primary drying endpoints were 3 to 4 times larger for non-annealed versus annealed samples. Bursac et al. showed that the depressurization technique decreased the standard variations between the vials by about 60%, indicating an improved batch homogeneity. Although ice nucleating agents can improve inter-vial uniformity, the best way to produce a homogeneous batch, is to directly control the ice nucleation temperature in all vials of a batch during freezing.
4.2 Product morphology
As the sublimation of the ice crystals leaves pores in the solute matrix, texture and porosity of the final, dried product is directly fixed by the details of ice growth in the freezing process. In general, it is proposed that the cake texture changes from a homogeneous, sponge-like structure for samples with a high degree of supercooling to a lamellar structure with a degree of orientation for samples with a low degree of supercooling. However, not only the degree of supercooling determines ice crystal morphology but also the freezing rate and time required for complete solidification is of importance. In general, freezing methods where supercooling exceeds 5°C freeze by global supercooling and result in a dispersed spherulitic morphology. The size of the spheroidal pores correlates directly with the degree of supercooling. Directional solidification is often observed at very high cooling rates or when ice nucleation is induced close to the equilibrium freezing point. This directional solidification shows lamellar plate morphology and the interface velocity is a major determinant.[12, 14]
The high freezing rate after slow cooling via shelf-ramped freezing results in very small spherulitic pores in the order of 100µm in diameter and a sponge-like matrix. In the two-step freezing process, the samples are first equilibrated in a super-cooled state across the whole sample volume. Thus, when shelf temperature is further increased rapid and homogeneous freezing occurs, resulting in small, uniform, spherulitic pores. For the pre-cooled shelf method, a lower and less consistent degree of supercooling was observed. An enormous temperature gradient might be observed inside the sample as the time for temperature equilibration is limited, leading to smaller pores at the bottom and large pores at the top. Liquid nitrogen immersion freezes by directional solidification in combination with a high freezing rate, resulting in small, lamellar, oriented pores. The temperature gradient during vertical freezing is more pronounced compared to the pre-cooled shelf method and thus, large chimney-like large lamellar structures are formed. The addition of an annealing steps promotes the formation of large spherulitic pores due to Ostwald ripening. For all methods that allow controlled nucleation at a temperature close to the equilibrium temperature freezing by directional solidification is assumed, which might result in large chimney-like pores. Vacuum induced freezing led to long, parallel, chimney like pores with a diameter of about 200µm. The controlled nucleation via depressurization of a 5% sucrose solution resulted in an increased pore diameter of 120µm compared to 50µm for shelf-ramped freezing. For samples that were frozen by ultrasound induced nucleation, large chimney like pores were observed at the top and smaller lamellar pores at the bottom. With electrofreezing at high nucleation temperatures large lamellar, highly oriented pores were detected. In the presence of AgI a mixed morphology of the cake is produced, with the trend of dominating lamellar structures at the top, where ice nucleation started, and spheroidal structures at the bottom. The addition of P. syringae almost eliminated supercooling and thus enabled a directional solidification with totally lamellar morphology. TBA influences the ice crystal habit and promotes the formation of TBA large, needle-shaped pores form the top to the bottom.
4.3 Primary and secondary drying performance
The primary drying rate respectively the sublimation during primary drying can be expressed by the following equation :
where dm/dt is the mass transfer rate for the water vapor, Po is the equilibrium vapor pressure over ice at the product temperature, Pc is the chamber pressure, Rp is the dry product layer resistance to vapor transfer and Rs is the resistance of the stopper. Rp is much larger than Rs, especially for samples of high concentration and high fill depth. According to this equation, the sublimation rate is directly correlated to the dry layer resistance of the product, which is determined by the freezing step related pore size of the product. In general, the smaller the pores in the solute matrix previously occupied by ice crystals, the greater is the resistance to water vapor flow from the product and the slower is the sublimation rate.
For secondary drying, the remaining unfrozen water, which can be about 20% for amorphous samples, requires the diffusion through the solid matrix to and a desorption step from the surface of the matrix. Thus, the sublimation rate in secondary drying strongly correlates with thickness and the surface area of the interstitial matrix.
Low supercooling close to the equilibrium freezing point results in the formation of large ice crystals, low surface area, and thus accelerated primary drying but slow secondary drying. In contrast, a high degree of supercooling during freezing, results in many small crystals and larger surface area, and thus slow sublimation but fast desorption. In general, for most formulations with a low solid content the decrease in primary drying dominates. Searles et al. showed that a 1% increase in ice nucleation temperature resulted in a 3% increase in drying times. Freezing processes that result in a pronounced supercooling like shelf-ramped freezing or two-step freezing require a substantially extended primary drying time compared to the other available methods.[5, 41] If freezing has to be performed on the shelf of the freeze-drier without the possibility for controlled ice nucleation, the pre-cooled shelf method can be used to reduce the drying time. Searles et al.  found that this method leads to an increase in drying rate by about 14% compared to shelf-ramped freezing. Similar results were obtained by Passot et al., who described an approx. 18% shorter sublimation time. However, annealing appears to be the more effective possibility to decrease drying rates in this case. Searles et al.  reported that annealing for only 30 min increased the primary drying rate by a factor of 3.5 because larger ice due to Ostwald ripening were formed. But it was also discussed that the formation of large holes on the cake surface of the annealed surface might additionally have facilitated sublimation. In general, higher annealing temperatures and longer annealing times correlate with a faster drying rate. But all drying rates were limited to a maximum value that was reached when the sublimation rate was no longer controlled by mass transfer but by energy transfer. Annealing was not always found to increase the drying rate, as the physical state of the sample might have an impact, as well. For example, it was observed that in systems that contain a crystalline phase, like mannitol/ trehalose/ sodium chloride, primary drying time was increased due to changes in the pore structure via changes in crystallinity. Freezing by immersion in liquid nitrogen has been shown to result in very small ice crystals and large surface area and thus, decreased primary drying rates compared to shelf-ramped freezing were observed. In addition, the non-vertical ice formation, obtained by ice growth starting from the wall of the vial, contributes to the restricted ice sublimation for samples frozen by liquid nitrogen immersion. The controlled nucleation via depressurization of a 5% sucrose solution resulted in a increased pore size, leading to a shortened drying time by 27% according to Bursac et al. . In comparison, Kramer et al.  showed that the vacuum induced surface freezing method led to the formation of large, chimney-like pores and thus, primary drying time was decreased by 25% for a 2% mannitol formulation and by 15% for a 2% sucrose formulation. Nakagawa et al. reported that the controlled ice nucleation by ultrasound close to the equilibrium freezing temperature (-2°C) increased the primary drying rate by 60% compared to samples that nucleated at lower nucleation temperatures (-8°C). An only 18% decrease in sublimation time for samples nucleated using ultrasound was observerd by Passot et al. . When the biogenic ice nucleation agent P. syringae was added to the samples a 30%  respectively 60%  increase in drying rate was experienced. In this case, the faster drying rate was not only a result of the lowered degree of supercooling but was also attributed to an increase in lamellar ice crystal content.  The addition of TBA was also found to increase the ice sublimation rate.  For example Kasraian et al.  found that the addition of 5% TBA to a 5% sucrose solution resulted in a ~10 fold decrease in product resistance and drying time. One reason is that TBA has the ability to modify the crystal habit of ice so that large needle-shaped ice crystals were formed resulting in a decreased mass transfer resistance. But also the fact that TBA itself has a high vapor pressure plays a role in the increase of sublimation rates.[74, 82-83] Daoussi et al.  reported a 10 to 30 times higher sublimation rate of a 90% TBA sample compared to traditional aqueous formulations.
Besides the direct correlation between drying rate and pore size, also the direction of ice crystal formation, the connection between the pores and the formation of a skin on the top of the product can influence the drying performance. For example it was found that for vertically oriented ice, obtain
Cite This Dissertation
To export a reference to this article please select a referencing stye below: