Nucleation - ETH Zürich - Homepage | ETH Zürich · 2 Nucleation Marco Mazzotti, 1Thomas Vetter,2...

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2 Nucleation Marco Mazzotti, 1 Thomas Vetter, 2 David R. Ochsenbein, 1 Giovanni M. Maggioni, 1 Christian Lindenberg 3 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 1 Institute of Process Engineering, ETH Zürich, Switzerland 2 School of Chemical Engineering and Analytical Science, University of Manchester, United Kingdom 3 Novartis Pharma AG, Switzerland 1

Transcript of Nucleation - ETH Zürich - Homepage | ETH Zürich · 2 Nucleation Marco Mazzotti, 1Thomas Vetter,2...

  • 2

    Nucleation

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

    2.1

    Introduction

    The term “nucleation” is used to describe the onset of the formation of a newphase from a parent phase [1, 2]. Examples of nucleation processes include theformation of vapor bubbles in a liquid phase, the formation of droplets froma vapor phase or from another liquid, as well as the formation of crystallineparticles from vapor, liquid or even another solid. In the following, we willfocus 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 thereforeof fundamental interest. From a processing perspective it is notoriously aphenomenon hard to control in batch crystallizers, a fact that increases theimportance of process design strategies that avoid or minimize nucleationthrough the addition of previously prepared seed crystals.

    It is useful to classify the events leading to the formation of nuclei highlyanisotropic into different types [3, 4]: primary homogeneous nucleation refersto the formation of nuclei from clear liquids; primary heterogeneous nucleationoccurs when nuclei are formed with the participation of foreign surfaces (suchas stirrers, crystallizer walls, or crystals of another form); secondary nucleationdescribes the formation of nuclei of one crystal form with the participationof already-present crystals of the same crystal form. While much of what is

    1Institute of Process Engineering, ETH Zürich, Switzerland2School of Chemical Engineering and Analytical Science, University of Manchester,

    United Kingdom3Novartis Pharma AG, Switzerland

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  • CHAPTER 2. NUCLEATION

    described in the following sections applies to all three types of nucleation, thereare 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 witheach other. For instance, starting from a clear solution containing a solute,it is possible that mixtures of different polymorphs are obtained during thecrystallization process when the nucleation kinetics of the different crystalforms are comparable. However, since different crystal forms exhibit differentstability, all forms but the most stable one, will ultimately convert into thestable form upon reaching thermodynamic equilibrium.

    Nevertheless, all pathways for nucleation share the fact that the formationof nuclei involves the concomitant creation of a new bulk phase and of aninterface. When there is a driving force for crystallization, the former leadsto a decrease in free energy of the system, while the latter increases it. Thisinterplay of contributions often leads to the presence of a substantial energybarrier for the creation of a nucleus. In such cases, the original state of theparent phase is not thermodynamically unstable, but rather metastable andthe rate of formation of nuclei, i.e., the nucleation rate, is finite. This impliesthat a purely thermodynamic understanding of the system, albeit essential,is insufficient to describe nucleation; a kinetic understanding of the system isadditionally required in order to answer questions pertaining to the dynamicsof the system. Furthermore, the effect of the operating conditions on bothaspects needs to be understood.

    Theories of nucleation aim at achieving this goal and a brief overview of themost important theoretical frameworks, experimental characterization tools,and applications shall be given in the following. Finally, this chapter willalso highlight how the use of seed crystals in a crystallization process allowsobtaining a desired crystal form, which is of importance when the performanceof 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 drivingforce for crystallization. We can express it as the difference in chemical potentialbetween a solute molecule in the solution at its current state, µ`, and in thesolution’s equilibrium state, µ∗` . In practical applications, it is more convenient

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  • 2.2. Homogeneous Nucleation

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

    ∆µ = µ` − µ∗` = kT lnS = kT ln(a`a∗`

    )≈ kT ln

    (c

    c∗

    )(2.1)

    where k is the Boltzmann constant, T is the temperature, a` represents theactivity of the solute in the supersaturated solution and a∗` is the activity ofthe solute in the solution’s equilibrium state (note that al = cγ, where c is theconcentration and γ its activity coefficient). While the last step introducedin Eq. (2.1) using concentrations instead of activities is only accurate forideal solutions, where γ ≈ γ∗ ≈ 1, or when γ/γ∗ ≈ 1 (which is a much lessstringent requirement), it represents a useful approximation as concentrationsare experimentally accessible quantities.

    Concerning the thermodynamics and kinetics of homogeneous nucleation, twotheories are of particular importance. The first one is the oldest and probablybest known theory of nucleation: classical nucleation theory (CNT). Originallydeveloped for the nucleation of droplets and bubbles, the CNT is also extensivelyapplied to crystals, which, in stark contrast to droplets/bubbles, exhibit a strongsupramolecular structure. The second theory we will discuss in this chapterrepresents a refinement of the CNT which recognizes this difference. It isoften referred to in literature as two-step nucleation theory (2-SNT)[5–7]; itsdevelopment took place mainly during the last three decades and was fueledby considerable progress in measurement devices allowing for the observationof ever smaller entities in solution[8–10].

    We will first introduce the two theories on a conceptual level before puttingthem on a more rigorous theoretical footing. A schematic illustration of the twotheories is shown in Figure 2.1 where different key states of the solution andof the newly forming phase are depicted. We depict these states in relation tothe number of molecules that are contained in the new phase and the numberof molecules with crystalline order in this phase. In both theories the initialstate for primary homogeneous nucleation is a clear, supersaturated solution(S > 1) containing a number of solute molecules that exhibit no crystallineorder (top left corner in Figure 2.1) and a final state consisting of one or moremacroscopic crystals (bottom right corner). However, the pathways connectingthe two states are rationalized in different ways in the CNT (blue arrows) and

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  • CHAPTER 2. NUCLEATION

    the 2-SNT (orange arrows); key states along the two pathways are discussed inthe following.

    Classical nucleation theory states that crystalline clusters are forming fromthe supersaturated solution through simultaneous fluctuations in density andorder. Building blocks of crystals (assumed here to be solute molecules) canattach to or detach from a cluster in a step-by-step fashion. When moleculesare attaching/detaching, the blue pathway laid out in Figure 2.1 is traveledreversibly 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 isenergetically unfavorable until a critical number of molecules is reached whileit is favored beyond this number. Two-step nucleation theory, on the otherhand, treats the evolution of density and structure of the newly formed phaseindependently from each other. Namely, the 2-SNT postulates that densityfluctuations first lead to the formation of disordered, liquid-like clusters (ordroplets), which exhibit a higher density than the original clear liquid. Onlyafter this step, crystalline order is achieved through rearrangement of themolecules within that cluster.

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  • 2.2. Homogeneous Nucleation

    number of molecules in new phase(s)

    numbe

    r of m

    olecules with

     crystalline orde

    r

    macroscopic crystal

    free molecules and crystalline nucleus

    free molecules and disordered, liquid‐like cluster

    molecules in solution

    free molecules and crystalline cluster 

    free molecules and crystalline nucleus inside 

    disordered cluster

    ...

    free molecules and crystalline cluster in liquid‐like cluster

    Figure 2.1: Conceptual picture of the mechanisms of nucleation accordingto classical nucleation theory (CNT; blue arrows) and to two step nucle-ation theory (2-SNT; orange arrows). Both theories consider a supersaturatedsolution as their starting point. According to both theories, clusters are form-ing/disintegrating through the attachment/detachment of building units. In theCNT, clusters are assumed to exhibit the final crystal structure immediatelywhen building units attach. In contrast, the 2-SNT assumes that nucleationproceeds through the formation of disordered, liquid-like clusters in a firststep, while the formation of structured clusters occurs from these droplets in asecond step. Upon reaching a critical cluster size – the so-called nucleus size– the attachment of further building units is energetically favored; ultimatelyleading to the formation of a macroscopic crystal.

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  • CHAPTER 2. NUCLEATION

    2.2.1

    Classical Nucleation Theory

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

    ∆Gn = −n∆µ+ σAn (2.2)

    where the bulk contribution has been expressed using the crystallization drivingforce, ∆µ (Eq. (2.1)) and the surface contribution is expressed through thesurface energy σ and the surface area of the cluster An. We may assumethat the cluster geometry can be characterised by a characteristic length L.The volume and the area of the cluster are then expressed as V = kvL3 andA = kaL2, 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 andka = π). The Gibbs free energy of a cluster of size L, at given temperature Tand supersaturation S, can then be expressed as:

    ∆G(L) = −kvvcL3∆µ+ σkaL2 = −

    kvL3

    vckT lnS + σkaL2 (2.3)

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

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  • 2.2. Homogeneous Nucleation

    Figure 2.2: Free energy required according to classical nucleation theory to forma cluster of size L. Each curve is drawn at constant values of supersaturationand temperature; the symbol on each curve marks the maximum of the freeenergy 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:

    Lc =2ka3kv

    σvckT lnS (2.4)

    The value of the corresponding free energy change is

    ∆Gc =(

    4k3a27k2v

    )σ3v2c

    k2T 2 ln2 S(2.5)

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

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  • CHAPTER 2. NUCLEATION

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

    J = AS exp(−∆GckT

    )= AS exp

    (− B

    ln2 S

    )(2.6)

    with

    B =(

    4k3a27k2v

    )σ3v2ck3T 3

    (2.7)

    The pre-exponential factor, J0 = AS, in this equation is typically seen as aproduct of the number of nucleation sites and the frequency of building blockattachment to the cluster. Within the literature discussing the CNT, severallimiting cases have been considered when deriving the attachment frequency [1,14], leading to slightly different functional dependencies pre-exponential factoron the operating conditions. However, the resulting expressions share their lineardependence 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, oftenleading to discrepancies of several orders of magnitude between experimentallymeasured data and theoretical predictions. This is often attributed to the pres-ence of small (sub-micron) particles or other impurities in solutions that actas nucleation sites, but other explanations exist as well (see section on 2-SNTbelow). While the presence of such “dust” particles and other heterogeneoussurfaces (stirrers, crystallizer walls) is often most pronounced on the kineticparameter A, it is also known to affect the thermodynamic parameter B inEq. (2.6). This will be discussed in Section 2.3.1. P ractitioners are there-

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  • 2.2. Homogeneous Nucleation

    fore often content to gather experimental data and treat A and B as fittingparameters.

    Nevertheless, it is important to realize that classical nucleation theory hasdelivered some mechanistic insight on the functional form of the nucleation rate,which can prove invaluable for process design. For instance, the observation ofa metastable zone, i.e., a zone in the phase diagram where nucleation is notobserved within a specified time-frame, is consistent with the nucleation ratelaw outlined in Eq. (2.6): the nucleation rate is rapidly increasing only after acertain threshold supersaturation is reached – before this threshold considerabletime can pass before nucleation is observed. Note that the metastable zone is apurely kinetic phenomenon and should not be thought of as a thermodynamicboundary.

    2.2.2

    Two-Step Nucleation Theory

    As shown in Figure 2.1, according to 2-SNT, the system evolves from an initialstate (the clear solution) to the final state (the crystals and the solution)passing through an intermediate metastable state. The difference between thereaction paths described by CNT and 2-SNT has also a profound effect onthe energetics of the two processes, as one can readily see in Figure 2.3. Thisfigure shows the Gibbs free energy change along a reaction coordinate η, whichrepresents both the cluster’s size and its structural properties. From the initialstate (∆G = 0), i.e., the clear solution, the system has to overcome two energybarriers of height ∆G∗1 and ∆G∗2 before ultimately reaching the final state.After the first energy barrier a metastable state is located, which representsthe liquid like cluster. It should be noted that the energy level of the liquidlike 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 thecrystallization 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 solutionsbefore the formation of crystals. The qualitative picture of the Gibbs freeenergy is different from the one obtained from CNT, which suggests thatthe mathematical form of the nucleation rate J may be different as well. Forcrystal nucleation of proteins in aqueous solutions, a phenomenological modelhas been derived under system specific assumption, and the correspondingnucleation rate has been estimated [15, 16]. Just as for CNT, also for this

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  • CHAPTER 2. NUCLEATION

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

    phenomenological model accurate ab initio calculations of the involved energiesare difficult, hence the authors took them as fitting parameters [16].

    Although the theory was initially conceived to describe nucleation of proteinsin solution [7, 15, 17–19], recent evidence and its formal generality suggestthat 2-step nucleation could also describe the crystallization of amino-acids,small organic molecules, and even inorganic compounds, some of which exhibitfeatures of a 2-step process[20–22]. It should be pointed out that the nature ofthe intermediate species formed (density, concentration, structure, etc.), as wellas 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 substantialenergy barriers, i.e., it is an energetically unfavorable process, hence it consti-

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  • 2.3. Heterogeneous and Secondary Nucleation

    tutes a “rare event” even when a considerable driving force for crystallizationexists. However, the presence of interfaces can promote nucleation. Industrialcrystallizers can rarely – if ever – be considered free from foreign surfaces (suchas impellers, microscopic “dust”, crystallizer walls, liquid-air interfaces/bubbles,etc.). Nucleation occurring in the presence of such foreign surfaces is referredto as heterogeneous nucleation. The adsorption of crystallizing material onthese surfaces lowers the critical free energy required for the formation of anucleus. The extent of this reduction is often rationalized to depend on thestructural similarity between the foreign surface and the crystal to be formedon 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 energyσef = Ψσ, with the effectiveness factor Ψ being between 0 and 1. Clearly, thisleads to a decreased value of B in Eq. (2.6) and hence an increased nucleationrate. In fact, this is a general behavior and the presence of any heterogeneoussurface increases the nucleation rate in comparison to the case of homogeneousnucleation. In other words, the supersaturation required to reach a certainthreshold nucleation rate is lower, hance the metastable zone for heterogeneousnucleation is narrower than for homogeneous nucleation.

    The nuclei formed through heterogeneous nucleation may either stick on thesurface on which they formed or they might be forcibly detached from thesurface if sufficient shear or mechanical force is exerted on them. In the formercase heterogeneous nucleation is generally a nuisance as it is a source of scalingand encrustation, while in the latter case the nuclei can grow into well-definedcrystals suspended in the solution. Unfortunately, the current state of theart does not allow to predict heterogeneous nucleation rates in a quantitativemanner from measured or simulated surface characteristics. In practice, thismeans that the parameters in Eq. (2.6) ought to be estimated from experimentaldata if a quantitative nucleation rate is desired and that the presence of anymaterial providing surfaces at which heterogeneous nucleation can occur needsto be controlled to gain meaningful and reproducible results.

    2.3.2

    Secondary nucleation

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

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  • 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 nucleationand the debate regarding the mechanistic interpretation and the mathematicaldescription of the phenomenon is far from being settled [25]. Nevertheless, mostauthors agree on its qualitative features and that there are a couple of distinctpathways leading to the formation of nuclei through secondary nucleation; theyare 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-impellercollision.

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

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  • 2.3. Heterogeneous and Secondary Nucleation

    or other crystals. Alternatively, they might stick on the surface of the parentcrystal, just as in the case of heterogeneous nucleation.

    A different set of mechanisms occurs when the particle surface is the directsource of new nuclei. It is noteworthy that the mechanical forces needed forthese other mechanisms are significantly higher than the one required for theboundary layer removal [28]. The new nuclei generate then by initial breeding,dendritic growth, and attrition. Of these three mechanisms, attrition is likelythe most common in industrial processes and was hence drawn in Figure 2.4b)and c). Initial breeding, typically observed in seeded crystallization, assumesthat fines have been produced during previous stages of crystallization, adheredto the larger seeds, and are eventually removed from the seeds due to fluid shearand/or collisions. Dendritic growth occurs when the diffusion of solute moleculesto the surface of a crystal is limiting the growth rate; it results in dendrites(“needles”) protruding from the orginal surface and can result in multiplybranched structures when undisturbed. However, in stirred conditions, the thindendrites are easily broken off, thus forming new nuclei. Since diffusion limitedcrystal growth is a necessary part of this mechanism, high supersaturationsare usually required for its occurrence, which avoided by process design (e.g.,slow cooling or anti-solvent addition), see also Section 2.6. Finally, attritiondescribes the formation of nuclei due to abrasion. The parent particle, uponcollision 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 beconsidered unchanged by each individual abrasion event. Clearly, multipleabrasion events on the same crystal lead to its destruction, unless the abradedcrystal can heal through growth.

    From the conceptual description of these mechanisms, it is clear that the fluiddynamics and several aspects of the crystals influence the rate of secondarynucleation. Relevant aspects of the crystals include: their underlying crystalstructure, their morphology, as well as the mechanical and chemical state ofthe exposed surfaces. Additionally, the supersaturation of the liquid phaseplays a decisive role in all the mentioned mechanisms: in the solute layerremoval mechanism the thickness of the layer increases with supersaturation[25], supersaturation governs whether dendritic growth occurs, and whetherthe attrition fragments survive in solution. The latter depends on whetherthey are larger or smaller than the critical size (c.f. Eq. (2.4)). All theseconsiderations point at an increase of secondary nucleation rate with increasingsupersaturation.

    The discussion presented above clearly highlights that obtaining a quantitativeand predictive description of secondary nucleation is considerably difficult and

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  • CHAPTER 2. NUCLEATION

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

    Jsec = kεa (S − 1)b g(φi, G) (2.8)

    where ε is the specific power input into the crystallizer, S is the supersaturation,and g is a function of the ith -moment φi of the crystal size distribution andof the crystal growth rate G, while a, b are fitting parameters. The functionalform 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 theoreticalframework for describing nucleation kinetics and expressing nucleation rates.Each theory depends on a certain number of parameters; although some ofthe parameters can be predicted from theory, it is customary to estimate allof them by fitting experimental data. However, the estimation of nucleationkinetics suffers from an important limitation: in most systems, nuclei are smalland their formation cannot be directly observed or otherwise detected. Indeed,one can easily calculate that an astonishing number of clusters with criticalsize would be needed to affect a system’s experimentally accessible properties,such as turbidity or concentration. Hence, only once the nuclei have grownlarger, nucleation events that happened earlier be detected.

    The time elapsed between the attainment of initial supersaturation and thedetection 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, tNand the growth time, tG, i.e., the time necessary to grow the new nuclei toa sufficient size. Hence, we can write: tD = tN + tG. As a consequence, theinformation contained in such experiments depends not only on the nucleationrate, but also on the crystal growth rate, G. It is thus clear that one cannot

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  • 2.4. Characterization of Nucleation

    estimate the nucleation rate J from detection experiments without a model orsome simplifying assumptions concerning crystal growth.

    The detection time also depends on the property that is monitored, as well asthe technique and the detection limit of the specific instrument that is usedto measure this property. The typically measured quantities in crystallizationfrom solution are conductivity, turbidity, and infra-red intensity, which can becorrelated to some property of the crystals (e.g., number of particles, averagesize, crystal volume fraction) or of the solution (concentration). For instance, atypical formula relating the detectable volume fraction of crystal in solution,αv, defined as the ratio between solid and system volume to the induction timeas[1, 29, 30]:

    tD =( 4αvkvJG3

    )1/4(2.9)

    where kv is the volume shape factor. In Eq. (2.9) supersaturation and tem-perature are assumed to be constant during the experiment. For the formerassumption to be valid, the detectable volume fraction αv must be sufficientlysmall. Note that this threshold value is also dependent on the specific experi-mental set-up used and on the substance system monitored. Several alternativedefinitions of detection time are possible as well, based on different monitoredproperties, e.g., the number of particles. Since the detection threshold is a vol-ume fraction, Eq. (2.9) implies that the detection time should be scale-invariant.This property is interesting since small systems allow for better mixing andmore uniform temperature and also require also smaller amounts of substancesto carry out induction time experiments.

    The induction time method has been applied to estimate the nucleation kineticsof many systems, particularly in lab-scale crystallizers (100 mL – 10 L), seefor example Schöll et al. [29], Lindenberg and Mazzotti [30]. In this type ofstudies, induction times were measured at different supersaturations, as shownin Figure 2.5. For each supersaturation a limited number of repetitions isusually carried out and significant scatter is observed in the detection times.The average value of the induction time at each supersaturation can then beconverted to a nucleation rate (using Eq. (2.9) and a previously determinedgrowth rate), which can be used to estimate parameters in a nucleation rateexpression, e.g., Eq. (2.6).

    The data scattering observed in the example is typical in this type of exper-

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  • CHAPTER 2. NUCLEATION

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

    Figure 2.5: A typical examples of detection time experiments in a 500 mLcrystallizer. α-L-glutamic acid was crystallized in water at constant supersatu-ration, 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].

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  • 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 isknown that strong local gradients can cause uncontrolled and non-reproduciblephenomena, e.g., hot spots where the supersaturation, and thus the nucleationrate, is lower.

    In order to avoid or at least minimize local gradients, induction time experimentshave been performed in increasingly smaller volumes over the past thirty years.Today, a considerably large record of data collected in crystallizers from 5 mLto droplets as small as nano- or even pico-liters is available in the literature.Concentration and temperature gradients, which might be uncontrolled sourcesof variability in larger scale experiments, are negligible in such systems. However,the stochasticity observed in the induction time measurements increases withdecreasing volumes, instead of decreasing. As an example, two data sets collectedin 1 mL isothermal reactors are shown in Figure 2.6, which clearly indicatethat —even in absence of gradients— the detection times are widely scattered.Indeed, comparing these results with those of Figure 2.5 highlights an evenbroader distribution of detection times.

    Similar observations have been reported for many other systems, both organicand inorganic, at different operating conditions, and, usually, with a broadnessof distribution inversely proportional to the size of the system. The evidencehas convinced researchers that such statistical behavior is not merely anexperimental accident, but an intrinsic feature of nucleation. In fact, nucleationshould be interpreted as a stochastic phenomenon whose non-deterministicnature becomes keenly apparent at small system scales.

    Classical nucleation theory, and also two-step nucleation theory, describe nucle-ation as an activated process in which an energy barrier needs to be overcometo go from the metastable, supersaturated solution to the stable, crystallinestate. When the energy barrier to be overcome between the two states becomessubstantial (larger than about 1 kT or roughly 2.5 kJ/mol at room tempera-ture), nucleation becomes a “rare event”. That energy barriers for nucleation liewell beyond this threshold, which can be inferred from experimental data andhas also been shown in molecular dynamic simulations[22]. Rare events exhibitan intrinsically stochastic nature, i.e., each realization of an event occurs at a

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  • CHAPTER 2. NUCLEATION

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

    a) b)

    Figure 2.6: Typical examples of detection time distributions observed in smallvolumes: a) acetaminophen crystallized in water in 1 mL vials by employinga constant cooling rate, but different solution concentration (red higher thanblue). Data from Kadam et al. [31]. The different curves correspond to twodifferent saturation temperatures (different solution concentrations): 60 ◦C(red), and 30 ◦C, b) p-aminobenzoic acid crystallized in acetonitrile in 1.5 mLunder isothermal conditions, but different initial supersaturations. Data fromSullivan 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 suchdistributions to establish reliable nucleation kinetics from the experimentaldata[15, 31, 33–36]. There are strong theoretical arguments that primarynucleation should be described by Poisson statistics and many experimentaldataset exhibit these statistics.

    In the Poisson model, for constant temperature and supersaturation, theprobability P of having formed at least one nucleus up to time tN, in a systemof 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 meanvalue and its standard deviation are inversely proportional not only to thevolume of the system, but also to the nucleation rate; since the latter increaseswith the supersaturation, this equation may explain why the data at lowersupersaturation 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 nucleationrates, not only for the case of Poisson distribution considered here, but forany statistical distribution, the detection time experiments and their analysismust satisfy some important requirements: First, the different experimentalrepetitions must be carried out under the same conditions; only then can theybe assumed to form part of the same statistical distribution. Second, since onecan only measure detection times, whereas the stochastic hypothesis concernsnucleation times, one must be able to link the detection times to the nucleationtimes. Hence, a mathematical model describing stochastic nucleation mustalso account for the growth of the nuclei into fully developed crystals. Third,a representative sample of the stochastic process must be gathered, i.e., alarge enough number of experimental points must be collected, so that theyform a representative sample of the detection time distribution underlying thestochastic 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 aspecific polymorph. However, many substances exhibit multiple polymorphs. Itis therefore no surprise that there is an extensive body of work dealing withthe order in which the different forms emerge in a process.

    Most prominently, Ostwald was the first to note that—conspicuously often—theleast stable polymorph appears to be the first to nucleate in a system, followedby the second-most unstable form, etc. [38]. This empirical observation, madeby many researchers for a large set of additional compounds, has since beennamed Ostwald’s rule of stages (OSR). While OSR does not represent a physicallaw and several counter examples exist (e.g., [39]), there have been varioustheories attempting to explain the underlying cause of this phenomenon. Ther-modynamic arguments have been brought forward [40], as well as argumentsthat claim properties of the solution structure to be relevant for polymorphselection [41].

    19

  • CHAPTER 2. NUCLEATION

    Here, we shall briefly present a line of reasoning based on nucleation kineticsfavored by a majority of authors [42–44], that has the advantage of beingcompatible with the observed supersaturation dependence encountered in vari-ous 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 particularpolymorphic forms are favored.

    Assuming that the nucleation kinetics of a given compound may be described byEq. (2.6), the logarithm of the ratio of the nucleation rates for two competingpolymorphs is given by:

    ln(JuJs

    )= ln

    (AuAs

    )+ ln

    (c∗sc∗u

    )+ Bs

    ln2(c/c∗s )− Bu

    ln2(c/c∗u)(2.11a)

    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),λ = ln(c∗s /c∗u) and βi = Bi/ ln2 Si.

    ln(JuJs

    )= α+ λ+ βs(c)− βu(c) (2.11b)

    In Eq. (2.11b) we have highlighted those terms that depend on changes in theliquid concentration, c. The behavior of the system is visualized in Figure 2.7;note that, by definition, c∗s < c∗u and λ < 0.

    It is easy to see that Eq. (2.11) approaches negative infinity for c → c∗u andthat it tends toward (α + λ) for large supersaturations. In other words, atconcentrations approaching the solubility of the unstable form, the nucleationof the stable form is more likely, while at higher concentrations the dominantform depends on the value of the kinetic parameters. If, for example, we havethat (α+λ) > 0, higher supersaturation will eventually always lead to preferrednucleation of the unstable form; see Figure 2.7a). If, on the other hand, thisinequality does not hold (cf. Figure 2.7b)), the unstable form can only dominatenucleation if Eq. (2.11) has a maximum (only the case when Bu ≤ Bs) and ifthat extremum lies above zero (Bu must be sufficiently smaller than Bs); theparticular conditions can be found easily by solving the associated system ofequations.

    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 alwaysfor increasing concentrations; right: α+ λ < 0; the nucleation of the unstablepolymorph may dominate only if Bs is sufficiently larger than Bu and withinintermediate concentrations. Figure adapted from [46].

    Naturally, the behavior of the system will be affected by temperature andan analogous analysis can be performed to determine its effect. To make anexample, 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 solubilityratio λ is only a weak function of T , it would then be beneficial to operate atelevated temperatures to produce the stable form and at lower temperaturesto manufacture the unstable polymorph.

    Summarizing, given primary nucleation kinetics of two or more polymorphs, itis comparably easy to identify operating conditions at which the nucleationof 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 mayeven be sufficient to decide upon a strategy for production. Furthermore, thevariety of possible cases is rather limited for simple kinetics, as can be seenfrom Figure 2.7.

    The above type of analysis retains its usefulness regardless of the type ofprocess under consideration, i.e., batch or continuous crystallization. Similarapproaches can be used for other types of nucleation kinetics that are dependenton other factors, including the choice of solvent, additives, templates, or themixing rate in the case of reactive crystallization [47, 48]; the effect of theseproperties on the polymorph selectivity may be much greater than that of

    21

  • CHAPTER 2. NUCLEATION

    supersaturation or temperature alone. Nevertheless, this approach hinges onthe fact that nucleation rates are assumed to be known. If this is not thecase, some prior kinetics estimation work—as, e.g., outlined in Section 2.4—isnecessary.

    If such studies lead to unsatisfactory results or nucleation during the crys-tallization step is ruled out as a viable option for other reasons, the logicalresponse is to attempt suppressing nucleation entirely, making instead use ofseed 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 controlcrystal properties and to ensure constant product quality such as crystallineform, particle size distribution and purity. Compared to their unseeded coun-terparts, seeded crystallization processes can be performed at relatively lowsupersaturations which is favorable for control of the afore-mentioned proper-ties.

    2.6.1

    Process Control

    As mentioned above, the metastable zone is the region in the phase diagram inwhich the supersaturation is sufficiently low to avoid spontaneous nucleation (ofa 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 withinthe metastable zone in order to avoid spontaneous nucleation; values of S atthe seeding point in industrial crystallization of organic compounds typicallyrange from 1.1 to 1.5. The seed crystals may be added as suspension to enablequick dispersion in the vessel and to avoid formation of lumps; furthermore,they should be smaller than the desired product crystals. Therefore, a particlecomminution step, such as dry milling or sieving, is often required in thepreparation of seeds. In industrial practice, it is also important to use seedcrystals with reproducible particle size distribution, in order to ensure lowproduct variability.

    The optimum amount and size of the seed crystals depends on the desiredparticle size of the product. Assuming the crystal size increases only by crystalgrowth and secondary effects such as agglomeration, breakage and secondarynucleation are negligible, the mean particle size of product crystals, Lp, can be

    22

  • 2.6. To Seed or Not to Seed?

    approximated by:

    Lp =(1 + Cs

    Cs

    )1/3Ls (2.12)

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

    Cs =[Seed Mass]

    [Yield] =ms

    mp −ms. (2.13)

    In Eq. (2.13), ms represents the mass of seed crystals and mp that of productcrystals [49].

    The total surface area of the seeds, which in turn depends on their size andamount, determines also the maximum rate of cooling or antisolvent additionwith which supersaturation is still kept a level that would trigger secondaryeffects. Kubota and co-workers proposed an experimental method to determinethe optimum conditions for seeded crystallization. First, experiments usingdifferent amount and size of seed crystals are conducted. Secondly, the experi-mental values of Lp/Ls are plotted versus the seed loading Cs and comparedto the ideal growth line given by Eq. (2.12), see Figure 2.9. Experimentalvalues of Lp/Ls below the ideal growth line may be caused by nucleation(low amount seeds, large seed crystals or high supersaturation caused by fastcooling or antisolvent addition) or breakage (brittle crystal, intense stirring)[50]. Values above the ideal growth line may be a result of agglomeration (highsupersaturation, insufficient stirring or high concentration of relatively smallcrystals) [51].

    Despite the fact that seeded crystallization processes generally are robust anddeliver constant product quality, variability may be introduced by impuritieshaving an effect on solubility and growth rate [52], or by changes in theequipment or process parameter such as stirrer type or stirring rate. It is alsoknown that the growth rate of crystals may not be constant over time andis impacted by defects in the crystal lattice as well as relative surface area ofdifferent crystal facets of the seed crystals [53].

    23

  • CHAPTER 2. NUCLEATION

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

    2.6.2

    Polymorphism Control

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

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

    2. The required polymorph is stable and the polymorph generated by pri-mary nucleation is metastable. As outlined in Section 2.5, this case, whichfollows Ostwald’s rule of stages, is relatively common. Seeding with thestable polymorph at low supersaturations and below the supersatura-tion of the metastable polymorph avoids occurrence of the metastablepolymorph 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 followingthe ideal growth line; triangles: product particle size below ideal growth line asa result of nucleation; squares: product particle size above ideal growth line asa result of agglomeration.

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

    4. The required polymorph is metastable and the polymorph generated byprimary nucleation is stable. Seeding cannot be employed to ensure onlythe metastable polymorph is obtained.

    “Disappearing” polymorphs, that is, those that have been manufactured untila certain point in time but subsequent attempts to produce this polymorphresult in a different form, are linked to situations 3 and 4, i.e., the desiredform is a metastable polymorph. Their disappearance can be explained by theoccurrence of a new more stable polymorph and the unintentional or universalseeding, which describes the presence of small amounts of the new more stablepolymorph 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 tofully pure material in a single process step. In reality, however, impurities,by-products or solvents can still be found in the crystallite. There are multiplereasons why the particle may not be pure, including absorption of impuritieson the crystal surface, incorporation into the crystal lattice, entrapment ofimpurities between agglomerated crystals, precipitation of the impurity asseparate crystals or amorphous particles.

    Impurities may be incorporated as point defects into the crystal lattice atequilibrium conditions and the incorporation can be characterized by a segrega-tion coefficient [56, 57]. In industrial crystallization, the amount of impuritiesfound in the crystallite can exceed the amounts predicted by the segregationcoefficient and is related to non-equilibrium growth processes resulting in in-clusions or defects. The tendency to form inclusions or defects increases withsupersaturation [56]. Therefore, seeded crystallization processes operated a lowsupersaturation may be employed to improve the purification effect.

    26

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    30

    2 Nucleation2.1 Introduction2.2 Homogeneous Nucleation2.2.1 Classical Nucleation Theory2.2.2 Two-Step Nucleation Theory

    2.3 Heterogeneous and Secondary Nucleation2.3.1 Heterogeneous nucleation2.3.2 Secondary nucleation

    2.4 Characterization of Nucleation2.4.1 Deterministic Nucleation Rates2.4.2 Stochastic Nucleation Rates

    2.5 Order of Polymorph Appearance – Ostwald's Rule of Stages2.6 To Seed or Not to Seed?2.6.1 Process Control2.6.2 Polymorphism Control2.6.3 Impurity control