Chemical Engineering Science 65 (2010) 5676–5686

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Chemical Engineering Science

0009-25

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ces

Modeling of crystal morphology distributions. Towards crystalswith preferred asymmetry

Jayanta Chakraborty a, Meenesh R. Singh a, Doraiswami Ramkrishna a,�,Christian Borchert b, Kai Sundmacher b

a Purdue University, School of Chemical Engineering, 480 Stadium Mall Dr., West Lafayette, IN 47907, USAb Max Plank Institute for Dynamics of Complex Technical Systems, Sandtorstrasse 1, D-39106 Magdeburg, Germany

a r t i c l e i n f o

Article history:

Received 29 January 2010

Received in revised form

16 March 2010

Accepted 17 March 2010Available online 27 March 2010

Keywords:

Population balance modeling

Crystal growth

Crystal symmetry

Crystallization

Group theory

09/$ - see front matter & 2010 Elsevier Ltd. A

016/j.ces.2010.03.026

esponding author. Tel.: +1 765 494 4066; fax

ail address: ramkrish@ecn.purdue.edu (D. Ram

a b s t r a c t

Exploitation of crystal symmetry is very important in formulation and efficient simulation of population

balance models for crystal morphology. This work presents the first population balance model for

morphology distribution considering the diversity of symmetry. In this model, we analyze the

symmetry of a population of crystals using group theory and divide the population into various

symmetry classes, which, in turn, is subdivided into various morphological forms. The internal

coordinate vector for any given crystal can be symmetry reduced and can be grouped into various sets

characterized by identical growth rates. It has been shown that the internal coordinate vector can be

represented in such a way that only one internal coordinate needs to be treated dynamically for each

set while all other coordinates remains invariant during growth. This leads to a very small number of

dynamic internal coordinates and the effective dimensionality of the problem becomes very small,

allowing simulation of a population of asymmetric crystals with minimal computational effort. It has

been shown using this model that the concentration of more symmetric crystals invariably increases

during the growth process. However, this natural gravitation of the crystal population towards more

symmetric forms can be controlled by manipulating the supersaturation which has been shown using

numerical examples.

& 2010 Elsevier Ltd. All rights reserved.

1. Introduction

We focus in this paper on cultivating, in a well-stirredcrystallizer, geometrically asymmetric crystals with a preponder-ance of faces having properties of specific interest. The propertiesare linked to an application as, for example, in the promotion ofbioavailability in pharmaceutical products, higher chemicalactivity of particles in synthetic catalysts, and so on. Asymmetriccrystals, however, are constantly gravitating towards symmetryso that sustenance of asymmetry calls for ways to forestall thismigration. We rely for this navigational effort only on themanipulation of supersaturation although other avenues haveexisted at least on an empirical basis particularly in industrialpractice. However, the methodology we adopt will set the trail formore comprehensive approaches in a rational manner whenquantitative understanding of other effects becomes available.

There is currently in the literature good understanding of therequired mathematical framework for analyzing the behaviorof a population of crystals growing in a stirred crystallizer

ll rights reserved.

: +1 765 494 0805.

krishna).

(Dirksen and Ring, 1991; Zhang and Doherty, 2004; Ma et al.,2008; Borchert et al., 2009). However, such efforts have relied onthe boons of symmetry so that the description of morphologyrequires only a very limited number of variables. The objectivethat we have set forth cannot, however, be met by imposing suchconstraints. We seek in this paper an approach built on exploitingbasic features of an asymmetric crystal that serve to minimizethe computational effort involved for cultivating asymmetriccrystals. The strategy for this is built on minimizing the number ofcrystal dimensions that vary during growth. The appreciation ofhow such a minimization is actually accomplished requiresgeometrical insight of a faceted crystal.

It has been shown that a faceted crystal may be interpreted asthe intersection of convex polyhedra, each bounded by a ‘‘family’’of faces characterized by the same growth rate (in a commonsupersaturation environment), viz., the rate at which each faceadvances from an arbitrary reference point in a direction normalto itself. As the orientation of each crystal face remains fixedrelative to crystal coordinates during growth, the interior anglesbetween faces of the polyhedron representing a particular familyis invariant with time. Since frequently some faces are slowgrowing relative to others, allowing the crystals to be viewed astwo dimensional objects, we restrict our demonstrations to such

J. Chakraborty et al. / Chemical Engineering Science 65 (2010) 5676–5686 5677

crystals for visual clarity. However, the methodology discussed inthis article is not restricted to the two dimensional crystals aloneand can readily be extended to three dimensional crystals byreplacing polygons by polyhedra.

Our methodology consists in exploiting the symmetryproperties of the polygons by elucidating the maximum numberof distinct time varying crystal dimensions. Towards this end, weprovide a theoretical development that begins with defining acrystal state vector comprising all crystal faces and progresstowards its description in terms of the minimum number ofdistinct time varying components in the vector. In what follows,various recent efforts for modeling crystal shape and morphologyis recalled before attending to the issues of main concern inthe paper.

2. Previous work

Control of crystal shape can be realized by changing theproperties of the solution, namely by using additives (Weissbuchet al., 1991; Sangwal, 2007), changing the solvent (Davey et al.,1982) or in the simplest way by manipulating supersaturation(Liu and Bennema, 1994; Liu et al., 1995; Boerrigter, 2002).Thus the control of supersaturation presents an avenue formanipulating crystal shape towards a desired direction. However,a quantitative model based understanding of crystal growth isessential in order to achieve desired control in commercialapplications.

Crystallizers are usually modeled using the population balanceequation (Randolph and Larson, 1971; Myerson, 1993; Ramkrishna,1985, 2000) considering only one internal variable. Often readilymeasurable quantities like volume, mass or diameter of theparticle is used as an internal variable depending on the methodof measurement used in the experiments. While such a simplisticone dimensional model is useful in predicting the size distributionof crystals, it has no means to predict crystal shape. Clearly, amulti-dimensional population balance model is required todescribe the distribution of shape and size of a population ofcrystals.

Multidimensional model of a crystal population often recog-nizes crystals in terms of more than one characteristic dimensionwithout invoking a faceted structure (Ma et al., 2002; Puel et al.,2003). However, recognition as a faceted structure is essential forapplications like catalysis where it is advantageous to promote aspecific face. Following the early work of Cardew (1985), Gadewarand Doherty (2004) considered crystals as faceted objects and asingle crystal model was formulated considering appearanceand disappearance of faces. Zhang and Doherty (2004) used thismultidimensional description of a single crystal in a populationbalance model with the assumption that crystals of the samesize must have the same shape. This assumption reduced themultidimensional description into a uni-dimensional PBE. Subse-quently, Zhang et al. (2006) have explored many interestingfeatures of the three dimensional single crystal model. Briesen(2006) described another technique for reduction of the twodimensional crystal population balance into a series of onedimensional PBEs.

Although the detailed morphological population balancemodel was formulated and solved (Ma et al., 2008; Wan et al.,2009; Borchert et al., 2009), quantitative modeling of realcrystallizers could not be achieved as an essential property ofparticulate solid, asymmetry was ignored. Asymmetry is animportant attribute of a particulate solid because along with size,it determines the specific surface for a given solid. The specificarea of a symmetric shape is always less than the specific area of acorresponding asymmetric shape. Modeling of asymmetry using

population balance has not been ventured so far as it is believedthat considering asymmetry will lead to an unmanageablenumber of dimensions in the model. For example, Ma et al.(2008) considered symmetric crystals of potash alum which have26 faces. They remarked that all 26 dimensions are needed ifsymmetry is not considered.

In this study, we model the growth process of asymmetriccrystals using a population balance model. We demonstrate thatthe dimensionality of the population balance equation for growthdoes not necessarily increase by considering a population ofasymmetric crystals. In order to formulate such a populationbalance model, we define and quantify the symmetry of a givencrystal. Subsequently crystals are classified into various classesaccording to their symmetry. For each class, a separate populationbalance equation is written. It is shown that with this approachthe dimensionality of the population balance equation does notexceed the number of independently growing faces.

This modeling effort also brings out the notion of evolution ofsymmetry of a population of crystals. It also shows that due togrowth, a population of crystals always travels towards a moresymmetric form. In the next section we shall discuss the model indetail, demonstrate the ideas using a numerical example, andpresent our conclusions.

3. Theory

Crystals are polyhedral objects which can have faces of morethan one kind. Each kind is characterized by a distinct arrange-ment of atoms/molecules which leads to a specific growth rate.The faces that have identical arrangement of atoms grow at thesame velocity and we shall refer to them as faces of the same

family. Faces of different families grow at different rates and thisdifference in growth rates leads to morphological evolution.

Crystals are usually modeled in terms of a set of distances froman origin, the choice of which is closely related to the symmetry ofthe crystal. Symmetry of an object refers to the existence of one ormore lines of symmetry about which rotation and/or reflectionleaves the object unchanged. Maximum symmetry usually has acenter of symmetry which can be chosen as the origin. For the lesssymmetric form, the choice of the origin depends on the extent ofsymmetry and will be discussed subsequently.

Now, consider a crystal comprising m families of faces with theith family having fi faces, then the total number of faces of thecrystal, denoted n, is given by

n¼Xmi ¼ 1

fi ð1Þ

so that the vector of distances of all faces from a suitably chosencenter, hARn, may be written as a partitioned vectorh� ½h1,h2, . . . ,hm�. Where hi represents the distances of faces ofthe ith family from the center and can be written as:hi � ½h1,i,h2,i, . . . ,hfi ,i�ARfi , i¼ 1,2, . . . ,m. If the growth rate of thekth face of the ith family is denoted _Hk,i, then from the definitionof the family, we have _H1,i ¼

_H2,i ¼ � � � ¼_Hfi ,i �

_Zi. If further thecrystal is symmetric, with the origin at the symmetry center, wewould have h1,i ¼ h2,i ¼ � � � ¼ hfi ,i � zi, i¼ 1,2, . . . ,m. In this case,the crystal state vector h may be replaced by the vector zARm.Hence, there is a reduction of dimensionality of the crystal statevector from n, which can be very large, to m which could besubstantially smaller. We show below that this reduction indimension is also enjoyed by asymmetric crystals, with a left-overtime-invariant state vector whose dimension would depend onthe extent of symmetry of the crystal.

Consider a completely asymmetric crystal with the ith familyhaving fi faces whose distances from some origin to be specified

J. Chakraborty et al. / Chemical Engineering Science 65 (2010) 5676–56865678

are given by h1,i,h2,i, . . . ,hfi ,i. By asymmetry, we imply thath1,iah2,ia � � �ahfi ,i. From the definition of the family

_Hj,i ¼_Hk,i ð2Þ

from which we have, recognizing the dependence of the crystalface locations on time

d

dtðhj,i�hk,iÞ ¼ 0 ð3Þ

It is to be noted that the growth rate of a particular face isassumed to be independent of the size of the crystal which is truefor most practical situations and is known as McCabe DL law.Now, integrating the above differential equation, we have

hj,i ¼ hk,iþcjk,i ð4Þ

where cjk,i is a constant of integration specific to the pair of jth andkth faces belonging to this ith family. Suppose we choose the kthface as a fixed reference face for the family and let zi � hk,i, then wehave hj,i ¼ ziþcjk,i and vectorially (suppressing the fixed index k)

hi ¼ zi1þci, 1� ½1,1, . . . ,1�ARfi , ciARfi : ð5Þ

Clearly, the time-variance of the vector hi is contained only in thescalar zi and therefore the number of differential equation neededto be solved in simulating the growth model is immensely reduced.However, the number density function is a density in Rfi , with fi�1quantities in ci which, while invariant with time (for a specificcrystal), can vary from crystal to crystal. Furthermore, the volumein hi space may be written as

dhi �Yfi

j ¼ 1

dhj,i ¼ dzi

Yfi

j ¼ 2

dcj,i ¼ dzi dci: ð6Þ

The argument presented above applies to each family of planes.However, the number of components in ci will depend onsymmetry, an issue to which we will return following a discussionof the symmetry of the crystal. The largest face should be taken asthe fixed reference face as it is the last face to disappear.

3.1. Symmetry of a crystal

The different arrangement of atoms on different planesoriginates from the different orientation of the planes that cutthe crystal lattice. Although an infinite number of such planes canbe imagined in a given crystal lattice, only a few of them areenergetically favorable and hence only those planes need to beconsidered in any practical situation (Boerrigter et al., 2002). Theorientation of the planes considered remains invariant duringgrowth and hence the angular relationship between any two facesremains unaltered. This feature can be used to represent crystalsin a unique and convenient way as in the discussion that follows.

(a) (b)

Fig. 1. Representation of an asymmetric crystal: (a) in terms of h-ve

Discussion of the symmetry of faceted crystals is simplified ifwe view such crystals as an intersection of polygons. If weconsider only a particular family of planes, it is easy to see thatsuch planes always form a polygon with known angles. Hence, acrystal can be thought of as a combination of intertwinedpolygons resulting from various families. We shall call thepolygons generating a crystal as generating polygons or simplygenerators. Such a representation of a crystal consisting of onlytwo families has been shown in Fig. 1.

In general, crystals can have complex shapes and a mathema-tical discussion and quantification of symmetry of such shapes isnot straightforward. But the symmetry of a crystal can bediscussed in terms of the symmetry of its generating polygons.The incentive for taking this approach lies in the ease with whichthe symmetry groups of polygons can be found. If the polygon isregular, its symmetry elements are readily available as thedihedral group (Lomont, 1959). If it is a non-regular polygon,symmetry exists for a few limited cases and symmetry groups forsuch cases can be found in a straightforward way. Hence, if we canestablish a mathematical relation between the symmetry groupsof the generating polygons to the symmetry of the crystal, we canreadily find the symmetry of any crystal.

Now, consider the two generators, a square and a diamond,each having four lines of symmetry as shown in Fig. 2(a) and (b).We are concerned with a crystal which is produced from thesetwo generators with a common geometric center. This situation isshown in Fig. 2(c). It is clear that the resulting crystal inherits allthe symmetry axes of the generators and we shall refer to thissituation as the case of ‘maximum symmetry’. Now let us examinehow various kinds of asymmetric shapes can be generated fromthese two generators. As the angle between any two kinds ofplanes is fixed during growth, the generators must be arranged insuch a way that this angle remains the same. Thus, differentarrangements differ only in the way one generator is translated

with respect to the other. Hence, in order to explore variouspossible symmetry situations, we may start with the mostsymmetric case and translate one generator with respect to theother.

Translation of the diamond with respect to the square canhappen in two possible ways: first, it may translate along any lineof symmetry, as shown in Fig. 2(d). In this case the crystal willremain symmetric against this line of symmetry but loses allother lines of symmetry. It may also translate along a directionwhich is not a line of symmetry of either object. In that case,shown in Fig. 2(e), the crystal becomes completely asymmetric.Overall, this visual exercise shows the symmetry group of thecrystal is the intersection of the symmetry groups of thegenerating polygons. This is an important conclusion and can beproved using group theory and is provided in Appendix.

(c)

ctor with respect to an origin; (b) and (c) in terms of polygons.

ab

c

d

12

3

a1b2

c3

d4

a1 2

b

3c

4d

1a2

b

3c

4d

4

(a) (b) (c)

(d) (e)

Fig. 2. Symmetry of the crystal in terms of the symmetry elements of its generating polygons: (a) regular diamond with four lines of symmetry, (b) square with four lines of

symmetry, (c) crystal generated with coincidence of all the lines of symmetry of the generators, (d) crystal generated with coincidence of only axis a of diamond with axis 1

of square, (e) crystal generated with no coincidence of symmetry axes.

Table 1Various combinations of generators.

RR IR

RD RR–RD IR–RD

ID RR–ID IR–ID

J. Chakraborty et al. / Chemical Engineering Science 65 (2010) 5676–5686 5679

3.2. Symmetry and the choice of origin

The choice of the origin is very important for a crystal ofreduced symmetry. It not only produces the minimum possibledimension for h or c, it also needs to be consistently defined fromcrystal to crystal. The choice is straightforward if the crystalcontains two or more lines of symmetry where the intersection ofthe lines of symmetry is the natural choice. If the crystal containsonly one line of symmetry, a suitable point can be chosen on thatline as an origin. For example, the crystal shown in Fig. 2(d), theorigin can be taken as the symmetry center of the rectangle thatalso lies on the line a1. Even if the crystal is apparentlyasymmetric, as in Fig. 1(a), we have a unique choice whichutilizes the symmetry of one of the generators. For this case, forexample, the origin can be placed on the symmetry center of therectangle.

3.3. Classification of a population of crystals into various symmetry

classes

From the previous discussion and Fig. 2 it emerges that thesymmetry of a crystal is solely dependent on the symmetry of itsgenerators and the way they are combined. Hence, to classify apopulation of crystals in terms of symmetry, we need to preparean exhaustive list of possible generators and all possiblecombinations of such generators. As the generating polygonshave fixed angles, there are only a limited number of suchgenerators for a given family. For example, as all the anglesbetween the planes of the rectangle family is p/2, it producespolygons which can either be regular (square) with four lines ofsymmetry or equiangular (rectangle) with two lines of symmetry.As in most situations the crystal is composed of two to threefamilies of planes and the number of faces in a particular family isknown, such combinations are readily obtained. For example, inthe case of rectangle and diamond, only four kinds of generatorsare possible: regular-diamond (RD), irregular-diamond (ID),

regular rectangle (RR) and irregular rectangle (IR). All possiblecombinations using these four generators are shown in Table 1.

For every combination of generators, the crystal can have adifferent number of symmetry axes depending on the way theyare combined. This can be explored in a similar fashion as shownin Fig. 2 and therefore we can classify the entire population intoseveral symmetry classes.

3.4. Using symmetry to determine minimum number of internal

coordinates

Any reflectional symmetry of a crystal has the potential toreduce the number of internal coordinates. A line of symmetrycreates images for each internal coordinates of a given family andif an internal coordinate becomes the image of another in thesame family, only one of them needs to be considered. If aninternal coordinate forms a distinct image, separate internalcoordinate is needed for it. If a face is normal to the line ofsymmetry, the internal coordinate corresponding to that face liesalong the line of symmetry. In this case, the internal coordinateand its image coincide and separate internal coordinate is alsoneeded for this case.

The minimum number of internal coordinates for a givenfamily can be found by using all lines of symmetry of the crystal ina recursive manner. Starting with the unreduced internalcoordinate vector, the first line of symmetry can be applied toget a reduced set. Then the second line of symmetry is applied onthe reduced set to reduce it further. This process continues until

J. Chakraborty et al. / Chemical Engineering Science 65 (2010) 5676–56865680

all symmetry axes are accounted for. We denote by f ji the number

of coordinates for the ith family after j lines of symmetry havebeen accounted for, so that f 0

i would represent the number ofcoordinates before accounting for any symmetry. We also denoteby f 0ji the total number of internal coordinates either forming adistinct image or lying along the line of symmetry for the jth lineof symmetry. Now, because a symmetry element divides thecrystal into two identical halves, the recurrence relation for f j

i isgiven by:

f ji ¼

f j�1i þ f 0ji

2ð7Þ

For a crystal with s lines of symmetry the minimum number ofinternal coordinates for ith family is therefore given by f s

i . In theforegoing discussion we will always describe a crystal usingminimum number of coordinates. The symmetry-reduced internalcoordinates for crystals with four and one line of symmetry areshown in Fig. 3.

a1

b2

c3

d4

a1

(a) (b)

Fig. 3. Reduction of number of internal co-ordinates by considering symmetry of

the crystal which is produced by combining regular diamond and square.

(a) Completely symmetric case: eight faces are represented by two internal

co-ordinates; (b) one symmetry axis case: eight faces are represented by using

five internal co-ordinates. The symmetry center of the square has been chosen

as center.

(a)

(e) (f)

(b)

Fig. 4. Various possible morphological forms that may be produced due to d

3.5. Morphological forms of a symmetry class

Higher growth rate of one face of a crystal with respect to itsneighboring faces often leads to disappearance of that face fromthe crystal transforming it into a vertex. Therefore, for the crystalsin the same symmetry class dimensionality of z and c can bedifferent if one or more faces become virtual. Such a differencecalls for subdivision of the population of crystals of a symmetryclass into different morphological forms.

Various morphological forms for a given symmetry class canbe generated from a parent morphological form, which containsmaximum number of faces, by allowing faces to become virtualdue to growth. All possible morphological forms in a givensymmetry class can be obtained by exhausting all possiblecombinations of internal coordinates and ensuring that theresulting crystal encloses a space. To enclose space, no twoadjacent internal coordinates of the crystal can have an angle of2p or larger. Fig. 4 shows all possible morphological forms for thesymmetry class that has one line of symmetry and composed ofthe two regular generators.

3.6. Dynamics of symmetry: accumulation of more symmetric forms

As discussed above, generation of one morphological formfrom another occurs due to transformation of a face from real tovirtual and vice versa. Such transformations may also lead tochange in the symmetry property. For example, if the rectangularface grows further in Fig. 4(f), a diamond is created which has fourlines of symmetry instead of one. Transformations among varioussymmetry classes and morphological forms for a population ofcrystals consisting of one, two and four lines of symmetry havebeen shown in Fig. 5. Various morphological forms of the samesymmetry class are enclosed in a thick lined box. Completelyasymmetric crystals and a few morphological forms have beenomitted for clarity but can be included without difficulty.

Population fluxes between various morphological forms andsymmetry classes are observed in a population of crystals due toreal to virtual or virtual to real transition of crystal faces. Thenature of the transition depends on the relative growth rates ofcrystal faces which in turn depend on the supersaturation.Various population fluxes for the current case have been shownin Fig. 5. We observe from Fig. 5 that two morphological forms in

(g) (h)

(c) (d)

isappearance of one or more faces for crystals of one line of symmetry.

1b 1e

2a

2b 2c 4a4b

4c1d

1c

1a

2

1

4

�1

�2

�3

�3

�4

Fig. 5. Various symmetry classes and morphological forms present in the system and their boundary flux. The species corresponding to the same symmetry class are

enclosed in a thick lined box.

Table 2Notations for various morphological forms.

Morphological form Notation Label

n2,b1ðz1 ,c,tÞ 2a

n2ðz,c,tÞ 2b

n2,b2ðz,tÞ 2c

n4,b3ðz2 ,tÞ 4a

n4ðz,tÞ 4b

n4,b4ðz1 ,tÞ 4c

J. Chakraborty et al. / Chemical Engineering Science 65 (2010) 5676–5686 5681

a given symmetry class may experience boundary flux in eitherdirection depending on supersaturation. For example, if thediamond face grows over the square face, 1a is created from 1band the opposite may occurs if the supersaturation changes. Onthe other hand, the boundary flux from one symmetry class toanother is strictly unidirectional. For example, 2a may begenerated from 1a as a result of disappearance of diamond faces.But if the process is reversed due to change in supersaturation, i.e.square faces start growing over the diamond faces; it gives rise to2b instead of 1a. Hence, if a crystal leaves a particular symmetryclass through a boundary, it can never enter into that class againand during the growth process, the concentration of moresymmetric morphological forms increases as a result of passageof population through various symmetry boundaries.

The detail with which morphology has been characterized mayperhaps appear to be excessive for practical applications. Forexample, the distinction between 1b, 2b and 4b may seemneedlessly elaborate. While this may be so, the need for thisdistinction is forced by the fact that the population balanceequations simplified by the use of symmetry are not the same forthe cited morphologies. We note that a regrouping of crystalmorphologies using more distinct types can of course be done tocompare with experimental observations.

3.7. Model equations

At this stage the PBEs for various morphological forms may beformulated. For brevity, we shall consider only the blocks 2 and 4shown in Fig. 5. Different symmetry classes will be represented byusing the number of symmetry axes as subscript to the populationand various morphological forms in a symmetry class isdistinguished by another subscript to the number density. Thesecond subscript represents the boundary on which a particularmorphological form occurs. If the morphological form does notoccurs on a boundary (e.g. 2b and 4b), only one subscript is used.For example, the morphological form 2a occurs on the boundaryb1 and therefore, the number density will be written as n2,b1

. Butthe morphological form 2b occurs in the domain itself and will be

written as n2. In a subsequent generalization we shall introducea more elaborate and more self-consistent notation. In thecurrent case, z has two components and we shall denote theinternal coordinates corresponding to square and diamond facesas z1 and z2 respectively. Only one c-coordinate is needed in thiscase (to represent species 2a and 2b), represented by c. Thenotations for various morphological forms are summarized inTable 2.

Treatment of morphological boundaries: At a morphologicalboundary, an internal co-ordinate becomes dependent on theco-ordinates of its neighbors which can be written as

hi,j ¼hi1 ,j1 sina1þhi2 ,j2

sina2

sina12ð8Þ

where the internal coordinate hi,j is dependent on its neighbors,hi1 ,j1

and hi2 ,j2 and a1 and a2 are the angles between hi1 ,j1,hi,j and

hi,j,hi2 ,j2respectively and a12 is the angle between hi1 ,j1 ,hi2 ,j2

. Wewill denote such relations for the ith boundary by bi. Notationsfor various boundaries are shown in Fig. 5. For the currentcase a1¼a2¼p/4 and a12¼p/2 for all internal coordinates.Therefore, the relations between the internal coordinates at

J. Chakraborty et al. / Chemical Engineering Science 65 (2010) 5676–56865682

various boundaries can be written as

b1 : z2 ¼2z1þcffiffiffi

2p

b2 : z2 ¼z1þcffiffiffi

2p

b3 : z2 ¼z1ffiffiffi

2p

b4 : z2 ¼2z1ffiffiffi

2p ð9Þ

Now, we have to consider the population fluxes across all theseboundaries. Population flux occurs across a boundary if thegrowth velocity is directed towards the boundary, i.e. _Z � nbi

Z0where nbi

is the unit normal towards the boundary bi. If thevelocity is directed away from the boundary, a discontinuouschange occurs in the number density at the boundary in whichthe population on the boundary moves into the higher dimen-sional domain. At this point ð _Z � nbi

¼ 0Þ, the number density inthe higher dimensional domain must satisfy an extra boundarycondition corresponding to the population at the boundary to bespecified below.

Consider the morphological form 2b for which the populationbalance equation in n2(z,c,t), can be written as

@n2ðz,c,tÞ

@tþ _Z � rzn2ðz,c,tÞ ¼ 0 ð10aÞ

This particular morphological form has two boundaries (b1 andb2) as shown in Fig. 5. These two boundaries will produce twoboundary conditions when the population moves away fromthese boundaries, given by

@_Z � nb1¼ 0 : nb1 ;z1 ,cn2ðz,c,tÞjb1

¼ n2,b1ðz1,c,tÞd z2�

2z1þcffiffiffi2p

� �� �

ð10bÞ

@_Z � nb2¼ 0 : nb2 ;z

n2ðz,c,tÞjb2¼ n2,b2

ðz,tÞdðc�ðffiffiffi2p

z2�z1ÞÞ ð10cÞ

We notice that the number density corresponding to the populationon a boundary is written as a density on the corresponding co-ordinate planes rather than on the boundary itself. Hence, it isnecessary to project the number density on the boundary to one onthe coordinate plane. The corresponding factor is given by nbi ;h

where bi and h represent the boundary and the coordinate planerespectively. We also need to establish dimensional consistencybetween number densities defined in the spaces of differentdimensionality using Dirac-delta function as shown in Eqs. (10b)and (10c). Similarly, the equations for the morphological forms 2aand 2c can be written as

2a: ð _Z � nb1Z0Þ

@n2,b1ðz1,c,tÞ

@tþ _Z1 � rz1

n2,b1ðz1,c,tÞ ¼ nb1 ;z1 ,c

_Z � nb1n2ðz,c,tÞjb1

ð11Þ

2c: ð _Z � nb2Z0Þ

@n2,b2ðz,tÞ

@tþ _Z � rzn2,b2

ðz,tÞ ¼ nb2 ;z_Z � nb2

n2ðz,c,tÞjb2ð12Þ

We notice that although the morphological form 4a occurs at aboundary of 2c, it does not imply a boundary conditioncorresponding to the receding boundary because the flux is oneway. The equations for the morphological forms 4a, 4b and 4c canbe written in a similar fashion

4a: ð _Z � nb3Z0Þ

@n4,b3ðz2,tÞ

@tþ _Z2 � rz2

n4,b3ðz2,tÞ ¼ nb3 ;z2

_Z � nb3n2,b2ðz,tÞjb3

þnb3 ;z2_Z � nb3

n4ðz,tÞjb3ð13Þ

4b:

@n4ðz,tÞ

@tþ _Z � rzn4ðz,tÞ ¼ 0 ð14aÞ

@_Z � nb3¼ 0� : nb3 ;z2

n4ðz,tÞjb3¼ n4,b3

ðz2,tÞdðz1�ffiffiffi2p

z2Þ ð14bÞ

@_Z � nb4¼ 0� : nb4 ;z1

n4ðz,tÞjb4¼ n4,b4

ðz1,tÞd z2�2z1ffiffiffi

2p

� �ð14cÞ

4c: ð _Z � nb4Z0Þ

@n4,b4ðz1,tÞ

@tþ _Z1 � rz1

n4,b4ðz1,tÞ ¼ nb4 ;z1

_Z � nb4n4ðz,tÞjb4

ð15Þ

Now, to complete the set of equations, we need to provide themass balance equation relating the concentration in the solutionphase with the population. As the volume change of dissolutionmay be considered small, the total dispersion volume can beconsidered constant. Hence, the concentration of solute in thesolvent phase, x(t), can be expressed in terms of populationdensity as

xðtÞ ¼mtotal�rVsðtÞ

1�VsðtÞð16Þ

where mtotal is the total mass of solid per unit volume (in solutionand as particles), which remains constant throughout the process.Vs is the volume of a particle per unit volume of crystallizer and ris the density of the solid. Vs can be expressed in terms ofpopulation density as

VsðtÞ ¼

Z Zz1 ,c

n2,b1ðz1,c,tÞVðz1,c,tÞdz1 dc

þ

Z Z Zz1 ,z2 ,c

n2ðz,c,tÞVðz,c,tÞdz dcþ

Z Zz1 ,z2

n2,b2ðz,tÞVðz,tÞdz

þ

Zz2

n4,b3ðz2,tÞVðz2,tÞdz2þ

Z Zz1 ,z2

n4ðz,tÞVðz,tÞdz

þ

Zz1

n4,b4ðz1,tÞVðz1,tÞdz1 ð17Þ

where V(z,c,t) is the volume of a crystal with internal coordinatesgiven by z,c, the expression for which can be found in Cardew(1985). The supersaturation is expressed as:

sðtÞ ¼ xðtÞ�x�

x�ð18Þ

where x* is the solubility of the solute. The growth rates of variousfaces are observed to have complex functionality with super-saturation and generally expressed as a combination of severalpower law functions of the type _Zi ¼ asb over the range ofsupersaturation. We shall discuss specific forms of growthfunctions in the next section. This complete set of equations canbe solved for a given initial asymmetric seed population anddetail morphological evolution can be traced.

4. Case study: evolution of a population of asymmetriccrystals

To simulate the growth of a population containing crystals withtwo types of faces, namely square and diamond, we need thesupersaturation dependent growth rates of both the faces. For thispurpose, we have taken growth functions which are similar to the110 and 011 faces of Acetamenophen crystals respectively (Risticet al., 2001) and are shown in Fig. 6. Other parameters neededfor simulations are shown in Table 3. The model equations(Eqs. (10)–(18)) contain both integral and partial differentialequations. The partial differential equations have been solved byusing the method of characteristics and an iterative approach has

Table 3Various parameters used in simulations.

Parameter Value

Density of solid 1200 kg/m3

Saturation concentration at T¼348K 1 kg/m3

Number of seed crystals 1.4678�105 m�3

Initial seed distribution n2(Z, c, 0) c (5�10�4, 5�10�4, 2.5�10�5,

s2¼10�8)

Initial volume fraction of seed 6.2608�10�6

Initial mass of seed crystals 7.5129�10�3 kg/m3

Final mass of crystalline solids 120�10�3 kg/m3

Parameters for Van’t Hoff’s plot

(equilibrium solubility vs.

temperature)

DSdiss

R ¼ 5:77 DHdiss

R ¼ 3674 K�1

Fig. 7. Evolution of the number fraction of various crystal populations: isothermal

case.

0 0.5 1 1.5 2 2.5 3 3.5x 105

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

Time (sec)

Rel

ativ

e S

uper

satu

ratio

n

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

1

2

0

0.5

1.5

2.5

3

3.5

4 x 10−8

Supersaturation (σ)

Gro

wth

Rat

e (m

/s)

DiamondRectangle

Fig. 6. Variation of growth rates of the diamond and square faces with relative

supersaturation.

J. Chakraborty et al. / Chemical Engineering Science 65 (2010) 5676–5686 5683

been taken to converge it with the mass balance equation. In thefollowing, we shall first consider a seeded batch cooling crystallizerunder isothermal condition and then discuss the effects of varioustemporal profiles of temperature during growth.

Fig. 8. Evolution of relative supersaturation in the batch crystallizer during crystal

growth under isothermal condition. The corresponding evolution of various crystal

populations have been shown in Fig. 7.

4.1. Seeded batch cooling crystallizer under isothermal condition

(T¼348 K)

In this crystallizer, the small seed particles of a given mass andsize distribution grow up to a specified final mass. During thisprocess the temperature is held constant and supersaturationdrops due to consumption of material. The initial seed has beenconsidered to be purely of morphological form n2(z,c,t) (2b),normally distributed in its domain with mean and variance givenin Table 3. During crystal growth, various boundary fluxes areobserved which lead to changes in the populations of variousmorphological forms. The changes in the zeroth moments ofvarious morphological forms have been considered as a suitablemeasure of evolution and expressed as the number fraction of agiven morphological form. The evolution of the zeroth momentsof various populations for the current case has been shown inFig. 7 where the fractional numbers have been plotted against therelative change in the mass of the crystals given byðmðtÞ�m0Þ=ðmf�m0Þ where m0 and mf are initial and final mass

of crystals and m(t) is the mass of crystals at any given time. Thecorresponding time evolution of supersaturation has been shownin Fig. 8.

It can be noted from Fig. 6 that the rectangular face growsfaster than the diamond face at initial supersaturation, (s�0.18).This give rise to a flux of population through the boundaries b2

and b3 which creates the morphological forms n2,b2ðz,tÞ and

n4,b3ðz2,tÞ respectively. This has been shown in the left most

portion of Fig. 7. As growth continues, the supersaturationdepletes and when the supersaturation reaches a value of�0.13, a switch in growth rate occurs (see Fig. 6). At this pointthe diamond face starts to grow at a faster rate than the squareface and a flip in the directionality of the flux occurs. Due to thisreverse flux, the total population of n2,b2

ðz,tÞ are added to thedomain of n2(z,c,t) in a discontinuous manner as shown in Fig. 7.Similarly, n4,b3

ðz2,tÞ convert into n4(z,t) instantaneously. Although

0 1 2 3 4 5 6 7x 106

347

347.5

348

348.5

349

349.5

350

350.5

351

Time (sec)

Tem

pera

ture

(K)

Fig. 10. Evolution of temperature during the growth of crystals in a crystallizer

while maintaining a constant supersaturation of s¼0.1.

J. Chakraborty et al. / Chemical Engineering Science 65 (2010) 5676–56865684

this condition also implies a flux through boundary b1, this fluxdoes not appear immediately. The reason for this is the following.When the population was moving towards and through theboundary b2, it has created a void in the domain of n2(z,c,t) nearboundary b1. Hence, when the population starts to move backtowards b1, it takes a finite time for the population to reach thatboundary and flux to start. Once the flux through boundary b1

starts, it continues until n2(z,c,t) is exhausted. Because theboundary b3 is impervious with respect to flux in this direction,the two and four symmetry line species remains unconnected forthe rest of the growth period and n4(z,t) transferred into n4,b3

ðz2,tÞgradually. This has been shown in the right most part of Fig. 7.

We notice that although we started with less symmetricmorphological form (n2(z,c,t)), we have created appreciableamount of more symmetric forms (n4,b3

ðz2,tÞ, n4(z,t), n4,b4ðz1,tÞ)

due to growth. The nature of growth can be controlled bycontrolling the supersaturation and if we could hold the super-saturation to such a value so that that _Z � nb3

¼ 0 we could avoidthe flux through boundary b3 and thereby creation of moresymmetric forms. Control over supersaturation can be readilyachieved by manipulating temperature and therefore it isstraightforward to control the asymmetry of a population whichwe shall demonstrate next.

4.2. Seeded batch crystallizer operating at a fixed value of

supersaturation (s¼0.1)

We notice from Fig. 6 that the growth rate of the rectangularsurface becomes smaller than that of the diamond surface belowrelative supersaturation 0.14 and therefore if we can keep thesupersaturation to a value lower than that, mostly the diamondface will grow and the population will traverse towards themorphology n2,b1

ðz1,c,tÞ from the parent form n2(z,c,t). In this caseno morphological form with four lines of symmetry is created. Theevolution of various morphological forms due to growth at a fixedsupersaturation of 0.1 is shown in Fig. 9. In this figure we noticethat only the less symmetric form n2,b1

ðz1,c,tÞ has been createdand no crystal with four lines of symmetry has been formed. Tokeep the supersaturation at a fixed value, the temperature hasbeen varied according to the saturation curve of the crystallinesolid. The temperature dependence of solubility is given by Van’t

Fig. 9. Evolution of the zeroth moment of various species while maintaining a

constant supersaturation of s¼0.1.

Hoff’s plot necessary parameters for which are provided inTable 3. The corresponding variation in temperature has beenshown in Fig. 10.

4.3. Control of symmetry related properties in a seeded batch

crystallizer

In many practical applications, symmetry related propertieslike surface area and sphericity are important. For example, forpharmaceutical powders, the bioavailability of a particular facemight be more than the other faces and higher fractional coverageof that face is desirable. In the current model, we consider therectangular face to have some favorable property and in thefollowing example we investigate the variation of the fractionalcoverage of the rectangular surface for different mode ofoperation of the crystallizer.

In Fig. 11 we have plotted the evolution of the fractionalsurface area of the rectangular surfaces for the isothermal casealong with two cases of fixed supersaturation. For the isothermalcase, the initial supersaturation is high and the growth rate ofrectangular surface is higher. Therefore, the relative importance ofthe rectangular face decreases until the supersaturation reachesbelow �0.14. After this point, the rectangular face dominates andreaches a value of 100%. If the value of the supersaturation is heldat a value higher than this critical value of 0.14, the rectangularsurface always grows faster than the diamond surface and itsmorphological importance reduces monotonically. Similarly, if thesupersaturation is held at a value higher than this critical level,the morphological importance increases monotonically to 100%.

Another important property of the particulate solid whichaffects downstream processing is sphericity. The evolution of theaverage sphericity of the crystal population for the three casesconsidered has been shown in Fig. 12. It may be noted that for allthree cases, the average sphericity of the population evolves to avalue which is close to that of a square from the higher sphericityof octagons. However, the transient behavior of the system is verydifferent depending on the supersaturation environment. For thecase of s¼0.1, the initial population quickly goes to the limitingvalue whereas for the case of s¼0.18, this approach is muchgradual and retains a much higher value of sphericity for aconsiderably long time. For the isothermal growth, the sphericity

0 0.2 0.4 0.6 0.8 10.88

0.89

0.9

0.91

0.92

0.93

0.94

0.95

0.96

0.97

0.98

Dimensionless Mass

Sph

eric

ity

σ=σ(t)σ=0.1σ=0.18

Fig. 12. Evolution of the average sphericity of the population for various

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Dimensionless Mass

Frac

tiona

l Cov

erag

e of

Rec

tang

ular

Sur

face

σ=σ(t)σ=0.1σ=0.18

Fig. 11. Fractional coverage of the rectangular surface for various growth

strategies.

J. Chakraborty et al. / Chemical Engineering Science 65 (2010) 5676–5686 5685

shows a minimum before rising again and finally mergingtowards the limiting value. In this case, the octahedron crystalpopulation first goes to less spherical forms like square. But as thesupersaturation drops and the growth rate trend reverses, theoctahedron is regenerated and the sphericity increases. However,it finally drops due to the production of rectangle from theoctagon in the boundary b1.

supersaturation environments.

5. Conclusions

We have demonstrated in this paper the morphologicalpopulation balance modeling for asymmetric crystals using anew approach. In this approach, the internal coordinate vector isclassified into various sets characterized by the growth rate. Onlyone dynamic coordinate is assigned for each set to describe theevolution while all other coordinates are written in a way so thatthey remain time invariant. To determine the minimum numberof internal coordinates required for a population of crystals with

varied symmetry, crystals are classified into several symmetryclasses and morphological forms. Although the theory requires amorphological characterization which may be too elaborate froma practical viewpoint, a suitable regrouping of populations canalways be made so that the classifications are experimentallyresolvable.

This strategy has been used to simulate a population ofasymmetric crystals efficiently. It has been shown that during thegrowth, the crystals of lower symmetry transferred into moresymmetric morphological forms due to appearance and disap-pearance of faces. However, the reverse is shown to be impossibleand therefore the number density of more symmetric crystalsmonotonically increases inside the crystallizer. We have demon-strated that it is possible to drive a population of asymmetriccrystals towards a desired symmetry by controlling the super-saturation. Different temperature profiles have been consideredand it has been shown that the control of sphericity and fractionalcoverage of a desired surface can be achieved using this model.

Acknowledgments

The authors (J.C., M.S. and D.R.) gratefully acknowledgefinancial support from Roche Pharmaceuticals and NSF Engineer-ing Research Center for Structured Organic Particulate Systems forconducting this research.

Appendix. Symmetry group of the region common to twopolygons

Let us consider two polygons P1 and P2 which can be thoughtof as an infinite set of points and hence the common region ofboth the polygons can be denoted by Pc where Pc � P1 \ P2. Thesymmetry groups of regular polygons are known as dihedralgroups and are well documented in the literature (Lomont, 1959).The matrix representations of such symmetry groups are alsoavailable. Now, let us assume that the cardinality of the symmetrygroups of P1 and P2 are n and m respectively. We can now denotethe symmetry group of P1 by

S1 ¼ fs11, . . . ,s1ng

and that of P2 by

S2 ¼ fs21, . . . ,s2mg

Let us assume that Pc is symmetric with respect to the operations1j where: s1jAS1�ðS1 \ S2Þ. Now, let us consider a point p suchthat pAPc and let us denote the image of the point p after thesymmetry operation s1j as p0. Now, the symmetry operation s1j

ensures that fp,p0gAP18p but as s01j=2S02, the condition fp,p0gAP2 isnot necessarily satisfied unless s01jAðS

01 \ S02Þ.

Hence, sij is a symmetry operation of Pc if s1jAðS1 \ S2Þ.

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