determining the growth mechanism of tolazamide by induction time measurement

Determining the Growth Mechanism of Tolazamide by InductionTime Measurement

Anuj Kuldipkumar,†,# Glen S. Kwon,*,† and Geoff G. Z. Zhang*,‡

Department of Pharmaceutical Sciences, School of Pharmacy, UniVersity of Wisconsin-Madison,777 Highland AVenue, Madison, Wisconsin 53705, and Solid State Sciences, Global PharmaceuticalR & D, Abbott Laboratories, 1401 Sheridan Road, North Chicago, Illinois 60064

ReceiVed April 14, 2006; ReVised Manuscript ReceiVed October 28, 2006

ABSTRACT: A turbidity measurement-based experimental method for the determination of the crystallization induction period isdescribed. The induction times for crystallization of tolazamide, an oral hypoglycemic agent, were measured over a range ofsupersaturation in both unseeded and seeded crystallization experiments. The measured induction times were then treated using theexpressions developed by van der Leeden et al., which relates the induction time and the supersaturation for various growth mechanismsin unseeded and seeded crystallization. On the basis of these analyses, the growth mechanism of tolazamide was identified astwo-dimensional nucleation-mediated growth. The analyses also afforded estimation of the kinetic parameters such as nucleationand growth rate constants. In addition, atomic force microscopy imaging of the major faces of tolazamide crystals corroborated thisgrowth mechanism.

Introduction

We have recently shown that an amphiphilic block copolymer,when present in the crystallization medium, was effective atpart per million (ppm) levels in changing the morphology/habitof tolazamide from a needle to a plate shape.1 We have alsoshown that the additive is present only on the surface of therecrystallized solid and can be removed easily by washing withan appropriate solvent.1 To understand the mechanism of thishabit modification, it is essential to first identify the crystalgrowth mechanism of tolazamide in the absence of anyadditives. In general, the crystal growth mechanism is deter-mined by measuring the growth rates and then fitting themeasured rates to the expressions describing the different crystalgrowth mechanisms. On the basis of the best fit of theexperimental data to the various expressions, the underlyinggrowth mechanism is identified.2,3 Some of the commontechniques for the measurement of crystal growth rates includemonitoring the change in crystal population (by methods thatmeasure particle size and number) and monitoring the individualface growth rates by optical microscopy.4 These techniques haveseveral disadvantages. When the growth rate of a particularsolute is estimated from an increase in the number of particles,erroneous results may be obtained if agglomeration of thecrystals occurs. Particle size measurements often approximatethe shape of the crystals with a sphere, while the actual crystalsmay be of other shapes such as acicular. When individual crystalface growth rates are measured, often the supersaturation usedis nominal as extensive secondary nucleation is observed at ahigh supersaturation. To overcome some of these disadvantages,it has been suggested to observe the development of surface

topology at the molecular level by the use of atomic forcemicroscopy (AFM), a technique that is often used as a nanoscaletechnique for the surface characterization of organic crystals.5

AFM has been used to investigate surface topography throughboth ex-situ and in-situ observations.6-8

Another technique to better understand nucleation and growthkinetics of molecules involves the measurement of the nucleationtime. The true nucleation time,tn, is defined as the time thatelapses from the creation of supersaturation in solution untilcritical nuclei form. This lag phase occurs as a consequence ofthe achievement of a steady-state distribution of sizes of thenuclei that are already present in solution or newly formed owingto the supersaturation that is created.9 However, it is not possibleto directly observe the critical nuclei as they form in solution,because they can only be observed after they have grown to acritical size. The induction time,tind, refers to the time thatelapses after the creation of supersaturation in solution until anew phase is detected and is the experimentally accessiblequantity. The induction time thus measured allows for aconnection to be made between nucleation theory and experi-mental investigation. Therefore, reliable methods for the deter-mination of induction time periods are important. Severaltechniques such as measurement of solution conductivity,10

intensity of transmitted light,11 electronic microscopy,12 fluo-rescence,13 and turbidity14 have been used for the experimentaldetermination of the induction time. Most of the aforementionedtechniques are more sensitive than visual detection of crystalsby use of an optical microscope. The induction time determinedexperimentally depends on the technique used to detect theformation of the new phase and hence is not a fundamentalproperty of the system being studied. The induction time isaffected by several parameters such as the initial supersaturation,temperature, pH, agitation speed, and the presence of additives/impurities.15-19

There are some studies reported in the literature, whereingeneral theoretical expressions have been derived for thedependence of the induction time on the supersaturation fordifferent crystal growth mechanisms.20,21 Use of these expres-sions allows for the identification of the underlying growthmechanism of a particular compound by measurement of theinduction time over a range of supersaturation. Moreover, these

* To whom correspondence should be addressed: (G.S.K.) Departmentof Pharmaceutical Sciences, School of Pharmacy, University of Wisconsin-Madison, 777 Highland Avenue, Madison, WI 53705, USA. Telephone:(608) 265-5183. Fax: (608) 262-5345. E-mail: [email protected].(G.G.Z.Z.) Solid State Sciences, Global Pharmaceutical R & D, AbbottLaboratories, 1401 Sheridan Road, North Chicago, IL 60064, USA.Telephone: (847) 937-4702. Fax: (847) 937-2417. E-mail:[email protected].

† University of Wisconsin-Madison.‡ Global Pharmaceutical R & D, Abbott Laboratories.# Current address: Vertex Pharmaceuticals Inc., 130 Waverly Street,

Cambridge, MA 02139-4242.

CRYSTALGROWTH& DESIGN

2007VOL.7,NO.2

234-242

10.1021/cg0602212 CCC: $37.00 © 2007 American Chemical SocietyPublished on Web 12/16/2006

expressions have been shown to be correct in identifying theoperating crystal growth mechanism of inorganic compoundsby use of other independent methods.22,23

In the present work, the crystallization of tolazamide wasstudied in the absence and presence of seed crystals. Theinduction times were experimentally determined as a functionof initial supersaturation by turbidity measurement using a UVspectrophotometer. The experimental data were then treatedusing the theoretical expressions for different crystal growthmechanisms relating thetind with the supersaturation. Theoperating crystal growth mechanism was identified based onthe fit of the experimental data to the theory. Furthermore, thecrystal growth mechanism thus determined was corroboratedby observing the surface topology of the crystals using AFM.

Materials and Methods

Materials. Tolazamide was purchased from Sigma (St. Louis, MO),sodium hydroxide was from Fisher Scientific (Fair Lawn, NJ), andhydrochloric acid was from EM Science (Gibbstown, NJ). Sodiumphosphate monobasic, monohydrate (EM Science, NJ) and sodiumphosphate dibasic, anhydrous (Fisher Scientific, NJ) were used toprepare the buffer solutions. All chemicals were of analytical gradeand were used without further treatment. Water was filtered through adouble-deionized purification system (Milli Q, Millipore Synthesis,Molsheim, France).

Determination of the Induction Time. The crystallization oftolazamide was performed by adjusting the pH of the medium. Allcrystallization experiments were carried out in jacketed glass beakersmaintained at 25°C by attaching to a circulating water bath. A 1%(w/v) tolazamide solution was prepared by dissolving the drug in 0.1N NaOH. The drug solution was then added to 50 mL of aqueousphosphate buffer (pH 5.9, 0.1 M). The pH of the phosphate buffer wasmaintained constant by the simultaneous addition of an equal amountof 0.1 N HCL. A syringe pump (model 11, Harvard Apparatus, Boston,MA) was used to add the solution at a rate of 3.3 mL/min. Agitationof the solution was achieved by use of a magnetic stirrer (CorningPC-320, Fisher Scientific, Pittsburgh, PA). A fiber optic probe attachedto a Cary 50 UV spectrophotometer (Varian, Inc., Walnut creek, CA)was used to measure the induction time. All solutions were filteredthrough 0.45µm membrane filters (Acrodisc, Pall Gelman Laboratory,Ann Arbor, MI) prior to use. The Cary 50 UV spectrophotometer wasprogrammed to measure the turbidity of the solution atλ ) 650 nm.In the seeded precipitation experiments, 250µg of ground tolazamideseed crystals was added to the crystallization medium prior to the startof the UV scan.

Preparation of Tolazamide Crystals of Appropriate Size. Toperform AFM experiments, tolazamide crystals of a large enough sizewere prepared by a slightly different crystallization procedure. Crystalsof tolazamide were obtained by addition of 0.48 mL of the stocksolution to 10 mL of the buffer solution in a scintillation vial (20 mL).Agitation of the medium was achieved by the use of a mechanical stirrer(Lightnin Labmaster, Fisher Scientific, Pittsburgh, PA) at a speed of500 rpm. Upon completion of addition, the vials were allowed to situnperturbed over the counter for a few months, after which crystalswere harvested.

Atomic Force Microscopy (AFM). A Nanoscope IV multimodeAFM (Digital Instruments, Inc., Santa Barbara, CA) was employed toimage the tolazamide crystals. All experiments were performed in airunder ambient conditions. Large enough crystals of tolazamide wereremoved from the mother liquor and washed three times with Milli-Qfiltered water (Millipore Synthesis, Molsheim, France) on a glass slideto remove any adhering particulates. They were then dried withKimwipes and mounted with epoxy on an AFM sample puck with theirorientation depending on the crystal face being studied. To minimizeany potential surface damage caused by the scanning AFM tip, tappingmode imaging was used in all the experiments. Silicon probes (DigitalInstruments, Inc., Santa Barbara, CA) with a cantilever length of 125µm, a nominal tip radius of curvature of 10 nm, and a resonantfrequency of∼300 kHz were utilized. In tapping mode height imageswere collected and analyzed with Nanoscope IV version 5.12 r5software (Digital Instruments, Inc., 1999). All images were acquired

at a scan rate of 1 Hz and a resolution of 256× 256 pixels. Largerarea images were routinely collected at the conclusion of eachexperiment to check for tip-induced artifacts. Under the minimal tip-sample forces applied, no obvious surface deformation was observed.

Results and Discussion

Induction/Lag Time Measurement using Turbidity. Ac-cording to the classical nucleation theory, the nucleation rate(J) is given by an expression as given below:

where A is the pre-exponential factor,k is the Boltzmannconstant,T is the absolute temperature,γ is the interfacial freeenergy,V is the molecular volume, andS is the supersaturation.This equation assumes that spherical particles are formed. Fortolazamide, this assumption is not valid, because our previouswork has demonstrated that tolazamide crystals are acicular inshape when formed in the absence of additives.1 Therefore, theequation was re-derived (see appendix) to accommodate othershapes as shown below:

where fs and fv are the surface and volume shape factors,respectively. Making the assumption that thetind is mainlycomposed oftn, which is inversely proportional to the nucleationrate, we get:

From eq 3, we see that an increase in the initial supersatu-ration will result in a decrease in the observed induction time.The induction time of tolazamide crystallization was determinedover a range of initial supersaturation (σ ) 1.02 - 3.82,σ )S - 1 ) c/s - 1, wherec is the concentration ands is thesolubility of tolazamide, respectively) by turbidity measurement.The spectrophotometer was programmed to measure the turbid-ity after the addition of the drug solution was completed. Asthe solution is clear initially, a minimum or baseline turbidityis recorded. However, as nuclei form and grow into macroscopiccrystals, the turbidity starts to increase. The result is a plot ofturbidity versus time from which thetind is determined bydrawing regression lines through the two distinct linear regionsand taking their point of intersection as its value. Representativeplots at different supersaturation are shown in Figure 1. As canbe seen from Figure 1a, at a low value of supersaturation thereis a prolonged lag/induction phase prior to the appearance ofcrystals in solution and an in increase in turbidity. As expected,this induction period is dramatically reduced as the initial valueof the supersaturation is increased (Figure 1b-d).

As mentioned earlier, the experimentally determined inductiontime is a composite of the time required for critical nucleiformation and their growth to a detectable size. Sohnel andMullin9 have shown that when the induction time is dominatedby the time required for critical nuclei formation, eq 3 is valid.To verify whether this was the case for tolazamide, logtind wasplotted versus (logS)-2 and is shown in Figure 2. From Figure2, it can be seen that the measured induction times fortolazamide crystallization follow the linear relationship givenby eq 3. However, there is a region of higher slope at highersupersaturation and a region of lower slope at lower supersatu-

J ) A exp( -16πγ3ν2

3k3T3(ln S)2) (1)

J ) A exp( -4fs3γ3ν2

27fv2k3T3(ln S)2) (2)

log tind ) K + R[log S]-2 (3)

Growth Mechanism of Tolazamide Crystal Growth & Design, Vol. 7, No. 2, 2007235

ration. Similar results have been reported for inorganic salts byother researchers.24,25 This change has been attributed to achange in the nucleation mechanism from homogeneous nucle-ation at high supersaturation to heterogeneous nucleation at lowsupersaturation. The reason for this change is the strongdependence of homogeneous nucleation rate on supersaturation.At high supersaturation, the homogeneous nucleation rate ishigh, and therefore dominates nucleation. At low supersaturation,on the other hand, the homogeneous nucleation rate is low, andhence heterogeneous nucleation predominates. In any bulksolution, many dust particles are present that can act asnucleation centers. It is impossible to remove all these dustparticles (even by repeated filtration through membranes), andas a result the longer the time lapse without homogeneousnucleation occurring in a supersaturated system, the greater thechance that these dust particles, as well as the wall of thecrystallization vessel, can act as heteronuclei for the crystals to

form. The interplay of homogeneous and heterogeneous nucle-ation is reflected in Figure 2.

The proportionality constant (R) in eq 3 is comprised of theshape factors, the interfacial free energy, the molecular volume,and temperature. If the values of the shape factors, the molecularvolume, and temperature are known, then the crystal/solutioninterfacial free energy can be extracted from the slope of theline corresponding to homogeneous nucleation. Since tolazamidecrystals were needlelike with large aspect ratios, the values ofthe shape factors were estimated to be about 26- 42 for fs,and 2.8- 10 for fv (see Appendix). The molecular volume wasestimated from the density and molecular weight of thetolazamide crystals. From the slope of the line correspondingto homogeneous nucleation, the interfacial free energy (γ)between tolazamide crystals and aqueous crystallization mediumwas determined to be 1.94-2.80 mJ/m2. Several authors haveused this technique to determine the value of this importantthermodynamic parameter for other organic26,27 and inorganicsystems.28-30 Moreover, induction time studies with inorganicsalts have found interfacial free energy values obtained usingthis method to be in good agreement with the values obtainedby other experimental techniques.28-30 This value ofγ compareswell with the values obtained by other authors for differentorganic materials such as acetaminophen (1.73 mJ/m2),27 urea(4.2-8.9 mJ/m2),26 lovastatin (1.45 mJ/m2),31 and asparagine(4.4 mJ/m2).31 The value ofγ can serve as an indicator of theability of a solute to crystallize from solution spontaneously,the higher its value the more difficult it is for the solute tocrystallize.

The turbidity measurement method used in this study providedan inexpensive and reliable technique for the measurement ofthe induction time. The spread of the data in Figure 2 is mainlydue to the stochastic nature of the nucleation process, ratherthan the experimental error. The experimentally determinedinduction times are invaluable in efficient crystallization systemdesign and process control. For instance, in crystallization by

Figure 1. Plots of turbidity versus time at (a)σ ) 1.02, (b)σ ) 1.58, (c)σ ) 2.13, and (d)σ ) 3.82 for the unseeded crystallization experiments.

Figure 2. Plot of logtind versus (logS)-2 for tolazamide crystallizationin the unseeded experiments. Values plotted are an average of sixindependent runs.S is calculated asc/s, where c refers to theconcentration of tolazamide in solution ands to its solubility,respectively. The data points represented by triangles correspond tolow supersaturation values, while the data points represented bydiamonds correspond to high supersaturation values.

236 Crystal Growth & Design, Vol. 7, No. 2, 2007 Kuldipkumar et al.

antisolvent addition, if the mixing times of the solvent andantisolvent streams can be controlled to be shorter than theinduction time, narrow and reproducible crystal size distributionsmay be achieved. Analysis of the induction times according toclassical nucleation theory may help in comprehending themechanism of nucleation. In addition, calculation of importantvariables such as the crystal/solution interfacial tension andnucleation and growth rates (as explained below) can beafforded.

Identification of Growth Mechanism of Tolazamide. Asmentioned earlier, we decided to use the expressions derivedby van der Leeden et al.20 to identify the growth mechanism oftolazamide crystals by fitting thetind data measured from seededand unseeded crystallization experiments over a range ofsupersaturation. The value ofFu was calculated by putting theappropriate values of the initial supersaturation and the measuredtind into either eq A 24 or A 26, respectively. Since tolazamidecrystals are needlelike, the value ofm (the dimensionality ofgrowth) is 1. Hencen (mν + 1) can take values of either 3/2 or2, depending on the value ofν (1/2 or 1). The calculated valuesof Fu are then plotted against either (lnS)-1 or (ln S)-2 and

depending on the goodness of fit the underlying growthmechanism identified. The plots for the data fit to the differentcrystal growth mechanisms are shown in Figure 3. The plot inFigure 3a shows the data for tolazamide crystals fit to the normalgrowth mechanism. In the next plot (Figure 3b), the data are fitto the spiral growth mechanism. The fit of the data to thesetwo mechanisms is obviously not good as can be seen from thelow correlation coefficients and the systematic deviation of thedata points from the fitted line. The plots in Figure 3c,d showthe data fit to the two-dimensional (2D) nucleation-mediatedgrowth and volume diffusion-controlled growth mechanisms,respectively. Again, the fit of the data to the volume diffusion-controlled mechanism is not good as can be seen from the lowcorrelation coefficient and the systematic deviation of the datapoints from the fit line. In contrast, the fit of the data is verygood to the 2D nucleation-mediated growth mechanism. Hence,it is concluded that this is the operating mechanism for thegrowth of tolazamide crystals under the experimental conditionsdescribed above.

In the seeded crystallization experiments, the experimentallyobserved induction times (ts) were significantly shorter at the

Figure 3. Plot of Fu versus (lnS)-2 or (ln S)-1 for tolazamide crystals in the case of unseeded crystallization experiments. The values ofFu werecalculated using eqs A24 and A26, respectively. Data are fit using (a) normal growth, (b) spiral growth, (c) 2D nucleation-mediated growth, and(d) volume diffusion-controlled growth.

Figure 4. Plot of (a) ln tind versus ln(S - 1) and (b)Fs versus 1/(lnS) in the case of seeded crystallization experiments. The value ofFs wascalculated using eq A28.


same initial supersaturation values when compared to theunseeded experiments (data not shown). This is expectedbecause the induction time is a composite of the time requiredfor the formation of the critical nuclei and their growth to adetectable size. In the presence of seed crystals, only growth isexpected to occur, and this results in much shorter inductiontimes being observed. Moreover, it is assumed in the case ofseeded crystallization that the crystals do not agglomerate andonly negligible secondary nucleation occurs. The sufficientlyshorter induction time in the seeded experiments implies thatthe effect of any additional nucleation onts is negligible. Theplot for the data fit to the different growth mechanisms is shownin Figure 4. As can be seen from Figure 4a, the slope of theline fit to ln ts versus ln(S- 1) is not-1 or -2, and hence themechanism of growth is not normal, volume diffusion-controlled, or spiral growth. The value ofFs was calculated byputting in the appropriate values of the initial supersaturationand the measuredts into eq A 28. The plot ofFs versus 1/(lnS)(Figure 4b) results in a straight line with a good correlationcoefficient. Hence, the mechanism of growth from the seededcrystallization experiments is identified as being 2D nucleation-mediated growth. This is the same as that identified for

tolazamide crystals from the unseeded crystallization experi-ments.

Since the growth mechanism was identified as being mediatedthrough the formation and spread of 2D nuclei, the value ofnis 2. From the slope of the line in Figure 4b, corresponding tothe seeded crystallization experiments, the specific edge freeenergyκ and the number of molecules in the 2D nucleus [n2D

) B2D/(ln S)2] were calculated. This resulted inB2D ) 13.4,κ) 1.64× 10-16 J/m andn2D varying from 10.3 to 27.2 for thesupersaturation range of 2.13-1.02. The number density of theseed crystals was calculated to be 5.74× 1010 m-3, and thevalue of R was estimated to be 1.08× 10-4 to 2.24× 10-4.Since the value ofAs is known from the intercept of the line inFigure 4b, the value of the growth rate constant (KG) wascalculated from eq A17 to be 3.13× 10-17 to 6.51× 10-17

m/s for the supersaturation range of 1.02-2.13. Using thesevalues forKG, the value of the growth rate (G) was calculatedas being 6.89× 10-20 to 3.12× 10-18 m/s. Similarly, from thecoefficient of the parabolic fit for the induction time data inunseeded crystallization experiments to the 2D nucleation-mediated growth mechanism, the value ofAu is known. Fromthis value ofAu, the value of the nucleation rate constant (KJ)

Figure 5. AFM height image (left) and height profile (right) of the 010 face of a tolazamide crystal. The height profiles are running from left toright through the height images. The scale on the height profiles on they-axis is in nm. From top to bottom images acquired are of the same areawhile zooming out (1× 1 - 10 × 10 µm).


can be calculated using eq A13 since the value ofKG is nowknown from the seeded crystallization experiments. The valueof KJ thus determined was 1.68× 1027 m-3 s and is very closeto the theoretically expected value of 1031 to 1036 m-3 s forhomogeneous nucleation in solutions.32

If the crystal/solution interface is rough at the molecular level,then numerous kink sites exist to which the incoming solutemolecules can attach and growth can take place by a normalgrowth mechanism. However, most often growth takes placevia layer growth wherein solute molecules attach at kink sitesfound on the edges of steps or 2D nuclei. This continues untilan entire layer is formed on the surface of the growing crystal,and then the process repeats for all further layers. Themechanism of growth can then be either due to the 2D nucleithat form or due to a growth spiral (as it provides an infinitenumber of steps). In the case of tolazamide, the formermechanism seems to operate under the experimental conditionsused here.

AFM Imaging of Tolazamide Crystals. In a recent review,Rodriguez-Hornedo et al.4 highlighted that although identifica-tion of the operating mechanism of crystal growth can be doneby fitting various equations to the experimental data, it issometimes difficult to discriminate between models. As a result,these authors suggested the use of techniques such as AFM toobserve the surface topography of the crystals to confirm thegrowth mechanism. Lechuga-Ballesteros et al.7 have reporteda study wherein these authors measured the growth rates ofprominent crystal faces ofL-alanine by observing changes incrystal dimensions. They concluded that the operating mecha-nism was due to spiral growth. Further, they confirmed thismechanism of growth for theL-alanine crystals by examinationof the surface topography of the major crystal faces using AFM.Although the growth mechanism of tolazamide was identifiedquite conclusively through the analyses of the experimentallymeasured induction times, we decided to observe the surfacetopography of the tolazamide crystal faces using AFM tocorroborate this growth mechanism.

Tapping mode was employed for obtaining images oftolazamide crystals. In this mode, the AFM tip is oscillated tobe in contact with the sample surface intermittently, therebyminimizing any damage associated with dragging the AFMprobe over the sample as seen in contact mode imaging at times.Detailed surface morphology of tolazamide crystals was ex-plored using AFM at a scale of 1-10 µm ex-situ and postgrowth. Some of the disadvantages of this include imaging ofthe crystals in a non-native environment (in the absence of themother liquor) and imaging after growth of the crystals iscomplete. Similar studies done with other organic crystalsystems have shown no significant differences in AFM imagesacquired in air and in aqueous solution.33 As-grown surfacetopography after the cessation of growth and after removal ofthe crystal from the growth solution represents the final stageof crystal growth and provides molecular information relatingto such growth processes.34

Figure 5 shows representative height images (left) and thecorresponding height profiles (right) acquired on the (010) faceof a tolazamide crystal. A relatively smooth surface is observedfrom the 1× 1 µm image. Since the surface of this face isatomically smooth (no steps or kinks), it is safe to concludethat the mechanism is not due to normal growth. Also, sinceno growth spirals are observed over this small scan area it canbe concluded that the mechanism of growth is not due to a spiraldislocation. This narrows down the growth mechanism as being

due to volume diffusion-controlled growth or mediated throughthe formation and spread of 2D nuclei.

When a 5× 5 µm scan is performed on the same area, largeprotrusions having a height of∼30 nm are observed (Figure5). Topographical variability of this magnitude must be at leastin part due to the growth mechanism. Further, a 10× 10 µmscan on the same area shows a larger number of such protrusionson this face. Also from this scan it can be seen that no apparentrestructuring of the crystal surface takes place. Similar protru-sions were observed in AFM images acquired on the other majorfaces of tolazamide crystals, namely, (100) and (001) (data notshown). From these images, it can be concluded that themechanism of growth of tolazamide crystals is due to the birthand spread of these 2D nuclei on the faces of the crystals. Thiscorrelates well with our earlier conclusion from the kineticanalyses oftind data.

Conclusions

The induction time for tolazamide crystallization was deter-mined over a range of supersaturation by turbidity measurement.When this measuredtind was analyzed according to classicalnucleation theory, two different nucleation mechanisms werefound to operate, homogeneous nucleation at high supersatu-ration and heterogeneous nucleation at low supersaturation.From the slope of the line corresponding to homogeneousnucleation, the crystal/solution interfacial free energy of tolaza-mide was estimated. The growth mechanism of tolazamidecrystals was identified as being 2D nucleation-mediated growthby kinetic analyses of thetind data obtained from unseededcrystallization experiments. This mechanism remained the samewhen seeded crystallization experiments were performed. Acombined analysis of the induction times from both unseededand seeded growth experiments afforded calculation of thenucleation and growth constants for tolazamide crystals. AFMimaging of the faces of grown tolazamide crystals providedinformation that was consistent with the identified growthmechanism.

Appendix

Derivation of eq 2. The overall excess free energy,∆G,between a small solid particle of solute and the solute in solutionis equal to the sum of the surface excess free energy (∆Gs) andthe volume excess free energy (∆Gv). For a non-sphericalparticle with an equivalent diameterd ()2r),

where fs and fv are the surface and volume shape factors,respectively,γ is the interfacial tension, and∆Gv is the freeenergy change of the transformation per unit volume.

The critical nucleus (rc) corresponding to the maximum∆Gvalue can be obtained by setting d(∆G)/dr ) 0. This yields:

From eq A3 and A4 we get:

∆Gs ) fsd2γ (A1)

∆Gv ) fvd3∆Gv (A2)

∆G ) 4fsγr2 + 8fvr3∆Gv (A3)

rc ) -fsγ

3fv∆Gv(A4)


where

The rate of nucleation can be expressed in the form ofArrhenius reaction velocity commonly used for the rate of athermally activated process:

Substituting the value of∆Gc from eq A5 into eq A7, eq 2is obtained.

Identification of the Crystal Growth Mechanism. For ourwork, we decided to use the expressions derived by van derLeeden et al.20 These authors have proposed a general expressionfor the induction time in unseeded precipitation (tu).35 Thisexpression reads:

whereJ is the nucleation rate,V is the volume of the system,R is the volume fraction of the new phase formed,G is thegrowth rate,an is a shape factor andn ) mV + 1 (m refers tothe dimensionality of growth and 0.5< ν < 1). The aboveequation contains two terms: the first one originates from themononuclear mechanism and the second one from the poly-nuclear mechanism. The first term is often negligible incomparison to the second. For three-dimensional (3D) nucle-ation, the formula for the steady-state nucleation rate is

whereKJ is the nucleation rate constant,S is the supersaturation,B is a constant composed of the shape factor (â), molecularvolume (n), and the specific surface free energy (γ) given byeq A 10 below:

The dependence of the growth rate on the supersaturation isgenerally of the form:

whereKG is the growth rate constant andf(S) is some functionof supersaturation depending on the growth mechanism. Com-bining eqs A8, A9, and A11 we get:

where

In the above equation, we can plug in the appropriatedependence of the growth rate on supersaturation dependingon the operative mechanism.

Similarly, in the case of seeded crystallization (wherein apredetermined amount of seed crystals are present at the startof the process), the induction time is given by

whereN (m-3) is the number density of the seed crystallitesadded to the solution,cm is a shape factor (two times the cross-sectional area for needle crystals), andR andG are as definedbefore.N can be calculated as shown in eq A15 below:

whereM is the mass of the added crystals,F is the density ofthe crystals,rs is the size of the seed crystals, andV is thevolume of the crystallization medium, respectively.

Plugging in the dependence of the growth rate on thesupersaturation from eq A 11 into eq A14 we get:

where

In the above equation, we can plug in the appropriatedependence of the growth rate on supersaturation dependingon the operative mechanism.

For a normal growth mechanism (ν ) 1), the growth rate isdetermined by the net flux of growth units to the crystal surfacewhich is proportional to (S - 1). As a result,

If growth is initiated by screw dislocations then a spiralgrowth mechanism (ν ) 1) is said to operate. In this case, thegrowth rate is determined by the density of steps on the surfaceof the crystal and the net flux of growth units to the surface.As a result,

However, in the above equation often the lnS is replaced by(S - 1) without causing any significant changes.36

If growth occurs by the formation and spread of numerous2D nuclei, then the mechanism of growth is said to be mediatedby these 2D nuclei (ν ) 1). In this case,f(S) is a complexfunction of the spreading velocity and the rate of nucleation ofthese 2D nuclei as given below:

where

â2D is a numerical 2D shape factor,κ is the specific edge free

ts )(R/cmN)1/mV

G(A14)

N ) 3M

4πFrs3V

(A15)

ts )As

f(S)(A16)

As )(R/cmN)1/mV

KG(A17)

f(S) ) (S- 1) (A18)

f(S) ) (S- 1) ln S (A19)

f(S) ) (S- 1)2/3S1/3 exp(-B2D

3 ln S) (A20)

B2D )â2Dκ

2a

(kT)2(A21)

∆Gc )4fs

3γ3

27fv2∆Gv

2(A5)

∆Gv ) - kT ln SV

(A6)

J ) A exp(-∆Gc

kT ) (A7)

tu ) 1JV

+ ( RanJGn-1)1/n

(A8)

J ) KJSexp( -B

ln2 S) (A9)

B ) âγ3ν2

(kT)3(A10)

G(S) ) KGf(S) (A11)

tu(S) ) Au[f(S)]-(n-1)/nS-1/n exp( B

n ln2 S) (A12)

Au ) ( RanKJKG

n-1)1/n(A13)


energy of the nucleus,a is the molecular area,k is the Boltzmannconstant, andT is the absolute temperature.

Lastly, growth can also be controlled by the rate of transportof growth units through the solution to the growing crystalsurface. In this case, a volume diffusion-controlled growthmechanism (ν ) 1/2) is said to operate. As a result,

This growth mechanism has a similarf(S) dependence as thenormal growth mechanism. The difference between the twomechanisms is in the value of the kinetic factorAu.

For all the above-mentioned expressions, prior knowledgeof the operating growth mechanism and the value of certainparameters such asn must be known to analyze the experimen-tally determined induction times. As an alternative, a fittingprocedure may by used to determine the value ofn and theoperating growth mechanism. This is described below.

For normal, spiral or volume diffusion-controlled growth, eqA12 may be rearranged into

where

Hence a plot ofFu(S) versus 1/(ln2 S) can be fitted by astraight line for different values ofn. Depending on the goodnessof fit (for instance theR2 value), the operating mechanism maybe identified as being either normal, spiral or volume diffusion-controlled growth.

For 2D nucleation-mediated growth, eq A12 may be rear-ranged to give

where

In this case, a plot ofFu(S) versus 1/(lnS) can be fitted by aparabolic dependence for different values ofn and dependingon the goodness of fit the value ofn determined.

Similarly, in the case of seeded precipitation, the experimen-tally determined induction time data can be plotted as lnts versusln(S - 1). If the result is a straight line with a slope of-1,then the operating growth mechanism is either normal or volumediffusion controlled. If the slope is-2 then the operatingmechanism is spiral growth. However, a different representationof the data is needed for a 2D nucleation-mediated growth. Inthis case,

where

Thus, a straight line is obtained whenFs is plotted versus1/(ln S).

As can be seen from eq A13, the analysis of data fromunseeded precipitation does not allow for the calculation of the

nucleation and growth prefactorsKJ andKG separately. On theother hand, from eq A17, it appears that analysis of the datafrom seeded precipitation allows for the full determination ofKG. Thus, a combined analysis of the induction time in seededand unseeded precipitation is necessary to obtain full informationaboutKG andKJ.

Table 1 shows the calculation of the shape factor andinterfacial free energy, Table 2 shows raw calculation of thenumber density of added crystals in the seeded experiments,and Table 3 shows the estimation of the kinetic parameters.

Acknowledgment. The authors thank the United StatesPharmacopeial (USP) Convention for the award of a predoctoralfellowship to A.K., Dr. Andrew D. Vogt (Abbott Laboratories)for help in analyzing the AFM data, and Dr. Ahmad Y. Sheikh(Abbott Laboratories) for helpful discussion.

References

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f(S) ) (S- 1) (A22)

Fu(S) ) ln Au + B

n ln2 S(A23)

Fu(S) ) ln{S1/n[f(S)](n-1)/ntu} (A24)

Fu(S) ) ln Au + (n - 1)B2D

3n ln S+ B

n ln2 S(A25)

Fu(S) ) ln[(S- 1)2(n-1)/3nS(n+2)/3ntu] (A26)

Fs(S) ) ln As +B2D

3 ln S(A27)

Fs(S) ) ln[S1/3(S- 1)2/3ts] (A28)

Table 1. Calculation of the Shape Factor and Interfacial FreeEnergy

area (m2) volume (m3) fs fvγ

(mJ/m2)

length (µm) 353.22× 10-10 1.22× 10-16 26.28 2.86 1.94width (µm) 3.5

thickness (µm) 1length (µm) 35

5.15× 10-10 4.29× 10-16 42 10 2.80width (µm) 3.5thickness (µm) 3.5a

a Since the thickness of the needle crystals could not be accuratelydetermined from microscopic image, a second value which can have amaximum equal to the width of the crystals was assumed.

Table 2. Calculation of Number Density of Added Crystals in theSeeded Experiments

mass of added seeds (g) 2.50× 10-4

density (g/m3) 1.33× 106

volume (m3) 5.00× 10-5

size of seed crystalsa (m) 2.50× 10-6

number density of seed crystals (m-3) 5.74× 1010

a The size (radius) of the seed crystals was estimated from SEM imageof the crystals (data not shown).

Table 3. Estimation of Kinetic Parameters

σ n2D κ (J/m) Ra KG (m/s) G (m/s) KJ(s/m3)

1.02 27.2 1.64× 10-16 1.08× 10-4 3.13× 10-17 6.89× 10-20 1.68× 1027

2.13 10.3 1.64× 10-16 2.24× 10-4 6.51× 10-17 3.12× 10-18 1.68× 1027

a Calculated as the volume fraction of the new phase formed (R ) Vmacro/V, where Vmacro is estimated from the amount of crystals formed at aparticular value of supersaturation (σ) and the density, andV refers to thevolume of the crystallization medium, respectively).


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CG0602212


determining the growth mechanism of tolazamide by induction time measurement

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