2 Nucleation

Marco Mazzotti,1 Thomas Vetter,2 David R. Ochsenbein,1 Giovanni M. Maggioni,1 Christian Lindenberg3

2.1 Introduction

The term “nucleation” is used to describe the onset of the formation of a new phase from a parent phase [1, 2]. Examples of nucleation processes include the formation of vapor bubbles in a liquid phase, the formation of droplets from a vapor phase or from another liquid, as well as the formation of crystalline particles from vapor, liquid or even another solid. In the following, we will focus on the formation of new crystalline particles from solution exclusively. Apart from the breakage of an already existing particle into two or more pieces, nucleation is the only mechanism generating new crystals and is therefore of fundamental interest. From a processing perspective it is notoriously a phenomenon hard to control in batch crystallizers, a fact that increases the importance of process design strategies that avoid or minimize nucleation through the addition of previously prepared seed crystals. It is useful to classify the events leading to the formation of nuclei highly anisotropic into diﬀerent types [3, 4]: primary homogeneous nucleation refers to the formation of nuclei from clear liquids; primary heterogeneous nucleation occurs when nuclei are formed with the participation of foreign surfaces (such as stirrers, crystallizer walls, or crystals of another form); secondary nucleation describes the formation of nuclei of one crystal form with the participation of already-present crystals of the same crystal form. While much of what is

1Institute of Process Engineering, ETH Zürich, Switzerland 2School of Chemical Engineering and Analytical Science, University of Manchester, United Kingdom 3Novartis Pharma AG, Switzerland

1 CHAPTER 2. NUCLEATION

described in the following sections applies to all three types of nucleation, there are slight differences that need to be accounted for. It is noteworthy that these different nucleation mechanisms can occur simulta- neously and to varying degrees throughout a crystallization process; moreover, the nucleation rates of different solids, e.g., polymorphs, may compete with each other. For instance, starting from a clear solution containing a solute, it is possible that mixtures of different polymorphs are obtained during the crystallization process when the nucleation kinetics of the different crystal forms are comparable. However, since different crystal forms exhibit different stability, all forms but the most stable one, will ultimately convert into the stable form upon reaching thermodynamic equilibrium. Nevertheless, all pathways for nucleation share the fact that the formation of nuclei involves the concomitant creation of a new bulk phase and of an interface. When there is a driving force for crystallization, the former leads to a decrease in free energy of the system, while the latter increases it. This interplay of contributions often leads to the presence of a substantial energy barrier for the creation of a nucleus. In such cases, the original state of the parent phase is not thermodynamically unstable, but rather metastable and the rate of formation of nuclei, i.e., the nucleation rate, is finite. This implies that a purely thermodynamic understanding of the system, albeit essential, is insufficient to describe nucleation; a kinetic understanding of the system is additionally required in order to answer questions pertaining to the dynamics of the system. Furthermore, the effect of the operating conditions on both aspects needs to be understood. Theories of nucleation aim at achieving this goal and a brief overview of the most important theoretical frameworks, experimental characterization tools, and applications shall be given in the following. Finally, this chapter will also highlight how the use of seed crystals in a crystallization process allows obtaining a desired crystal form, which is of importance when the performance of a product strongly depends on the manufactured crystal form.

2.2 Homogeneous Nucleation

In order to describe the formation of nuclei, we must first define the driving force for crystallization. We can express it as the difference in chemical potential between a solute molecule in the solution at its current state, µ`, and in the ∗ solution’s equilibrium state, µ` . In practical applications, it is more convenient

2 2.2. Homogeneous Nucleation

to express this driving force in terms of the supersaturation S. These quantities can be related by:

∗ a` c ∆µ = µ` − µ` = kT ln S = kT ln ∗ ≈ kT ln ∗ (2.1) a` c

where k is the Boltzmann constant, T is the temperature, a` represents the ∗ activity of the solute in the supersaturated solution and a` is the activity of the solute in the solution’s equilibrium state (note that al = cγ, where c is the concentration and γ its activity coefficient). While the last step introduced in Eq. (2.1) using concentrations instead of activities is only accurate for ideal solutions, where γ ≈ γ∗ ≈ 1, or when γ/γ∗ ≈ 1 (which is a much less stringent requirement), it represents a useful approximation as concentrations are experimentally accessible quantities. Concerning the thermodynamics and kinetics of homogeneous nucleation, two theories are of particular importance. The first one is the oldest and probably best known theory of nucleation: classical nucleation theory (CNT). Originally developed for the nucleation of droplets and bubbles, the CNT is also extensively applied to crystals, which, in stark contrast to droplets/bubbles, exhibit a strong supramolecular structure. The second theory we will discuss in this chapter represents a refinement of the CNT which recognizes this difference. It is often referred to in literature as two-step nucleation theory (2-SNT)[5–7]; its development took place mainly during the last three decades and was fueled by considerable progress in measurement devices allowing for the observation of ever smaller entities in solution[8–10]. We will first introduce the two theories on a conceptual level before putting them on a more rigorous theoretical footing. A schematic illustration of the two theories is shown in Figure 2.1 where different key states of the solution and of the newly forming phase are depicted. We depict these states in relation to the number of molecules that are contained in the new phase and the number of molecules with crystalline order in this phase. In both theories the initial state for primary homogeneous nucleation is a clear, supersaturated solution (S > 1) containing a number of solute molecules that exhibit no crystalline order (top left corner in Figure 2.1) and a final state consisting of one or more macroscopic crystals (bottom right corner). However, the pathways connecting the two states are rationalized in different ways in the CNT (blue arrows) and

3 CHAPTER 2. NUCLEATION

the 2-SNT (orange arrows); key states along the two pathways are discussed in the following. Classical nucleation theory states that crystalline clusters are forming from the supersaturated solution through simultaneous fluctuations in density and order. Building blocks of crystals (assumed here to be solute molecules) can attach to or detach from a cluster in a step-by-step fashion. When molecules are attaching/detaching, the blue pathway laid out in Figure 2.1 is traveled reversibly depending on the rates of molecule attachment and detachment. We will show in Section 2.2.1 that the attachment of molecules to a cluster is energetically unfavorable until a critical number of molecules is reached while it is favored beyond this number. Two-step nucleation theory, on the other hand, treats the evolution of density and structure of the newly formed phase independently from each other. Namely, the 2-SNT postulates that density fluctuations first lead to the formation of disordered, liquid-like clusters (or droplets), which exhibit a higher density than the original clear liquid. Only after this step, crystalline order is achieved through rearrangement of the molecules within that cluster.

4 2.2. Homogeneous Nucleation

number of molecules in new phase(s)

molecules in solution free molecules and disordered, liquid‐like cluster order

free molecules and free molecules and crystalline crystalline cluster cluster in liquid‐like cluster crystalline

... with

molecules

of free molecules and free molecules and crystalline nucleus crystalline nucleus inside disordered cluster number

macroscopic crystal

Figure 2.1: Conceptual picture of the mechanisms of nucleation according to classical nucleation theory (CNT; blue arrows) and to two step nucleation theory (2-SNT; orange arrows). Both theories consider a supersaturated solution as their starting point. According to both theories, clusters are forming/disintegrating through the attachment/detachment of building units. In the CNT, clusters are assumed to exhibit the ﬁnal crystal structure immediately when building units attach. In contrast, the 2-SNT assumes that nucleation proceeds through the formation of disordered, liquid-like clusters in a ﬁrst step, while the formation of structured clusters occurs from these droplets in a second step. Upon reaching a critical cluster size – the so-called nucleus size – the attachment of further building units is energetically favored; ultimately leading to the formation of a macroscopic crystal.

5 CHAPTER 2. NUCLEATION

2.2.1 Classical Nucleation Theory

Classical nucleation theory dates back to the work of Gibbs, Becker and Döring, Volmer and many others [11, 12]. The interested reader can ﬁnd the full derivation, historical details, and various technical aspects of CNT in the classical works of Kashchiev and deBenedetti [1, 13]. In order to describe the thermodynamic aspects of CNT, we derive the energy diﬀerence between a solution and a solution containing a cluster of n molecules. As hinted at in the introduction, the total change in the system free energy, ∆Gn, that is, the work of cluster formation, is given by the sum of two terms: an energy gain due to formation of a crystalline bulk phase and an energy loss caused by the formation of an interface between the new and the parent phase. We may thus write:

∆Gn = −n∆µ + σAn (2.2) where the bulk contribution has been expressed using the crystallization driving force, ∆µ (Eq. (2.1)) and the surface contribution is expressed through the surface energy σ and the surface area of the cluster An. We may assume that the cluster geometry can be characterised by a characteristic length L. 3 The volume and the area of the cluster are then expressed as V = kvL and 2 A = kaL , respectively. kv and ks are the volume and surface shape factors (for cube of side L, kv = 1 and ka = 6; for sphere of diameter L, kv = π/6 and ka = π). The Gibbs free energy of a cluster of size L, at given temperature T and supersaturation S, can then be expressed as:

3 kv 3 2 kvL 2 ∆G(L) = − L ∆µ + σkaL = − kT ln S + σkaL (2.3) vc vc

where vc is the molecular volume. From this equation, it is clear that ∆G is zero for L = 0, goes through a maximum and decreases monotonically afterwards. This behavior is shown in Figure 2.2 for several supersaturations. The maximum on each curve represents an unstable equilibrium, implying that a cluster of such a critical size, Lc, has an equal probability of growing into a full crystal and of dissolving back into solvated molecules. A cluster of critical

6 2.2. Homogeneous Nucleation

Figure 2.2: Free energy required according to classical nucleation theory to form a cluster of size L. Each curve is drawn at constant values of supersaturation and temperature; the symbol on each curve marks the maximum of the free energy and the corresponding critical size Lc.

size is typically referred to as a nucleus. Lc may be computed from Eq. (2.3) by setting d∆G/dL = 0, thus yielding:

2ka σvc Lc = (2.4) 3kv kT ln S

The value of the corresponding free energy change is

3 3 2 4ka σ vc ∆Gc = (2.5) 2 2 2 2 27kv k T ln S

The term between brackets in Eq. (2.5) is a constant dependent on the nucleus geometry: for a sphere, it equals 16π/3; for a cube, it is 32. Analogous to the reaction coordinate used in chemical systems, the cluster size L can be interpreted as the characteristic coordinate for nucleation, which describes the evolution of the system energy, ∆G, during the formation and evolution of

7 CHAPTER 2. NUCLEATION

a cluster. Using this analogy, we can interpret ∆Gc as the activation barrier to nucleation and the nucleus as a transition state. As already visible from Figure 2.2, we see that Eq. (2.5) and Eq. (2.4) indicate that higher supersaturations lead to lower energy barriers for nucleation and smaller nuclei sizes. This ﬁnding is crucial, because according to transition state theory, the rate of passing through a transition state is proportional to the exponential of the height of the c energy barrier that needs to be crossed. In other words, the nucleation rate J, i.e., the number of nuclei formed in a solution per unit time and unit volume, is found to be:

∆G B J = AS exp − c = AS exp − (2.6) kT ln2 S with

3 3 2 4ka σ vc B = 2 3 3 (2.7) 27kv k T

The pre-exponential factor, J0 = AS, in this equation is typically seen as a product of the number of nucleation sites and the frequency of building block attachment to the cluster. Within the literature discussing the CNT, several limiting cases have been considered when deriving the attachment frequency [1, 14], leading to slightly diﬀerent functional dependencies pre-exponential factor on the operating conditions. However, the resulting expressions share their linear dependence on the concentration in solution (and hence the supersaturation, as explicitly written in Eq. (2.6)). The number of nucleation sites is, however, notoriously hard to predict, often leading to discrepancies of several orders of magnitude between experimentally measured data and theoretical predictions. This is often attributed to the presence of small (sub-micron) particles or other impurities in solutions that act as nucleation sites, but other explanations exist as well (see section on 2-SNT below). While the presence of such “dust” particles and other heterogeneous surfaces (stirrers, crystallizer walls) is often most pronounced on the kinetic parameter A, it is also known to aﬀect the thermodynamic parameter B in Eq. (2.6). This will be discussed in Section 2.3.1. P ractitioners are there-

8 2.2. Homogeneous Nucleation

fore often content to gather experimental data and treat A and B as ﬁtting parameters. Nevertheless, it is important to realize that classical nucleation theory has delivered some mechanistic insight on the functional form of the nucleation rate, which can prove invaluable for process design. For instance, the observation of a metastable zone, i.e., a zone in the phase diagram where nucleation is not observed within a speciﬁed time-frame, is consistent with the nucleation rate law outlined in Eq. (2.6): the nucleation rate is rapidly increasing only after a certain threshold supersaturation is reached – before this threshold considerable time can pass before nucleation is observed. Note that the metastable zone is a purely kinetic phenomenon and should not be thought of as a thermodynamic boundary.

2.2.2 Two-Step Nucleation Theory

As shown in Figure 2.1, according to 2-SNT, the system evolves from an initial state (the clear solution) to the final state (the crystals and the solution) passing through an intermediate metastable state. The difference between the reaction paths described by CNT and 2-SNT has also a profound effect on the energetics of the two processes, as one can readily see in Figure 2.3. This figure shows the Gibbs free energy change along a reaction coordinate η, which represents both the cluster’s size and its structural properties. From the initial state (∆G = 0), i.e., the clear solution, the system has to overcome two energy ∗ ∗ barriers of height ∆G1 and ∆G2 before ultimately reaching the final state. After the first energy barrier a metastable state is located, which represents the liquid like cluster. It should be noted that the energy level of the liquid like cluster is not necessarily higher than that of the solution, as in Figure 2.3, but could also be lower. Two-step nucleation theory (2-SNT) was originally developed to describe the crystallization of proteins in aqueous solutions, such as lyzozyme in water[15] for which CNT did not agree well with experimental observations. Specifically, for these systems liquid-like clusters were observed in supersaturated solutions before the formation of crystals. The qualitative picture of the Gibbs free energy is different from the one obtained from CNT, which suggests that the mathematical form of the nucleation rate J may be different as well. For crystal nucleation of proteins in aqueous solutions, a phenomenological model has been derived under system specific assumption, and the corresponding nucleation rate has been estimated [15, 16]. Just as for CNT, also for this

9 CHAPTER 2. NUCLEATION

Figure 2.3: A qualitative representation of the energy proﬁle of a cluster as a function of a nucleation coordinate, η, in 2-SNT. The nucleation coordinate encompasses both the cluster size and its degree of crystallinity.

phenomenological model accurate ab initio calculations of the involved energies are diﬃcult, hence the authors took them as ﬁtting parameters [16]. Although the theory was initially conceived to describe nucleation of proteins in solution [7, 15, 17–19], recent evidence and its formal generality suggest that 2-step nucleation could also describe the crystallization of amino-acids, small organic molecules, and even inorganic compounds, some of which exhibit features of a 2-step process[20–22]. It should be pointed out that the nature of the intermediate species formed (density, concentration, structure, etc.), as well as a consistent formulation that applies to all systems is still a topic of debate.

2.3 Heterogeneous and Secondary Nucleation

2.3.1 Heterogeneous nucleation

Forming nuclei through homogeneous nucleation entails crossing substantial energy barriers, i.e., it is an energetically unfavorable process, hence it consti-

10 2.3. Heterogeneous and Secondary Nucleation

tutes a “rare event” even when a considerable driving force for crystallization exists. However, the presence of interfaces can promote nucleation. Industrial crystallizers can rarely – if ever – be considered free from foreign surfaces (such as impellers, microscopic “dust”, crystallizer walls, liquid-air interfaces/bubbles, etc.). Nucleation occurring in the presence of such foreign surfaces is referred to as heterogeneous nucleation. The adsorption of crystallizing material on these surfaces lowers the critical free energy required for the formation of a nucleus. The extent of this reduction is often rationalized to depend on the structural similarity between the foreign surface and the crystal to be formed on it [3]. This is often assumed to lead to a reduction in surface energy, i.e., the surface energy σ in Eq. (2.5) is replaced with an effective surface energy σef = Ψσ, with the effectiveness factor Ψ being between 0 and 1. Clearly, this leads to a decreased value of B in Eq. (2.6) and hence an increased nucleation rate. In fact, this is a general behavior and the presence of any heterogeneous surface increases the nucleation rate in comparison to the case of homogeneous nucleation. In other words, the supersaturation required to reach a certain threshold nucleation rate is lower, hance the metastable zone for heterogeneous nucleation is narrower than for homogeneous nucleation. The nuclei formed through heterogeneous nucleation may either stick on the surface on which they formed or they might be forcibly detached from the surface if sufficient shear or mechanical force is exerted on them. In the former case heterogeneous nucleation is generally a nuisance as it is a source of scaling and encrustation, while in the latter case the nuclei can grow into well-defined crystals suspended in the solution. Unfortunately, the current state of the art does not allow to predict heterogeneous nucleation rates in a quantitative manner from measured or simulated surface characteristics. In practice, this means that the parameters in Eq. (2.6) ought to be estimated from experimental data if a quantitative nucleation rate is desired and that the presence of any material providing surfaces at which heterogeneous nucleation can occur needs to be controlled to gain meaningful and reproducible results.

2.3.2 Secondary nucleation

Primary nucleation produces new nuclei of the substance being crystallized independently of the presence of such crystals in the system. Secondary nucleation, on the contrary, is the phenomenon leading to the formation of new nuclei because of the prior presence of fully grown crystals in the suspension. It is interesting to note that nuclei formed through secondary nucleation ex-

11 CHAPTER 2. NUCLEATION

hibit the same crystal structure as the parent crystal [23, 24] Over the years, multiple mechanisms have been proposed to describe secondary nucleation and the debate regarding the mechanistic interpretation and the mathematical description of the phenomenon is far from being settled [25]. Nevertheless, most authors agree on its qualitative features and that there are a couple of distinct pathways leading to the formation of nuclei through secondary nucleation; they are conceptually depicted in Figure 2.4.

Figure 2.4: Conceptualization of diﬀerent secondary nucleation mechanisms: a) boundary layer mechanisms; b) crystal-crystal collision; c) crystal-impeller collision.

In the first mechanism (depicted in Figure 2.4a)), secondary nuclei form in the boundary layer surrounding each crystal in suspension, or on the crystalline surface itself [25–27]. The details are governed by the specific physico-chemical properties of the system considered and by the crystallization conditions. In the boundary layer, new nuclei form through an activated process similar to nuclei formed through primary nucleation. The energy barrier for this process, however, is believed to be much lower since the particle surface acts as a catalyst for nuclei formation and the surface exhibits the same crystal structure as the new crystals being formed. Once formed the nuclei might be removed from the boundary layer by fluid shear or collisions of the crystal with impeller, walls

12 2.3. Heterogeneous and Secondary Nucleation

or other crystals. Alternatively, they might stick on the surface of the parent crystal, just as in the case of heterogeneous nucleation. A different set of mechanisms occurs when the particle surface is the direct source of new nuclei. It is noteworthy that the mechanical forces needed for these other mechanisms are significantly higher than the one required for the boundary layer removal [28]. The new nuclei generate then by initial breeding, dendritic growth, and attrition. Of these three mechanisms, attrition is likely the most common in industrial processes and was hence drawn in Figure 2.4b) and c). Initial breeding, typically observed in seeded crystallization, assumes that fines have been produced during previous stages of crystallization, adhered to the larger seeds, and are eventually removed from the seeds due to fluid shear and/or collisions. Dendritic growth occurs when the diffusion of solute molecules to the surface of a crystal is limiting the growth rate; it results in dendrites (“needles”) protruding from the orginal surface and can result in multiply branched structures when undisturbed. However, in stirred conditions, the thin dendrites are easily broken off, thus forming new nuclei. Since diffusion limited crystal growth is a necessary part of this mechanism, high supersaturations are usually required for its occurrence, which avoided by process design (e.g., slow cooling or anti-solvent addition), see also Section 2.6. Finally, attrition describes the formation of nuclei due to abrasion. The parent particle, upon collision with other crystals (Figure 2.4b)) or with parts of the crystallizer (Figure 2.4c)), produces fines so small that the particle particle itself can be considered unchanged by each individual abrasion event. Clearly, multiple abrasion events on the same crystal lead to its destruction, unless the abraded crystal can heal through growth. From the conceptual description of these mechanisms, it is clear that the fluid dynamics and several aspects of the crystals influence the rate of secondary nucleation. Relevant aspects of the crystals include: their underlying crystal structure, their morphology, as well as the mechanical and chemical state of the exposed surfaces. Additionally, the supersaturation of the liquid phase plays a decisive role in all the mentioned mechanisms: in the solute layer removal mechanism the thickness of the layer increases with supersaturation [25], supersaturation governs whether dendritic growth occurs, and whether the attrition fragments survive in solution. The latter depends on whether they are larger or smaller than the critical size (c.f. Eq. (2.4)). All these considerations point at an increase of secondary nucleation rate with increasing supersaturation. The discussion presented above clearly highlights that obtaining a quantitative and predictive description of secondary nucleation is considerably difficult and

13 CHAPTER 2. NUCLEATION

has not yet been achieved. Nevertheless, the present qualitative understanding has helped to formulate empirical and semi-empirical rate expressions of secondary nucleation that are able to describe speciﬁc sets of data[25]. The expressions can be summarized as:

a b Jsec = kε (S − 1) g(φi,G) (2.8) where ε is the specific power input into the crystallizer, S is the supersaturation, th and g is a function of the i -moment φi of the crystal size distribution and of the crystal growth rate G, while a, b are fitting parameters. The functional form of g is system-specific as well.

2.4 Characterization of Nucleation

2.4.1 Deterministic Nucleation Rates

Classical nucleation theory and 2-step nucleation theory provide a theoretical framework for describing nucleation kinetics and expressing nucleation rates. Each theory depends on a certain number of parameters; although some of the parameters can be predicted from theory, it is customary to estimate all of them by fitting experimental data. However, the estimation of nucleation kinetics suffers from an important limitation: in most systems, nuclei are small and their formation cannot be directly observed or otherwise detected. Indeed, one can easily calculate that an astonishing number of clusters with critical size would be needed to affect a system’s experimentally accessible properties, such as turbidity or concentration. Hence, only once the nuclei have grown larger, nucleation events that happened earlier be detected. The time elapsed between the attainment of initial supersaturation and the detection of crystals is defined as the detection, or induction time, tD. Con- ceptually, this time can be thought of as the sum of the nucleation time, tN and the growth time, tG, i.e., the time necessary to grow the new nuclei to a sufficient size. Hence, we can write: tD = tN + tG. As a consequence, the information contained in such experiments depends not only on the nucleation rate, but also on the crystal growth rate, G. It is thus clear that one cannot

14 2.4. Characterization of Nucleation

estimate the nucleation rate J from detection experiments without a model or some simplifying assumptions concerning crystal growth. The detection time also depends on the property that is monitored, as well as the technique and the detection limit of the speciﬁc instrument that is used to measure this property. The typically measured quantities in crystallization from solution are conductivity, turbidity, and infra-red intensity, which can be correlated to some property of the crystals (e.g., number of particles, average size, crystal volume fraction) or of the solution (concentration). For instance, a typical formula relating the detectable volume fraction of crystal in solution, αv, deﬁned as the ratio between solid and system volume to the induction time as[1, 29, 30]:

1/4 4αv tD = 3 (2.9) kvJG

where kv is the volume shape factor. In Eq. (2.9) supersaturation and temperature are assumed to be constant during the experiment. For the former assumption to be valid, the detectable volume fraction αv must be sufficiently small. Note that this threshold value is also dependent on the specific experimental set-up used and on the substance system monitored. Several alternative definitions of detection time are possible as well, based on different monitored properties, e.g., the number of particles. Since the detection threshold is a volume fraction, Eq. (2.9) implies that the detection time should be scale-invariant. This property is interesting since small systems allow for better mixing and more uniform temperature and also require also smaller amounts of substances to carry out induction time experiments. The induction time method has been applied to estimate the nucleation kinetics of many systems, particularly in lab-scale crystallizers (100 mL – 10 L), see for example Schöll et al. [29], Lindenberg and Mazzotti [30]. In this type of studies, induction times were measured at different supersaturations, as shown in Figure 2.5. For each supersaturation a limited number of repetitions is usually carried out and significant scatter is observed in the detection times. The average value of the induction time at each supersaturation can then be converted to a nucleation rate (using Eq. (2.9) and a previously determined growth rate), which can be used to estimate parameters in a nucleation rate expression, e.g., Eq. (2.6). The data scattering observed in the example is typical in this type of exper-

15 CHAPTER 2. NUCLEATION

iments, and a rather poor reproducibility of detection time experiments has been consistently reported in the literature. Some theories concerning the origin and the eﬀects of such scattering have been provided, and they are discussed in the following Section 2.4.2.

Figure 2.5: A typical examples of detection time experiments in a 500 mL crystallizer. α-L-glutamic acid was crystallized in water at constant supersaturation, at three diﬀerent temperatures: 25 ◦C (black), 35 ◦C (blue), and 45 ◦C (red). The bars associated to each point are given by the standard deviation. Data from Lindenberg and Mazzotti [30].

16 2.4. Characterization of Nucleation

2.4.2 Stochastic Nucleation Rates

One possible cause of poor reproducibility worth investigating are the ﬂuid- dynamic problems related to mixing, as well as heat and mass transfer lim- itations. These issues are well-known in chemical reactor design, and it is known that strong local gradients can cause uncontrolled and non-reproducible phenomena, e.g., hot spots where the supersaturation, and thus the nucleation rate, is lower. In order to avoid or at least minimize local gradients, induction time experiments have been performed in increasingly smaller volumes over the past thirty years. Today, a considerably large record of data collected in crystallizers from 5 mL to droplets as small as nano- or even pico-liters is available in the literature. Concentration and temperature gradients, which might be uncontrolled sources of variability in larger scale experiments, are negligible in such systems. However, the stochasticity observed in the induction time measurements increases with decreasing volumes, instead of decreasing. As an example, two data sets collected in 1 mL isothermal reactors are shown in Figure 2.6, which clearly indicate that —even in absence of gradients— the detection times are widely scattered. Indeed, comparing these results with those of Figure 2.5 highlights an even broader distribution of detection times. Similar observations have been reported for many other systems, both organic and inorganic, at diﬀerent operating conditions, and, usually, with a broadness of distribution inversely proportional to the size of the system. The evidence has convinced researchers that such statistical behavior is not merely an experimental accident, but an intrinsic feature of nucleation. In fact, nucleation should be interpreted as a stochastic phenomenon whose non-deterministic nature becomes keenly apparent at small system scales. Classical nucleation theory, and also two-step nucleation theory, describe nucleation as an activated process in which an energy barrier needs to be overcome to go from the metastable, supersaturated solution to the stable, crystalline state. When the energy barrier to be overcome between the two states becomes substantial (larger than about 1 kT or roughly 2.5 kJ/mol at room temperature), nucleation becomes a “rare event”. That energy barriers for nucleation lie well beyond this threshold, which can be inferred from experimental data and has also been shown in molecular dynamic simulations[22]. Rare events exhibit an intrinsically stochastic nature, i.e., each realization of an event occurs at a

17 CHAPTER 2. NUCLEATION

random time, but the ensemble of all possible realizations follows a statistical distribution, as it is observed in detection time experiments.

a) b)

Figure 2.6: Typical examples of detection time distributions observed in small volumes: a) acetaminophen crystallized in water in 1 mL vials by employing a constant cooling rate, but different solution concentration (red higher than blue). Data from Kadam et al. [31]. The different curves correspond to two different saturation temperatures (different solution concentrations): 60 ◦C (red), and 30 ◦C, b) p-aminobenzoic acid crystallized in acetonitrile in 1.5 mL under isothermal conditions, but different initial supersaturations. Data from Sullivan et al. [32].

Various models have hence been proposed to reconcile theory and experiments, to identify the statistical distributions describing the data, and to use such distributions to establish reliable nucleation kinetics from the experimental data[15, 31, 33–36]. There are strong theoretical arguments that primary nucleation should be described by Poisson statistics and many experimental dataset exhibit these statistics. In the Poisson model, for constant temperature and supersaturation, the probability P of having formed at least one nucleus up to time tN, in a system of volume V and of nucleation rate J, is:

P (tN) = 1 − exp (−JV tN) (2.10)

18 2.5. Order of Polymorph Appearance – Ostwald’s Rule of Stages

The distribution of nucleation times is described by Eq. (2.10) and its mean value and its standard deviation are inversely proportional not only to the volume of the system, but also to the nucleation rate; since the latter increases with the supersaturation, this equation may explain why the data at lower supersaturation appears more scattered than those at higher supersaturation (cf Figure 2.6, but also Figure 2.5). To prove the stochastic hypothesis of nucleation and to estimate nucleation rates, not only for the case of Poisson distribution considered here, but for any statistical distribution, the detection time experiments and their analysis must satisfy some important requirements: First, the diﬀerent experimental repetitions must be carried out under the same conditions; only then can they be assumed to form part of the same statistical distribution. Second, since one can only measure detection times, whereas the stochastic hypothesis concerns nucleation times, one must be able to link the detection times to the nucleation times. Hence, a mathematical model describing stochastic nucleation must also account for the growth of the nuclei into fully developed crystals. Third, a representative sample of the stochastic process must be gathered, i.e., a large enough number of experimental points must be collected, so that they form a representative sample of the detection time distribution underlying the stochastic process[37].

2.5 Order of Polymorph Appearance – Ostwald’s Rule of Stages

One of the goals in many manufacturing processes is to consistently obtain a specific polymorph. However, many substances exhibit multiple polymorphs. It is therefore no surprise that there is an extensive body of work dealing with the order in which the different forms emerge in a process. Most prominently, Ostwald was the first to note that—conspicuously often—the least stable polymorph appears to be the first to nucleate in a system, followed by the second-most unstable form, etc. [38]. This empirical observation, made by many researchers for a large set of additional compounds, has since been named Ostwald’s rule of stages (OSR). While OSR does not represent a physical law and several counter examples exist (e.g., [39]), there have been various theories attempting to explain the underlying cause of this phenomenon. Ther- modynamic arguments have been brought forward [40], as well as arguments that claim properties of the solution structure to be relevant for polymorph selection [41].

19 CHAPTER 2. NUCLEATION

Here, we shall brieﬂy present a line of reasoning based on nucleation kinetics favored by a majority of authors [42–44], that has the advantage of being compatible with the observed supersaturation dependence encountered in various systems [45], and of revealing a clear, systematic path for process design. Fundamentally, this interpretation implies that depending on the solution envi- ronment, certain pre-critical cluster conﬁgurations associated with particular polymorphic forms are favored. Assuming that the nucleation kinetics of a given compound may be described by Eq. (2.6), the logarithm of the ratio of the nucleation rates for two competing polymorphs is given by:

J A c∗ B B ln u = ln u + ln s + s − u (2.11a) ∗ 2 ∗ 2 ∗ Js As cu ln (c/cs ) ln (c/cu)

The labels u and s here refer to the less stable and more stable form, respectively, Eq. (2.11a) can be recast in a more compact form by deﬁning α = ln(Au/As), ∗ ∗ 2 λ = ln(cs /cu) and βi = Bi/ ln Si.

Ju ln = α + λ + βs(c) − βu(c) (2.11b) Js

In Eq. (2.11b) we have highlighted those terms that depend on changes in the liquid concentration, c. The behavior of the system is visualized in Figure 2.7; ∗ ∗ note that, by definition, cs < cu and λ < 0. ∗ It is easy to see that Eq. (2.11) approaches negative infinity for c → cu and that it tends toward (α + λ) for large supersaturations. In other words, at concentrations approaching the solubility of the unstable form, the nucleation of the stable form is more likely, while at higher concentrations the dominant form depends on the value of the kinetic parameters. If, for example, we have that (α+λ) > 0, higher supersaturation will eventually always lead to preferred nucleation of the unstable form; see Figure 2.7a). If, on the other hand, this inequality does not hold (cf. Figure 2.7b)), the unstable form can only dominate nucleation if Eq. (2.11) has a maximum (only the case when Bu ≤ Bs) and if that extremum lies above zero (Bu must be sufficiently smaller than Bs); the particular conditions can be found easily by solving the associated system of equations.

20 2.5. Order of Polymorph Appearance – Ostwald’s Rule of Stages

Figure 2.7: Behavior of the ratio Ju/Js as a function of the concentration c. Left: α + λ > 0; the nucleation of the unstable polymorph dominates always for increasing concentrations; right: α + λ < 0; the nucleation of the unstable polymorph may dominate only if Bs is suﬃciently larger than Bu and within intermediate concentrations. Figure adapted from [46].

Naturally, the behavior of the system will be affected by temperature and an analogous analysis can be performed to determine its effect. To make an example, if α is found to exhibit a strong negative correlation with temperature, that is, ∂α/∂T 0. For a given solute concentration and assuming the solubility ratio λ is only a weak function of T , it would then be beneficial to operate at elevated temperatures to produce the stable form and at lower temperatures to manufacture the unstable polymorph. Summarizing, given primary nucleation kinetics of two or more polymorphs, it is comparably easy to identify operating conditions at which the nucleation of a particular form is favored based on some simple analysis of the system; knowledge of the pre-factors and the relative stability (α and λ) alone may even be sufficient to decide upon a strategy for production. Furthermore, the variety of possible cases is rather limited for simple kinetics, as can be seen from Figure 2.7. The above type of analysis retains its usefulness regardless of the type of process under consideration, i.e., batch or continuous crystallization. Similar approaches can be used for other types of nucleation kinetics that are dependent on other factors, including the choice of solvent, additives, templates, or the mixing rate in the case of reactive crystallization [47, 48]; the effect of these properties on the polymorph selectivity may be much greater than that of

21 CHAPTER 2. NUCLEATION

supersaturation or temperature alone. Nevertheless, this approach hinges on the fact that nucleation rates are assumed to be known. If this is not the case, some prior kinetics estimation work—as, e.g., outlined in Section 2.4—is necessary. If such studies lead to unsatisfactory results or nucleation during the crystallization step is ruled out as a viable option for other reasons, the logical response is to attempt suppressing nucleation entirely, making instead use of seed crystals to drive the production of a speciﬁc polymorph.

2.6 To Seed or Not to Seed?

Seeding is widely used in the chemical and pharmaceutical industry to control crystal properties and to ensure constant product quality such as crystalline form, particle size distribution and purity. Compared to their unseeded coun- terparts, seeded crystallization processes can be performed at relatively low supersaturations which is favorable for control of the afore-mentioned properties.

2.6.1 Process Control

As mentioned above, the metastable zone is the region in the phase diagram in which the supersaturation is suﬃciently low to avoid spontaneous nucleation (of a given polymorph), illustrated, e.g., by the blue and red regions in Figure 2.8. In seeded crystallization processes, the seed crystals should be added within the metastable zone in order to avoid spontaneous nucleation; values of S at the seeding point in industrial crystallization of organic compounds typically range from 1.1 to 1.5. The seed crystals may be added as suspension to enable quick dispersion in the vessel and to avoid formation of lumps; furthermore, they should be smaller than the desired product crystals. Therefore, a particle comminution step, such as dry milling or sieving, is often required in the preparation of seeds. In industrial practice, it is also important to use seed crystals with reproducible particle size distribution, in order to ensure low product variability. The optimum amount and size of the seed crystals depends on the desired particle size of the product. Assuming the crystal size increases only by crystal growth and secondary eﬀects such as agglomeration, breakage and secondary nucleation are negligible, the mean particle size of product crystals, Lp, can be

22 2.6. To Seed or Not to Seed?

approximated by:

1/3 1 + Cs Lp = Ls (2.12) Cs

where Ls is the size of the seed crystals and Cs is the seed loading, deﬁned as

[Seed Mass] ms Cs = = . (2.13) [Yield] mp − ms

In Eq. (2.13), ms represents the mass of seed crystals and mp that of product crystals [49]. The total surface area of the seeds, which in turn depends on their size and amount, determines also the maximum rate of cooling or antisolvent addition with which supersaturation is still kept a level that would trigger secondary effects. Kubota and co-workers proposed an experimental method to determine the optimum conditions for seeded crystallization. First, experiments using different amount and size of seed crystals are conducted. Secondly, the experimental values of Lp/Ls are plotted versus the seed loading Cs and compared to the ideal growth line given by Eq. (2.12), see Figure 2.9. Experimental values of Lp/Ls below the ideal growth line may be caused by nucleation (low amount seeds, large seed crystals or high supersaturation caused by fast cooling or antisolvent addition) or breakage (brittle crystal, intense stirring) [50]. Values above the ideal growth line may be a result of agglomeration (high supersaturation, insufficient stirring or high concentration of relatively small crystals) [51]. Despite the fact that seeded crystallization processes generally are robust and deliver constant product quality, variability may be introduced by impurities having an effect on solubility and growth rate [52], or by changes in the equipment or process parameter such as stirrer type or stirring rate. It is also known that the growth rate of crystals may not be constant over time and is impacted by defects in the crystal lattice as well as relative surface area of different crystal facets of the seed crystals [53].

23 CHAPTER 2. NUCLEATION

Figure 2.8: Solubility and metastable zone of a stable polymorph (blue) and a metastable polymorph (red) for the case of a monotropic system. Solid line: solubility; dashed lines: limit of the metastable zone. The blue diamond depicts the seeding point for obtaining the stable polymorph and the red triangle the seeding point for the metastable polymorph.

2.6.2 Polymorphism Control

Seeding can also be employed to control the polymorphic form of a compound produced by crystallization. In principle, there are four possible situations that may occur in practice [54]:

1. The desired polymorph is stable and identical to the polymorph generated by primary nucleation. In this case seeding may be mainly utilized to improve process robustness as described above, but not to control the polymorphic form.

2. The required polymorph is stable and the polymorph generated by primary nucleation is metastable. As outlined in Section 2.5, this case, which follows Ostwald’s rule of stages, is relatively common. Seeding with the stable polymorph at low supersaturations and below the supersaturation of the metastable polymorph avoids occurrence of the metastable polymorph in the process, see Figure 2.8.

24 2.6. To Seed or Not to Seed?

Figure 2.9: Seed chart. Solid line: ideal growth line; diamonds: product following the ideal growth line; triangles: product particle size below ideal growth line as a result of nucleation; squares: product particle size above ideal growth line as a result of agglomeration.

3. The required polymorph is metastable and is that which is also generated by primary nucleation. Seeding with the metastable form may avoid the occurrence of the stable form. However, there is a high risk of a polymorph transformation during the process since the supersaturation of the stable form is always higher (for monotropic systems or below the transition temperature for enantiotropic systems), see Figure 2.8.

4. The required polymorph is metastable and the polymorph generated by primary nucleation is stable. Seeding cannot be employed to ensure only the metastable polymorph is obtained.

“Disappearing” polymorphs, that is, those that have been manufactured until a certain point in time but subsequent attempts to produce this polymorph result in a diﬀerent form, are linked to situations 3 and 4, i.e., the desired form is a metastable polymorph. Their disappearance can be explained by the occurrence of a new more stable polymorph and the unintentional or universal seeding, which describes the presence of small amounts of the new more stable polymorph in all laboratories or production facilities [55].

25 CHAPTER 2. NUCLEATION

2.6.3 Impurity control

Crystallization is a very efficient separation process potentially leading to fully pure material in a single process step. In reality, however, impurities, by-products or solvents can still be found in the crystallite. There are multiple reasons why the particle may not be pure, including absorption of impurities on the crystal surface, incorporation into the crystal lattice, entrapment of impurities between agglomerated crystals, precipitation of the impurity as separate crystals or amorphous particles. Impurities may be incorporated as point defects into the crystal lattice at equilibrium conditions and the incorporation can be characterized by a segregation coefficient [56, 57]. In industrial crystallization, the amount of impurities found in the crystallite can exceed the amounts predicted by the segregation coefficient and is related to non-equilibrium growth processes resulting in inclusions or defects. The tendency to form inclusions or defects increases with supersaturation [56]. Therefore, seeded crystallization processes operated a low supersaturation may be employed to improve the purification effect.

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