Extended Abstract of PhD Thesis: Optimizing the Crystallization

of Pharmaceutical Compounds Using Additives

Dr Thomas Vetter

March 14, 2014

1 Summary

The distinguishing feature of crystallization is the for-mation of products that are both highly pure and inthe solid state. However, purity is not the only impor-tant aspect in crystallization, i.e., it is a well-knownfact that the quality of a crystalline product also de-pends on its size and shape distribution. These prop-erties are influenced by many characteristics of a crys-tallization process, such as the supersaturation, thetemperature and the chemical environment in whichthe crystals are formed. The main goal of my PhDthesis was to understand how a different chemical en-vironment changes the crystal size and shape and howthis chemical environment can be altered such that adesired crystal size and shape distribution is obtainedfrom a crystallization process.

While this work focuses on crystallization processesfrom solution and more specifically on batch cool-ing crystallization, the findings and techniques couldbe generalized to continuous manufacturing processesor to other types of crystallization processes, such asmelt crystallization. The influence of solvents and ad-ditional shape modifiers (so called additives) on thecrystal morphology and the particle size distributionwas investigated using a widespread array of tech-niques and on various size scales (from the molecularlevel to the process level) which led to the followingresults:

• A simple screening procedure was developed thatallows finding combinations of solvents and addi-tives altering crystal morphology successfully. Thisscreening procedure was carried out for the modelcompound urea and generalized to other com-pounds by defining screening heuristics.

• The interaction of additive molecules with crystalsurfaces was investigated on a molecular level usingmolecular dynamics simulations on the model sys-tem urea (crystal), water (solvent) and biuret andacetone (additive molecules). This study yielded indepth insight on the mechanisms of crystal growthand how the presence of the additive molecules

changed the evolution of crystal growth of urea.The results of this investigation were found to beconsistent with the results of the screening proce-dure mentioned above.

• Nucleation and growth kinetics in the presence ofadditives were estimated from experimental dataand a population balance equation model for thesmall molecular pharmaceutical Ibuprofen in thepresence of the polymeric additive Pluronic F127in solvent mixtures of ethanol and water.

Another focal point of my PhD thesis was theuse of models to describe crystallization processesover multiple time- and size-scales, i.e., from theshort-term stochastic phenomenon of nucleation,to the growth of crystals, to Ostwald ripeningprocesses, which typically progress slowly. To thisaim, a comparison between two types of models wascarried out: models based on classical populationbalance equations and models based on the kineticrate equation. The results of the two models werecompared in a wide range of operating conditionswhile considering nucleation, crystal growth andOstwald ripening simultaneously. The advantagesand disadvantages of the two types of models werethoroughly assessed.

In the following the key results of my PhD thesis [1]will be briefly summarized in Sections 2–5.

2 Development of a screeningprocedure for additives andsolvents

The factors influencing the outcome of a crystalliza-tion process in terms of crystal structure and crys-tal morphology can be grouped into two categories:chemical composition of the solution from which par-ticles are formed and the process operating condi-tions. While the former includes the effect of sol-vents and additives, the latter includes the effects of

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supersaturation, temperature and the hydrodynam-ics of the crystallizer (e.g., stirring rate, mixing condi-tions). This work focuses on the latter set of influencefactors. Despite the many recent advances made inthe prediction, rationalization and modeling of crys-tal morphology, an experimental investigation is - atleast with standard tools and methods – often easierto apply and faster than its predictive alternatives.

These properties already made screening tech-niques a standard procedure for the discovery ofsolid state forms (polymorphs, solvates, etc.) in thepharmaceutical industry, as recently reviewed byAaltonen et al. [2]. Despite this widespread use ofHTS to find solid state forms, little attention hasbeen paid to the fact that the existing HTS equip-ment can additionally be used for the screening ofdifferent crystal morphologies. The use of additivesto change the morphology of product crystals duringHTS appears to be a promising extension, whichcould be performed concurrently with the solid stateform screening that is already in place. Therefore,a general screening procedure for additives andsolvents was formulated, taking both the solid formand crystal morphology into account. To this end,a series of batch cooling crystallization experimentswith various combinations of solvents and additiveswas performed for the model substance urea. Batchcooling crystallization was chosen as the method togenerate supersaturation and form crystals becauseof its relatively simple implementation and its highreproducibility. Furthermore, the generation of awhole population of crystals guarantees a morerobust investigation of the resulting crystal shapesthan single crystal experiments.

After carrying out the screening procedure (detailsin Chapter 2 of the PhD thesis), the product par-ticles were characterized by shape and solid stateform (solvatism, polymorphism, . . . ) using light mi-croscopy and differential scanning calorimetry andX-ray diffraction analysis, respectively. This yieldedseveral “hits” in terms of additive and solvent combi-nations that allowed to reduce the aspect ratio of ureacrystals in a consistent manner. In particular, thepresence of amide groups in the additive moleculesand polar solvents was found to yield the best results.For the sake of brevity, the results of this screen-ing procedure are not repeated here and the inter-ested reader is referred to Chapter 2 of my PhD the-sis. Subsequently, the results were interpreted witha view on the surface chemistry of different facetsof urea crystals and the molecular structure of theadditive molecules. To generalize the screening pro-cedure heuristics were defined that will allow to applythe screening procedure to any system under investi-

gation.

3 Uncovering Molecular De-tails of Urea Crystal Growthin the Presence of Additives

While a screening procedure can produce usefulresults based on direct inference from experimentaldata, it is in no way predictive and essentially relieson educated guesses what combination of additivesand solvents might be successfully used to improvethe shape of crystals (the alternative is an exhaustivescreening of many possible combinations). Giventhe steady increase in computational power andthe ever continuing development of appropriatesimulation algorithms and methods for enhancedsampling, molecular simulations have emerged as aviable approach to build a comprehensive picture ofthe molecular phenomena involved in crystallizationprocesses [3].

In my PhD thesis and the corresponding pub-lication [4] this was illustrated by performingstandard and enhanced sampling molecular dynam-ics simulations of the crystal growth of urea and itsinteraction with different additives. More specificallya full atomistic description of the surface dynamicsand thermodynamics of the fast-growing {001} andthe slow-growing {110} faces was provided. Thecrystal growth of urea from water and methanolsolutions was recently modeled by Piana et al. whoused a combination of Molecular Dynamics (MD)and kinetic Monte Carlo (kMC) simulations, whichenabled an accurate description of the shape of agrowing crystal [5, 6]. However, the accuracy ofthis approach relied on the delicate extraction ofappropriate rates for the kMC simulations fromthe MD simulations, which typically involves sometuning. Contrary to this, a direct atomistic descrip-tion of crystal growth mechanisms in the presenceof additives using molecular dynamics simulationswas sought in my PhD thesis. Such a moleculardescription of the effects of additives on crystalgrowth is a challenging topic that has seldom beentackled in the literature; with the studies reportedtypically only exploring the very limited time scaleof a few nanoseconds [7–9]. However, it is wellestablished that simulations in the hundreds ofnanoseconds are needed to properly observe the(relatively) rare events that characterize the crystalsurface dynamics, such as the formation of 2D nucleion a crystal surface or the adsorption/desorption ofadditive molecules.

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As an alternative to running exceedingly long sim-ulations, one can enhance the occurrence of suchrare events through the use of enhanced samplingmethods. In this work well tempered (WT) metady-namics simulations were used to sample adsorptionand desorption events, thus allowing for a quanti-tative estimation of the associated free energies forboth urea and additives on individual faces of ureacrystals. WT metadynamics is a state-of-the-art en-hanced sampling MD technique [10], which enablesan efficient sampling of rare events and a convergentestimation of free energies.

In order to study the face-dependent growthdynamics (through standard MD) and to estimatethe free energy of adsorption (through WT metady-namics), slabs of crystalline urea exposing a singleface to a solution consisting of urea, water andadditive molecules were prepared. Due to periodicboundary conditions on each side of the box sucha simulation can be imagined as the simulation ofan infinite plane exposing a specific facet (either afast-growing {001} or a slow-growing {110} facet)on both sides of the slab.From a standard MD simulation the evolution ofthe number of crystalline and the number of liquidmolecules can be tracked over time (details arereported in Chapter 3 of my PhD thesis and therespective publication [4]), which is shown for twoexemplary simulations in Figure 1. The increase incrystalline molecules, as well as the total numberof urea molecules in solution are shown. In bothsimulations crystal growth depletes the numberof urea molecules in the liquid phase leading to astationary state where no net growth is observed.However, the evolution up to this stationary stateis entirely different for the {001} and the {110}face: For the fast-growing {001} face a continuousprofile is observed, while the simulation of theslow-growing {110} face exhibits a stepwise profile.This can be interpreted as the signature of twodifferent growth mechanisms, i.e., rough growthfor the fast-growing and 2D nucleation for theslow-growing face. In the case of crystal growthby 2D nucleation the formation of a nucleus onthe otherwise macroscopically flat crystal surface isthe rate-limiting step. After the formation of sucha 2D nucleus the growth proceeds quickly until acomplete layer of the crystal is completed, whichcauses this two-step process to start anew. Thisfinding was subsequently confirmed by analyzing thegrowth behavior of individual crystal layers evolvingduring the simulation, i.e., it was found that inthe case of the fast-growing facet multiple crystallayers are growing at once, which makes the crystalmacroscopically rough, while on the slow-growing

0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.2time [ s]

0

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N C,N

L

NC{001}

NL {001}

NC{110}

NL {110}

a)

b)

zz

Figure 1: MD simulations of {001} and {110} ureafacets in water (red and blue lines, respectively):evolution of the urea molecules incorporated intothe crystal (solid lines) and of the number of ureamolecules in solution (dashed lines).

Table 1: Free energies of adsorption and S{001},{110}for acetone, biuret and urea.

molecule ∆Gads,{001} ∆Gads,{110} S{001},{110}[kcal/mol] [kcal/mol] [-]

acetone −0.66 ± 0.38 −1.47 ± 0.45 0.26biuret −4.55 ± 1.02 −2.48 ± 0.43 32.44urea −3.22 ± 0.83 −1.7 ± 0.39 12.98

facet only one crystal layer is growing at a time.Additionally, the presence of different growth mech-anisms on these two faces is corroborated by atomicforce microscopy images reported in the literature [5].

In order to investigate the effect of additives oncrystal growth, their adsorption behavior on theafore-mentioned crystal surfaces was investigated. Tothis end, WT metadynamics simulations were per-formed with the goal of extracting the free energyof adsorption from them. These facet-dependent en-ergies were determined for three types of molecules:for the urea molecule itself, for biuret and for ace-tone, with the two foreign molecules representing aneffective and an ineffective additive (based on theirability to influence crystal growth such that compacturea crystals are obtained).

The obtained free energies of adsorption are re-ported in Table 1. This data was rationalized us-ing the face-selectivity, S{001},{110}, defined as theratio of the adsorption equilibrium constants whichis reported in the same table. The face-selectivityallows to quantify the preferential interaction of a

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given molecule between two specific crystal faces. Inparticular the values of ∆Gads and S{001},{110} showthat the interaction of the biuret molecule with theurea crystal faces is much stronger than that of theacetone molecule and also that the selectivity of thebiuret for the fast {001} face is remarkable. Thisevidence allows us to demonstrate the selective ad-sorption of biuret molecules on the {001} faces ofthe urea crystal and thus to rationalize why biuretis such an excellent growth inhibitor for these faces:biuret has a higher probability to interact with thefast face than with the slow one and at the sametime has a free energy of adsorption comparable tourea itself on the same face. These two propertiesare caused by the capability of biuret to fill a latticesite on the {001} faces nearly perfectly, which was in-vestigated in detail in the corresponding publication[4]. It was discovered that the biuret molecule ad-sorbs in five different preferential configurations onthe {001} faces, each of them different in distancefrom the solid/liquid interface and its orientation.

The study of the paradigmatic case of urea on amolecular levels allowed to identify some key ingre-dients that could be used in the design of additivescapable of avoiding the formation of needle-shapedcrystals. High affinity and high selectivity for thefast growing face of a needle crystal emerged as cru-cial for a potential shape-affecting effect. These fea-tures are inherently related to the structure of theadditives, that should exhibit moieties capable of re-versibly bind the lattice sites exposed on the fastgrowing crystal face thus limiting its growth rate.

4 Characterization of the nu-cleation and growth ratesof Ibuprofen crystals in thepresence of the polymeric ad-ditive Pluronic F127

In order to investigate the effect of additives onthe crystallization kinetics of a whole population ofcrystals in a quantitative manner and under realisticconditions present in a batch crystallizer a studyon the crystallization kinetics of the pharmaceuticalIbuprofen in the presence of the polymeric additivePluronic F127 (PF127) was conducted during myPhD thesis. While less specific than “tailor-made”additives, polymeric additives have two distinct ad-vantages: they are very unlikely to be incorporatedinto the crystal lattice of a compound since theydiffer from the solute molecules both structurallyand in size and they are often already approved as

excipients in the formulation of pharmaceuticals, sothat their use in a manufacturing process could beaccomplished without overcoming major regulatoryhurdles.Early studies theorized that polymers adsorb on thecrystal faces and thus inhibit or slow down crystalgrowth. It was for example reported that the poly-mers form net-like structures on the crystal surfacesduring crystal growth.[11] The influence of polymericadditives on the nucleation rate was also investigatedand their retarding effect on the nucleation wasexplained by interactions of the polymer moleculeswith the solute molecules through hydrogen bonding,therefore hindering their precipitation.[12] Polymericadditives also influence viscosity, thereby possiblyintroducing mass transfer limitations to crystalgrowth and nucleation. However, in most of thestudies published on the subject the phenomena ofnucleation and crystal growth are not separated,thus making it rather difficult to directly infer themechanism of action for the additive investigated, sothat the authors frequently resorted to qualitativeobservations rather than a quantitative investigationof the effect.

The crystallization kinetics of Ibuprofen in thepresence of Pluronic F127 were quantitatively inves-tigated in my PhD thesis using a two-step procedure:first, the crystal growth kinetics are independentlyinvestigated by designing seeded isothermal batch ex-periments at moderate supersaturation levels and ahigh seed loading. This precludes the occurrence ofa large number of nucleation events, so that the con-sumption of solute from the liquid phase can be en-tirely attributed to the growth of seed crystals, henceeffectively decoupling the phenomena of nucleationand crystal growth. Second, unseeded cooling crys-tallization experiments are conducted from which theprimary (heterogeneous) and secondary nucleationrate can be quantitatively estimated.

Focusing on crystal growth initially, a number ofseeded isothermal desupersaturation experimentswas conducted for which the evolution of the soluteconcentration in the liquid phase was monitoredusing in-situ attenuated total reflectance infraredspectroscopy (ATR-FTIR) coupled with an appro-priate calibration model (details reported in Chapter4 of my PhD thesis and in the correspondingpublication [13]). The experimental dataset coversdifferent supersaturation levels, temperatures andconcentrations of the additive molecule. A processmodel based on the population balance equation andconstitutive equations for the growth rate equationand the solubility was then fitted to the experimen-tal data by minimizing the difference between the

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modeled and the experimental concentration pro-files. Since there is a variety of growth rate modelsdescribed in the literature, a model identificationstep was performed by comparing the goodness offit of each of these models to the experimental data.It was concluded that the growth rate mechanismof Ibuprofen in the presence of Pluronic F127 in thesupersaturation and temperature range investigatedfollows a birth and spread (i.e., 2D nucleation)mechanism. The so-obtained growth rate is shownfor different additive concentrations in Figure 2.It is worth making two remarks. First, the neteffect of the additive’s presence is the decrease ofthe growth rate. This is due to the adsorption ofthe additive, which hinders growth. Secondly, suchgrowth rate decrease could be due to two mecha-nisms caused by the presence of the additive, namelyeither hindrance of surface diffusion or hindranceof solute incorporation in the kink sites blocked bythe additive; these two mechanisms might coexist.While analyzing the latter would require a differenttype of experimental investigation beyond the scopeof this work, the former mechanism can be discussedby considering its likely dependence on the viscosityof the environment in a similar way as the diffusivityin the solution. The diffusion coefficient is predictedby the Stokes-Einstein equation to be inversely pro-portional to the viscosity in solution. By measuringthe viscosity of ethanol/water mixtures containingdifferent concentrations of PF127, it was foundthat the presence of PF127 in solution increasesthe viscosity and therefore decreases the diffusioncoefficient (and thereby also decreases the surfacediffusion coefficient). Therefore, the conclusion wasdrawn that the difference in growth rate shownin Figure 2 stems mainly from a reduced surfacediffusion coefficient, which was in turn interpretedas an effect of the adsorption of PF127 moleculeson the surface of the crystals, thereby hinderingthe diffusion of the solute molecules on the crystalsurface. The nature of this adsorption seems to beisotropic, since comparing the final crystals obtainedfor runs carried out with different concentrations ofPF127 did not show any appreciable differences incrystal habit. While this might not be a proof forthe isotropicity of the adsorption between the PF127molecules and the crystal surfaces, it is at least anindication that the adsorption cannot be stronglyanisotropic.

After the determination of the crystal growth rate,the nucleation kinetics of Ibuprofen in the presence ofPF127 were investigated using unseeded batch cool-ing crystallization experiments. To this end, a setof experiments using different cooling rates (thereby

Figure 2: Growth rate of Ibuprofen crystals in depen-dence of supersaturation and temperature for concen-trations of 0, 0.04 and 0.08 kg PF127/kgsolvent.

also altering the average supersaturation level) andadditive concentrations was performed. Experimen-tal data was again recorded in the form of concen-tration profiles measured by ATR-FTIR. Addition-ally, the particle size distribution at the end of theprocess was characterized using a Coulter Multisizer.Employing two measurement devices rather than onewas deemed necessary because the estimation of nu-cleation rates from concentration profiles is notori-ously inaccurate because crystals of very small sizehave only a minute impact on these profiles. Thepopulation balance equation model described abovewas extended to incorporate primary (heterogeneous)and secondary nucleation and was then again fittedto the experimental data, i.e., the difference betweenthe model and the experimental data was minimizedby adjusting the kinetic parameters in the nucleationrate expressions. Two different sources of experimen-tal data will never be in perfect agreement with eachother because they are subject to different experimen-tal errors. Hence, a procedure to identify the optimaltrade off between the two measurements was used inthe estimation of nucleation rate parameters. Thethusly estimated primary and secondary nucleationrates for different operating conditions are reportedin Figure3a and Figure 3b, respectively. It can clearlybe seen that the estimated primary nucleation ratesare smaller for higher concentrations of Pluronic F127over a wide range of supersaturations and tempera-tures with the primary nucleation rate for 0.08 kgPF127/kgsolvent being more than an order of mag-nitude lower than the nucleation rate in absence ofthe polymeric additive, while the nucleation rate at

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0.04 kg PF127/kgsolvent is an intermediate case in thewhole supersaturation and temperature range inves-tigated. Considering the secondary nucleation rateon the other hand (cf. Figure 3b), one sees that thatthe estimated secondary nucleation at high values of∆c = (S−1)c∗ is only slightly affected by the additionof PF127. However, the addition of PF127 strikinglyreduces the influence of ∆c on the secondary nucle-ation rate.In contrast to the study of the crystal growth ratepresented above the constitutive equations used to fitthe experimental data were of an empirical nature asfirst principle expressions did not fit the experimen-tal data reasonably well. One should therefore refrainfrom an overinterpretation of the obtained parame-ter values (hence they are not further discussed here).However, from the identified decreased primary nu-cleation rate, one can conclude that nucleation in thepresence of PF127 is inhibited. Whether this inhi-bition is due to a change in the interfacial tensionbetween crystal and solution or a hindered diffusionof solute molecules (as in the case of the growth rate,see above) or some other reason could not be conclu-sively determined. The resulting effect in the contextof cooling crystallization experiments from clear so-lution is that the metastable zone broadens with theaddition of PF127, which might widen the operatingrange of crystallization processes that try to avoidthe generation of additional nuclei by homogeneousnucleation.

5 Modeling Nucleation, crystalgrowth and ostwald ripeningin crystallization processes

Crystallization can essentially be described as a com-bination of several mechanisms: the formation of thenew phase (nucleation), growth of crystals and sec-ondary effects such as agglomeration and breakage.The combination of these effects determines the evo-lution of the particle size distribution (PSD) untilthe supersaturation of the solution is depleted. Whenstarting from an initially solid-free system, nucleationis dominant at the beginning of the process, whentypically the difference in chemical potential betweensolution and the solid phase, or supersaturation, ishigh. Crystal growth becomes significant as soon ascrystals are present in the suspension and will domi-nate the decrease of supersaturation after some time.After the depletion of supersaturation, a mechanismknown as Ostwald ripening (also referred to as coars-ening, aging, or simply ripening) takes over and fur-ther influences the evolution of the particle size dis-

(a)

(b)

Figure 3: Fitted nucleation rate of Ibuprofen for con-centrations of 0, 0.04 and 0.08 kg PF127/kgsolvent.The nucleation rate is split into its primary and sec-ondary part. (a) Primary nucleation rate in depen-dence of supersaturation and temperature (b) sec-ondary nucleation rate in dependence of the surfacearea of all crystals, µ2, and ∆c = (S − 1)c∗

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tribution, while breakage and agglomeration can con-tinue to act on the crystals.

Though very powerful in numerous applications,classical population balance equation models cannotdescribe Ostwald ripening and nucleation simultane-ously. This is because the nucleation models assumea constant critical size, while the Ostwald ripeningmodels require the critical size to be a function ofthe supersaturation. This problem is often resolvedby using two separate population balance equationmodels to describe a complete crystallization pro-cess: one to describe nucleation and growth and oneto describe Ostwald ripening using a size-dependentgrowth rate (see for example Iggland and Mazzotti[14]). This situation is unsatisfactory, both conceptu-ally and practically, since the basic mechanisms be-hind both nucleation, growth and ripening are thesame and in some applications, e.g., precipitation ofmultiple polymorphs [16], nucleation and growth orripening occur for different solid phases at the sametime. A model which includes all these steps has beenpresented by Kashchiev [15] (which will be termed thekinetic rate equation model (KRE) in the following).The KRE model describes particles based on attach-ment and detachment of single molecules, or of clus-ters of molecules, according to the pseudo-reactionscheme

An +Ajg(n,j)−−−−−−⇀↽−−−−−−

h(n+j,n)An+j (1)

Here, An denotes a crystal of size n, g(n, j) is therate constant of the two-particle attachment “reac-tion” of a crystal of size n to a crystal of size j, andh(n+ j, n) is the rate constant of the one-particle de-tachment of a crystal of size j from a crystal of sizen+ j. It is noteworthy that the attachment reactionaccounts for both crystal growth (when n or j is 1)and agglomeration, whereas the detachment reactionaccounts for both dissolution (when n or j equals 1)and breakage. Nucleation is not described explicitlyas in the population balance equation model (whereit is described using a boundary condition at a fixedsize), but occurs naturally as a result of the interplayof the ensemble of reactions in Equation 1, providedattachment and detachment rates are properly de-fined.

In my PhD thesis (Chapter 6) and the correspond-ing publication [17] a thorough comparison betweenpopulation balance equation models and the above-mentioned model was presented. To this end, dedi-cated simulations for nucleation, crystal growth andOstwald ripening were performed in a wide range ofprocess conditions and the behavior of the two mod-els was confirmed to be well-aligned. In a final setof simulations complete crystallization processes were

simulated, i.e., from the nucleation of particles wellinto the time when Ostwald ripening occurs. Suchsimulations are unique to the above-presented modeland could not be performed using a classical popu-lation balance equation model. An example of sucha simulation is shown in Figure 4, where the parti-cle size distribution, ñỸ , is plotted as a contour plotagainst dimensionless time and the size (in numberof molecules). Two points are noteworthy about thisplot: first, in this plot a horizontal slice of the contourplot would yield the PSD at a certain instant of timeand second, the dashed black line represents the criti-cal particle size over time. The simulated process canthus be interpreted to consist of several phases: ini-tially, there are no particles present as the whole PSDis below the critical size (some people refer to theseparticles as “embryos”), then these subcritical parti-cles overcome the critical size and nucleation occurs.The concomitant nucleation and growth decreases thesupersaturation in solution, which causes the criticalsize to increase according to classical nucleation the-ory. At even later times, a focusing of the size dis-tribution is observed, i.e., a bimodal PSD is formed,containing both subcritical clusters and a populationof particles with sizes around the critical size. Notethat the focusing phase is rather short and that it isdirectly followed by Ostwald ripening. While this isunusual, it is simply caused by the small size of theprecipitated particles (around 104 molecules). Ost-wald ripening can be easily identified in this plot bythe simultaneous evolution of the critical size andthe PSD. This example shows that the KRE modelis able to describe the whole crystallization processfrom nucleation to Ostwald ripening including sub-critical particles.In the corresponding publication [17] further simula-tions under different process conditions are presentedand the parameters in both models types are inves-tigated in detail. While the classical population bal-ance equation approach is an irreplaceable tool fora wide variety of processes, the unifying descriptionof the KRE model presented above has conceptualand practical merits for certain applications. Froma conceptual point of view, the mechanisms of nucle-ation, crystal growth and Ostwald ripening should bedescribed as different aspects of the same fundamen-tal driving force (the difference in chemical potential)since they all involve the transfer of solute moleculesfrom a disordered liquid phase to an ordered crys-talline phase. Consequently, the mechanisms shouldalso be described in a consistent fashion. It is ourstrong belief that such a unifying, continuous descrip-tion of these mechanisms, without artificially decou-pling them, can be achieved by implementing andsolving the KRE model.

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size, ñ

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Figure 4: Example simulation using the kinetic rateequation model displayed in a plot of dimensionlesssize vs. time. The color of the contours relates to thelogarithmic value of the PSD and the black dashedline is the critical size.

From a practical perspective, the way the KREmodel has been formulated allows for the accuratedescription of crystals below the critical size also dur-ing process stages where nucleation is present (oreven dominating). In the light of recent studies thathave shown that agglomeration of subcritical crys-tals might be an important aspect in nucleation [18],the possibility to model these effects for the subcriti-cal crystals is an important feature. Although notinvestigated in my PhD thesis, such a mechanismcould be implemented in a model based on the ki-netic rate equations, while an implementation in aclassical PBE model that describes nucleation to oc-cur directly at the critical size would only be reflectedin a correction of the nucleation rate (i.e., it wouldbe increased), which is an unsatisfying oversimplifi-cation of the underlying physics.

6 Importance and impact

The influence of impurity or additive molecules ismanifold in crystallization and touches the main areaof crystallization (the purification aspect), as well asother areas, due to their influence on the particle sizeand shape distribution. In my PhD thesis progresswas made with respect to the latter aspect. To inves-tigate this demanding problem a variety of techniques

(screening, predictive techniques, estimation of kinet-ics, etc.) was used from the molecular to the processlevel.While the characterization of kinetics and screeningtechniques will continue to play an important role incrystallization in the near future (and my PhD thesiscontained its fair share of them, see Sections 2 and 4),progress has also been made on modeling techniquesthat are based on first principles and describe crys-tallization on the molecular level (see Section 3). Theprogress on these aspects presented above has beenachieved during my PhD thesis, however, further in-sights were gained in later studies. For instance, theresults presented in Section 3 were essential to thecreation of a more predictive study carried out onurea in a variety of solvents [19]. This study com-bined predictions performed using WT metadynam-ics simulations with a detailed crystallization modeland compared it to experimental findings. It wasfound that our predictions were in very good agree-ment with experimental findings, i.e., the modelswhere able to predict crystal shapes ranging fromneedles to perfect tetraeders from a variety of sol-vents. This line of work is currently continued byDr Matteo Salvalaglio and continues to be a success-ful cooperation between the research groups of Prof.Mazzotti and Prof. Parrinello.The difficulties encountered in the quantitative es-timation of nucleation rates due to its stochasticityand the underlying complicated phase diagram of theIbuprofen/water/ethanol system (see Chapter 5 inmy PhD thesis for details) has provided the initialmotivation for two PhD projects currently ongoingin the research group of Prof. Mazzotti.Parallel to the works presented above, I was involvedin the design, construction and evaluation of a mea-surement device capable of measuring the shape of apopulation of particles in an on-line fashion. [20–22].An improved version of this setup [23] is currentlyused by three PhD students in the measurement ofmultidimensional growth rates of crystals [24], the di-rect quantification of agglomerates and various otherapplications.Summarizing, my PhD thesis covered a wide subjectarea, produced some interesting results and sparkedresearch leads which are currently being followed byseveral PhD students and one postdoc in the researchgroup of Prof. Mazzotti.

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References

[1] Vetter, T.; “Optimizing the crystallization ofpharmaceutical compounds using additives”,PhD Thesis, 2012, doi: 10.3929/ethz-a-007623356.

[2] Aaltonen, J.; Allesø M.; Mirza, S.; Koradia, V.;Gordon, K. C.; Rantanen, J., Solid form screen-ing – A review. European Journal of Pharma-ceutics and Biopharmaceutics 2009, 71 (1), 23–37.

[3] Anwar, J.; Zahn, D., Uncovering MolecularProcesses in Crystal Nucleation and Growthby Using Molecular Simulation. AngewandteChemie International Edition 2011, 50 (9),1996–2013.

[4] Salvalaglio, M.; Vetter, T.; Giberti, F.; Maz-zotti, M.; Parrinello, M. “Uncovering MolecularDetails of Urea Crystal Growth in the Presenceof Additives”, Journal of the American Chem-ical Society 2012, 134 (41), 17221–17233.

[5] Piana, S.; Reyhani, M.; Gale, J. D., Simulatingmicrometre-scale crystal growth from solution.Nature 2005, 438, 70–73.

[6] Piana, S.; Gale, J. D., Understanding the barri-ers to crystal growth: Dynamical simulation ofthe dissolution and growth of urea from aque-ous solution. Journal of the American ChemicalSociety 2005, 127, 1975–1982.

[7] Yin, Y.; Chow, P. S.; Tan, R. B. H., Molec-ular Simulation Study of the Effect of VariousAdditives on Salbutamol Sulfate Crystal Habit.Molecular Pharmacology 2011, 8 (5), 1910–1918.

[8] Zhu, W.; Romanski, F. S.; Meng, X.; Mitra, S.;Tomassone, M. S., Atomistic simulation studyof surfactant and polymer interactions on thesurface of a fenofibrate crystal. European Jour-nal of Pharmaceutical Science 2011, 42 (5),452–461.

[9] Aschauer, U.; Spagnoli, D.; Bowen, P.; Parker,S. C., Growth modification of seeded calciteusing carboxylic acids: Atomistic simulations.Journal of Colloid and Interface Science 2010,346 (1), 226–231.

[10] Barducci, A.; Bussi, G.; Parrinello, M., Well-Tempered Metadynamics: A Smoothly Con-verging and Tunable Free-Energy Method.Physical Review Letters 2008, 100, 020603.

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[13] Vetter, T.; Mazzotti, M.; Brozio, J., “Slowingthe Growth Rate of Ibuprofen Crystals Usingthe Polymeric Additive Pluronic F127” CrystalGrowth & Design 2011, 11 (9), 3813–3821.

[14] Iggland, M.; Mazzotti, M., Population BalanceModeling with Size-Dependent Solubility: Ost-wald Ripening, Crystal Growth & Design 2012,12, 1489–1500.

[15] Kashchiev, D., Nucleation, Butterworth-Heinemann, Burlington, USA, 2000.

[16] Ozkan, G.; Ortoleva, P., A mesoscopic modelof nucleation and Ostwald ripening/stepping:Application to the silica polymorph system,Journal of Chemical Physics 2000, 112, 10510–10525.

[17] Vetter, T.; Iggland, M.; Ochsenbein, D.R.;Hänseler, F.S.; Mazzotti, M. “Modeling Nucle-ation, Growth, and Ostwald Ripening in Crys-tallization Processes: A Comparison betweenPopulation Balance and Kinetic Rate Equa-tion”, Crystal Growth & Design 2013, 13 (11),4890–4905.

[18] Yuk J. M.; Park, J.; Ercius, P.; Kim, K.; Helle-busch, D. J.; Crommie, M. F.; Lee, J. Y.; Zettl,A.; Alivisatos, A. P Science 2012, 336, 61–64.

[19] Salvalaglio, M.; Vetter, T.; Mazzotti, M.; Par-rinello, M. “Controlling and Predicting Crys-tal Shapes: The Case of Urea”, AngewandteChemie International Edition 2013, 52 (50),13369–13372.

[20] Kempkes, M.; Vetter, T.; Mazzotti, M. “Mea-surement of 3D particle size distributions bystereoscopic imaging”, Chemical EngineeringScience 2010, 65 (4), 1362–1373.

[21] Kempkes, M.; Vetter, T.; Mazzotti, M. “Moni-toring the particle size and shape in the crystal-lization of paracetamol from water”, ChemicalEngineering Research & Design 2010, 88 (4),447–454.

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[22] Schorsch, S.; Vetter, T.; Mazzotti, M. ‘Measur-ing multidimensional particle size distributionsduring crystallization”, Chemical EngineeringScience 2012, 77, 130—142.

[23] Schorsch, S.; Ochsenbein, D.R.; Vetter, T.;Morari, M.; Mazzotti, M. “High accuracy on-line measurement of multidimensional parti-cle size distributions during crystallization”,Chemical Engineering Science 2014, 105, 155–168.

[24] Ochsenbein, D.R.; Schorsch, S.; Vetter, T.;Mazzotti, M.; Morari, M. “Growth Rate Esti-mation of β L-Glutamic Acid from Online Mea-surements of Multidimensional Particle SizeDistributions and Concentration”, Industrial& Engineering Chemistry Research 2014, doi:10.1021/ie4031852.

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Vetter-EFCE Excellence Award 2014-Doherty.pdfDepartment of Chemical EngineeringUniversity of CaliforniaSanta Barbara, CA 93106-5080