Precipitation and Transformation of the Three Polymorphs of d

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Ind. Eng. Chem. Res. 2010, 49, 5854–5862

Precipitation and Transformation of the Three Polymorphs of D-Mannitol Jeroen Cornel, Piran Kidambi, and Marco Mazzotti* Institute of Process Engineering, ETH Zurich, Sonneggstrasse 3, 8092 Zurich, Switzerland

In this work, the thermodynamic and kinetic aspects of D-mannitol’s three polymorphs and their transformations are addressed. Scalable and reproducible syntheses of the solid-state forms are presented, and a mathematical model to describe a three-polymorph transformation is discussed in detail. It is shown that, for certain initial solute concentrations, the R polymorph nucleates first and then transforms through the β form to the γ form. Moreover, a supersaturation threshold to nucleate the R form has been observed, and below this threshold value, the γ form nucleates. A theoretical analysis demonstrates that this behavior can be predicted and that this experimental observation is physically sound. 1. Introduction Polymorphism is crucial in the fine chemical and pharmaceutical industry because different polymorphs have different physical and chemical properties, such as solubility and reactivity.1-3 A solid understanding of the thermodynamic and kinetic mechanisms is essential for the development and control of a crystallization process that involves polymorphs. In previous works, we discussed in detail the kinetic mechanisms governing L-glutamic acid’s polymorph transformation. By elucidation of the secondary nucleation kinetics of the stable polymorph through an extensive set of seeded transformation experiments monitored by in situ Raman spectroscopy, a fully predictive process model could be established.4,5 In other works, the polymorph transformation of carbamazepine’s three solidstate forms has been discussed, but a model of the process was not developed.6,7 In this work, a system involving three solid-state forms, i.e., R, β, and γ D-mannitol, is studied. In the experimental part, polymorph transformation experiments monitored using in situ Raman spectroscopy show that the most unstable R polymorph transforms through the β form to the thermodynamically stable γ polymorph. This happens only beyond a supersaturation threshold because for lower supersaturation levels the γ form nucleates directly and no transformation is observed. In the theoretical part, an analysis is presented to explore whether such behavior is physically consistent and under which conditions it is expected to occur. 2. Experimental Section 2.1. Materials and Syntheses of Polymorphs. D-Mannitol, acetone, and ethanol were purchased from Fluka (Buchs, Switzerland) and used without further purification. Deionized water was obtained from a Milli-Q Advantage A10 water purification system from Millipore (Zug, Switzerland). Numerous recipes to obtain different solid-state forms of 8-13 D-mannitol are reported in the literature. Note also that, unfortunately, the three polymorphs have been called differently in the literature. Recently, with the purpose of reducing the risk of ambiguity, an overview of the nomenclature has been provided.8 However, it was found that most methods were difficult to reproduce and yielded only a few milligrams of the desired polymorph. The syntheses of the R and β polymorphs * To whom correspondence should be addressed. Phone: +41-446322456. Fax: +41-44-6321141. E-mail: marco.mazzotti@ ipe.mavt.ethz.ch.

of D-mannitol described in this work were found to be reproducible and scalable. The R polymorph was obtained through an antisolvent precipitation.8 To this aim, 50 mL of acetone was added to a stirred solution of 9 g of D-mannitol dissolved in 50 mL of water at room temperature. Crystals formed after approximately 30 s and were allowed to grow for 3 min, after which they were filtered and dried at 35 °C at atmospheric pressure. The β polymorph was obtained from melt crystallization.14 Approximately 24 g of γ D-mannitol was heated to 170 °C to form a melt and was then allowed to crystallize through a natural cooling profile to room temperature. The crystals were collected from the beaker and were ground. The γ form is commercially available and was used without further purification. 2.2. Experimental Equipment. All experiments were performed in a 250 mL glass reactor that was temperaturecontrolled, was equipped with a four-blade glass impeller operated at 250 rpm, and allowed for insertion of one in situ immersion probe. In situ Raman spectra were recorded using a RXN1 spectrometer from Kaiser Optics (Ecully, France) equipped with a 400 mW laser diode at 785 nm and a thermoelectrically cooled CCD detector. In situ measurements in crystal suspensions were recorded using a ball-type immersion probe connected using a fiber optic. Ex situ characterization techniques such as microscopy and X-ray diffraction are described in detail elsewhere.15 2.3. Experimental Procedures. The polymorph transformation experiments described in this work can be classified into unseeded and seeded transformation experiments. In an unseeded transformation experiment, a supersaturated solution was created by pouring a hot solution into a cold reactor, whose jacket temperature was gradually increased until the temperature of the solution reached the operating temperature of 10 °C, where it was left to nucleate at constant temperature. It must be emphasized that primary nucleation during cooling was never observed. In a seeded transformation experiment, a wellcharacterized population of dry seed crystals was added to a saturated solution at constant temperature. 3. Process Model The process model for the polymorph transformation of is based on population balance equations (PBEs) for the individual solid-state forms combined with proper constitutive equations for nucleation, growth, and dissolution. D-mannitol

10.1021/ie9019616  2010 American Chemical Society Published on Web 05/24/2010

Ind. Eng. Chem. Res., Vol. 49, No. 12, 2010

The PBE for a perfectly mixed batch crystallizer, assuming size-independent growth and neither agglomeration nor breakage, reads as ∂ni ∂ni )0 + Gi ∂t ∂L

(1)

where L denotes the particle size, t is the time, ni represents the number density of the crystals of form i, i.e., its particle size distribution (PSD), Gi is its crystal growth rate, and i ) R, β, γ in this and the following equations. The solute material balance, which defines the solute concentration c(t), can be written as

∑

dc ) -3 kv,iFiGi dt i)R,β,γ

∫

∞

0

L2ni dL

(2)

where kv, i and Fi denote the volume shape factor and the crystal density of form i, respectively. In the case of D-mannitol’s polymorphs, we have chosen kv ) 0.02, i.e., a typical value for needles, and F ) 1510 kg m-3, which is the value that has been shown to apply to all solid-state forms at 100 K.9 The following initial and boundary conditions apply to the PBEs and to the material balance: ni(0, L) ) ni,0(L)

(3)

Ji ni(t, 0) ) Gi

(4)

c(0) ) c0

(5)

c Si ) ci*

(6)

with ci* being the solubility of form i. The solubilities in water of the R, β, and γ forms at 10 °C equal 205.5, 146.6, and 123.6 g kg-1 of water, respectively. The PBEs were solved using the method of moments.16,17 The jth moment of the PSD of a population of crystals of form i is defined as µi,j(t) )

∫

0

Ljni(t, L) dL

(7)

The time derivative of the 0th moment can be calculated as

{

dµi,0 Dini,t*(Li*), ) J i, dt

if Si e 1 if Si > 1

|∫ | t

ti*

(

ni(t, L) ) ni,0 L -

∫ G (τ) dτ) + r (L) t

0

i

i

(11)

where ri(L) is defined as

{

Ji(ξ) , ri(L) ) Gi(ξ) 0,

if (L - L0) ∈ [0, ui(0, t)]

(12)

otherwise

Di dt

(8)

(9)

In eqs 8 and 9, ti* represents the time where form i starts to dissolve, i.e. ti* ) min{t(Si e 1)}

ui(0, t) )

∫ G (τ) dτ t

0

(13)

i

In eq 12, ξ is the root of the following equation:

∫ G (τ) dτ - (L - L ) ) 0 t

ξ

i

0

(10)

(14)

which is calculated numerically.17,19 The time derivatives for all higher moments are dµi,j ) jGµi,j-1 where j g 1 dt

(15)

Using the second and third moments, the solute mass balance equation (2) can be recast as dµi,3 dc ) -3 kv,iFiGiµi,2 ) kv,iFi dt dt i)R,β,γ i)R,β,γ

∑

∑

(16)

The solid concentration of the crystals of form i can be calculated using the third moment: mi(t) ) kv,iFiµi,3(t)

where Di represents the dissolution rate of form i and the characteristic length Li* can be calculated as Li* )

In the case of a polymorph transformation seeded with crystals of form i, ti* equals 0 and ni, ti* represents ni, 0, i.e., the PSD of the seed crystals; however, in the case of an unseeded polymorph transformation, the R and β forms nucleate and grow initially, and after the supersaturation is consumed, they dissolve. At the point in time where the dissolution of each form starts, i.e., the point where Si e 1 for the first time, the PSD has to be calculated in order to apply eq 8. It must be noted that the R and β forms have different solubilities and hence tR* and tβ* are different, i.e., tR* < tβ*. One way to reconstruct the PSD is through fitting splines (piecewise polynomials) using a finite number of moments; the details of this method can be found elsewhere.16,18 A second and more elegant way to reconstruct a PSD in the case where both nucleation and size-independent crystal growth take place is the following application of the Laplace transform.17 Assuming that the values of the nucleation and growth rates over time are available, e.g., obtained after application of the method of moments, the PSD of form i at a given time t can be calculated as follows:

and ui(0, t) is given by

where c0, Ji, and ni, 0(L) denote the initial solute concentration, the nucleation rate, and the initial PSD for form i, respectively. It is worth noting that in an unseeded process ni,0(L) equals 0. The supersaturation with respect to form i, Si, required to calculate the nucleation and growth rates is defined as

∞

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(17)

and by combining eq 16 with eq 17, one obtains c(t) ) c0 +

∑

i)R,β,γ

mi,0 -

∑

mi(t)

(18)

i)R,β,γ

where mi, 0 denotes the mass of seed crystals of form i, i.e., mi(0). Equations 8 and 15, for j ) 1-3, form a system of 12 coupled ordinary differential equations combined with the mass balance equation (18), which can be solved once the initial values of the four moments of the PSD of the three forms and of the solute concentration are fixed. The ode23 solver in the MAT-

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Figure 1. Optical (a, c, and e) and scanning electron microscopy (b, d, and f) images of the three solid-state forms.

LAB programming environment was found to be robust and fast in solving this problem. The presented mathematical model uses the method of moments and the Laplace transformation technique to calculate efficiently the time evolution of the solid-state and solute concentrations. This approach represents an application of a novel and efficient method to numerically reconstruct PSDs by using the Laplace transform technique that has recently been proposed by others.17 The model is stable and requires a reasonable computational effort (approximately 15 s using a Pentium 4 computer with a 3 GHz processor and 1 GB of RAM). 4. Experimental Results This section is divided into two parts. First, the characterization of D-mannitol’s three polymorphs is discussed. Then, a series of precipitation and transformation experiments are presented. 4.1. Characterization of Polymorphs. Optical and scanning electron microscopy images of the three solid-state forms are

shown in Figure 1, where attention should be paid to the different magnifications. All forms exhibit a needlelike morphology; hence, microscopy cannot be used to identify the polymorphic form, as could be done in other cases.4,5 Parts a and b of Figure 2 show the powder X-ray diffraction patterns and the Raman spectra of the three polymorphs, respectively, which are distinct and in good agreement with data obtained by others.8,11,14 Using these techniques, the different polymorphs can be well discriminated. The solubilities of the three solid-state forms in water as a function of the temperature are shown in Figure 3. The solubility of the most stable γ form (closed circles) has been determined gravimetrically and is in good agreement with the solubility data reported by O’Sullivan and Glennon (open circles).10 To estimate the solubility of the β form, β crystals were added to a solution saturated with respect to the γ form until no further dissolution occurred. The solubility of the R form was estimated gravimetrically by sampling a suspension of R crystals 2-3 min after the onset of nucleation in a precipitation through crash cooling. The presence of exclusively R crystals prior to filtration


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Table 1. Summary of the Unseeded Transformation Experiments and the Precipitated Polymorpha initial supersaturation

solid form

S0,R

S0,β

S0,γ

initial

final

1.68 1.50 1.41 1.36 1.32 1.29 1.28 1.27 1.26 1.20 1.02 0.87

2.35 2.10 1.98 1.91 1.85 1.81 1.79 1.78 1.77 1.68 1.43 1.22

2.79 2.49 2.34 2.26 2.19 2.14 2.13 2.11 2.09 2.00 1.70 1.45

R R R R R R R R γ γ γ γ

γ γ γ γ γ γ γ γ γ γ γ γ

All experiments were performed at 10 °C and repeated at least twice. a

Figure 2. Powder X-ray diffraction patterns (a) and Raman spectra (b) of the three solid-state forms. The intensities have been scaled to enhance the comparison.

Figure 3. Measured solubilities of the three solid-state forms in water as a function of the temperature. The solubility data for the γ form from the literature10 (open circles) and measured in this work (closed circles) are in good agreement.

was verified using in situ Raman spectroscopy. Figure 3 shows that this system of polymorphs is monotropic in the temperature range studied, i.e., 5-25 °C. 4.2. Precipitation and Transformation Experiments. In this section, polymorph transformation experiments monitored using in situ Raman spectroscopy will be presented in detail. To enhance the comparison of different experiments, the measured Raman spectra have been normalized to unit length.

Table 1 summarizes the unseeded experiments that were performed and the nucleated solid-state forms that were observed in each of these experiments. Note that all experiments were carried out at 10 °C and repeated at least twice. 4.2.1. Unseeded Formation of r and Transformation to β and γ. Parts a and b of Figure 4 show the time-resolved in situ Raman spectra recorded during an unseeded polymorph transformation with an initial supersaturation S0,R of 1.27 at 10 °C. During the induction period, no crystals are present and the Raman spectra result from the supersaturated solution. After approximately 8000 s, the R form nucleates, resulting in strong signals at 1144 and 1051 cm-1. Subsequently, a polymorph transformation occurs, and at the end of the experiment, only the thermodynamically stable γ polymorph is present. Figure 4b indicates a peak at 1025-1035 cm-1, which can only be explained by the presence of the β polymorph, as can be seen in Figure 2b. On this basis, we propose that this solventmediated polymorph transformation occurs from the R form through the β form to the γ polymorph. It must be noted, however, that an independent validation through powder X-ray diffraction measurements of samples drawn during the transformation could not be made. Figure 4b indicates that the β form transforms quickly into the γ form and this transformation inevitably takes place also during drying of the samples prior to powder X-ray diffraction analysis. Besides, the amount of the β form present might be too small, i.e., lower than the detection limit of this technique. 4.2.2. Seeded Transformation of β to γ. To confirm the validity of the assumption of the transformation through the three solid-state forms, seeded transformation experiments were performed where the β form was added to a saturated solution at 10 °C. Parts a and b of Figure 5 show the time-resolved Raman spectra during one of these experiments. It can readily be seen that the β form transforms quickly into the γ form, as indicated by the peak shift from 1030 to 1036 cm-1. Moreover, the single peak at 1130 cm-1 evolves into two peaks at 1119 and 1136 cm-1, also indicating that the β form transforms to the γ polymorph. 4.2.3. Unseeded Formation of γ. Parts a and b of Figure 6 show time-resolved Raman spectra when starting with an initial supersaturation S0,R of 1.26, i.e., a value 0.01 lower than those in the case of Figure 4 at 10 °C. It can be observed that the characteristic signals of the R and β forms are below the detection limit. Assuming that this implies that the two forms are indeed absent, this means that the γ form nucleated directly. Numerous repetitions of the experiments shown in

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Figure 4. Unseeded polymorph transformation at 10 °C with an initial supersaturation S0,R equal to 1.27 going through the R, β, and γ forms. Note that part b is a magnification of the range on the right side in part a.

Figure 5. Seeded polymorph transformation at 10 °C, i.e., S0,β equal to 1.0, going through the β and γ forms. Note that both parts a and b are magnifications of the characteristic ranges of the Raman spectra.

Figure 6. Precipitation of the γ form at 10 °C with an initial supersaturation S0,R equal to 1.26. Note that part b is a magnification of the range on the right side in part a.

Figures 4 and 6 were made, resulting always in the same observations. In all unseeded experiments with S0,R g 1.27, the R form always nucleated and then transformed, whereas in all unseeded experiments with S0,R e 1.26, the γ form always nucleated directly, as summarized in Table 1. In the following, this supersaturation value is considered to be the threshold beyond which the metastable R form nucleates first and is detectable with the Raman spectrometer. Underneath this threshold, only the γ form is detected, whereas the other

two forms are not visible. Whether they are present in very small amounts or they are indeed absent cannot be concluded from the measurements reported here. 5. Discussion 5.1. Ostwald’s Rule of Stages. The experiments reported in sections 4.2.1 and 4.2.3 demonstrate that only at high supersaturation the R form nucleates first, as suggested by


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Figure 7. Nucleation rates of each polymorph as a function of the solute concentration. Part d corresponds to the case that was observed in the unseeded transformation experiments using D-mannitol.

Ostwald’s rule of stages, which predicts that the most unstable polymorph nucleates first and then transforms stepwise to the thermodynamically stable polymorph. On the contrary at low supersaturation the stable γ polymorph nucleates directly, which is a violation of Ostwald’s rule of stages. Examples of similar violations have been observed experimentally and reported for different polymorphic systems.22-24 Cardew and Davey20,21 discussed this issue in the case of compounds exhibiting two polymorphs and demonstrated that compliance with or violation of the rule of stages depends on the nucleation kinetics, more specifically, on the relative magnitude of the parameters in the nucleation rate expressions. Following their analysis, the primary nucleation rate of the ith polymorph is expressed as:

(

Ji ) ai exp -

bi (µ - µsi )2

)

(19)

where µ ) µ(c) is the chemical potential of the solute in solution at the given concentration c (or at the given supersaturation) and µsi is the chemical potential of the ith polymorph’s crystals, which is a given value at the chosen temperature. The parameters ai and bi are all positive constants. Cardew and Davey gave the precise conditions on the nucleation rate parameters of the two polymorphs 1 and 2 (2 being the stable one) that lead to the nucleation of one before the other. In the Appendix of this manuscript, it is demonstrated that there are three possible cases depending on the parameter values (the quantity b* is defined in the Appendix by eq A-7):

I. a1 > a2 and either b1 g b2 or b1 < b2. The less stable polymorph nucleates faster than the more stable polymorph only beyond a certain supersaturation threshold. II. a1 e a2 and b1 > b2b*. The more stable polymorph always nucleates faster than the less stable one. III. a1 e a2 and b1 < b2b*. The less stable polymorph nucleates faster than the more stable one in an intermediate range of supersaturation values, whereas the opposite holds for both low and very large supersaturation values. This shows that Ostwald’s rule of stages applies, if at all, only in certain ranges of supersaturation values. However, there are combinations of parameters for which it never applies. A system with three polymorphs such as that of D-mannitol can be viewed as consisting of three pairs of polymorphs, i.e., R-β, R-γ, and β-γ, to each of which the three cases above can apply. When considering the relative magnitude of the nucleation rate parameters, there are six permutations of the sequence of the ai values, i.e., aR, aβ, aγ, and six permutations of the sequence of the bi values; this leads to 36 possible combinations, some of which are, however, mutually exclusive. Each of the feasible combinations implies that one or the other polymorph nucleates first in certain supersaturation ranges. While a comprehensive analysis is beyond the scope of this paper, it is however worth considering the few specific examples illustrated in Figure 7. Figure 7 shows the nucleation rates of three solid-state forms as a function of the chemical potential of the solute for different sets of nucleation parameters. Figure 7a shows the case where aR < aβ, aR < aγ, and aβ < aγ and bi are the same for all forms;

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Table 2. Kinetic Phenomena and Their Expressions Used To Yield the Simulation Results Shown in Figure 8a mechanism

rate

R nucleation R growth R dissolution β nucleation β growth β dissolution γ nucleation γ growth

JR ) 1 × 10 exp(-3/ln SR) GR ) 2 × 10-6(SR - 1)1.2 DR ) 1 × 10-4(SR - 1) Jβ ) 1 × 108 exp(-5/ln2 Sβ) Gβ ) 2 × 10-6(Sβ - 1)1.2 Dβ ) 1 × 10-4(Sβ - 1) Jγ ) 1 × 105 exp(-3/ln2 Sγ) Gγ ) 2 × 10-6(Sγ - 1)1.2 12

units 2

m-3 s-1 m s-1 m s-1 m-3 s-1 m s-1 m s-1 m-3 s-1 m s-1

a The solubilities of the R, β, and γ forms were set to 200, 150, and 120 g kg-1 of solvent, respectively. The crystal densities and the volume shape factors for all solid-state forms equaled 1510 kg m-3 and 0.02, respectively.

i.e., for the three different pairs, case II applies and the γ form has the highest nucleation rate at all concentration levels. For the same ai values as those in Figure 7a, Figure 7b illustrates the case where bR ) bβ , bγ; i.e., for the pairs R-β and R-γ, case II applies, whereas for the pair β-γ, case III applies. Indeed, it can be observed that for certain concentrations a metastable form, in this case β, has the highest nucleation rate. Parts c and d of Figure 7 show examples where aR > aβ, aR > aγ, and aβ > aγ; on the basis of the analysis reported in the Appendix, there is in both cases a value of the solute concentration, i.e., c′, above which Ostwald’s rule applies; i.e., the R form nucleates first and then transforms stepwise to the β form and to the γ form. The case in Figure 7c is interesting because all three forms have a supersaturation where they nucleate first. In Figure 7d, only forms R and γ can nucleate directly; this is a combination of parameters that is consistent with the trend observed in the experiments with D-mannitol. As a matter of fact, considering the three cases above and in order to be consistent with the experimental behavior, one concludes that the nucleation rate parameters must be such that the pairs R-β and R-γ belong to case I, whereas the pair β-γ can belong to any of the three cases. This demonstrates that in order to estimate the model parameters for the D-mannitol system the reported experiments are not sufficient, and more ad hoc measurements would be required. It is, however, possible to select a set of parameters that satisfy the requirements above and to use the process model discussed in section 3 to simulate the outcome of a polymorph transformation experiment. This can be done in order to verify that the experimental observations can be reproduced at least qualitatively. The kinetic expressions and their selected parameters are reported in Table 2 and correspond to the behavior illustrated in Figure 7d. It must be emphasized that these parameter values were not estimated based on timeresolved Raman data but were selected such that the

calculated behavior agreed qualitatively with the experiments; i.e., only forms R and γ can nucleate directly, and a certain supersaturation threshold has to be overcome to nucleate the R form. It must also be noted that the growth and dissolution rates for the three polymorphs were assumed to be the same and secondary nucleation mechanisms were not taken into account. Figure 8 shows three simulations, for three different initial solute concentrations, selected in order to be larger or smaller than the two critical values c′ and c′′ highlighted in Figure 7d but always larger than the solubility of the least stable R form. Figure 8a shows that for a low initial supersaturation, i.e., S0,R < c′′, the γ form nucleates, whereas the solid concentrations of the R and β forms are negligible. For an intermediate initial supersaturation, i.e., c′′ < S0,R < c′, Figure 8b shows that the R form nucleates first and then transforms to the γ form; throughout this transformation, the β polymorph is formed but remains present at low concentration. Finally, Figure 8c shows that for even higher supersaturations, i.e., S0,R > c′, the R form nucleates and transforms through the β form to the γ form. The experimental observations described above indicate that in the case of D-mannitol c′′ equals 1.26cR*. 5.2. Concluding Remarks. On the basis of the theoretical analysis presented above, it can be concluded that the experimental behavior of D-mannitol, namely, the existence of a supersaturation threshold, below which the most stable form nucleates and above which the least stable form nucleates, is physically sound. This is another violation of Ostwald’s rule of stages, which is observed in the case of a system with three polymorphs, as was earlier observed for other systems with two polymorphs. Our analysis, which is the extension of an earlier one to three polymorphs,20,21 demonstrates also that the behavior of such three-polymorph systems can be very different depending on their nucleation rate kinetics. The population balance model described in section 3 can be used to simulate and predict the evolution of polymorph transformation experiments. In the case of D-mannitol, the simulation results agreed, at least qualitatively, with the experimental observations. For such a complex system, a complete and accurate kinetic model, as presented earlier,5 would be very difficult to develop in practice and impossible using only the precipitation and solid-phase transformation experiments presented here. One would need detailed kinetic information, i.e., kinetic expressions and related parameters, about nucleation, growth, and dissolution rates, obtained through independent experiments in which these phenomena are decoupled. It is also worth noting that the experimental observation of a specific polymorph depends not only on its nucleation rate but also on its growth rate relative to the others.20,21 In

Figure 8. Simulated solute and solid concentrations for different initial solute concentrations obtained using the process model discussed in section 3 and the kinetic expressions given in Table 2.


other words, a certain polymorph might precipitate early but grow slowly and soon transform and never reach a concentration or a particle size that makes it visible through the in situ monitoring tool used. Besides, the observation of a certain polymorph inherently depends on the in situ monitoring technique and its sensitivity. Appendix This section studies Ostwald’s rule of stages from theoretical and quantitative points of view. The goal is to derive precisely the conditions on the nucleation rate parameters of two polymorphs of the same chemical species, which lead to the nucleation of one before the other, as reported earlier by Cardew and Davey.20,21 Let us consider polymorphs 1 and 2, where polymorph 2 is the more stable form, having lower solubility. In general, the primary nucleation rate of both polymorphs at a given, constant temperature can be written as

(

Ji ) ai exp -

bi (µ - µsi )2

)

where i ) 1, 2

(A-1)

where µ ) µ(c) is the chemical potential of the solute in solution at the given concentration c (or at the given supersaturation) and µis is the chemical potential of the ith polymorph’s crystals, which is a given value at the chosen temperature. These fulfill the obvious constraints µ > µs1 > µs2; the (positive) difference between the chemical potential in solution and that of the crystals is the driving force for crystal formation and growth. The parameters ai and bi are all positive constants. The objective is to find the constraints on the parameters ai and bi that lead to J1 > J2, i.e., the condition corresponding to the fulfillment of Ostwald’s rule of stages, which states that the less stable polymorph nucleates first. Using eq A-1, this corresponds to ln J1 ) ln a1 -

b1 (µ -

µs1)2

> ln a2 -

b2 (µ - µs2)2

) ln J2

(A-2) By defining m ) µ - µs2, which is a function of supersaturation and the parameters γ ) ln(a1/a2) g or < 0 and σ ) µs1 - µs2 > 0, eq A-2 can be recast as

Figure A-1. f(m) defined by eq A-3 in the cases γ > 0 (a) and γ e 0 (b).

f(m) ) γ +

b2 m

-

2

b1 (m - σ)2

>0

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(A-3)

Note that this inequality makes sense only as long as m g σ, i.e., only if the solution is supersaturated with respect to both polymorphs. The function f(m) has a vertical asymptote at m ) σ, where f f -∞, and it approaches γ as m f +∞. Its derivative is f ′(m) ) -

2b2 3

+

m

2b1 (m - σ)3

(A-4)

which has only one real root at m0 )

σ 1 - b1/3

(A-5)

where b ) b1/b2. This is in the domain of interest if and only if b < 1, i.e., if and only if b1 < b2. In this case, the function f(m) has the maximum f(m0) ) γ +

b2(1 - b1/3)3 σ2

(A-6)

which is larger than zero when

[ ( )]

b < b* ) 1 +

γσ2 b2

1/3 3

(A-7)

Note that the right-hand side of the last equation, b*, is larger than, equal to, or smaller than 1 if γ is positive, null, or negative, respectively; b* can be written as (1 - a/c)3, when a ) σ/b21/2 and c ) -(a/γ)1/3 are defined as Cardew and Davey did.20,21 Let us consider first the case a1 > a2, i.e., γ > 0 (see Figure A-1a). In this case, f(m) has one root in the domain of interest, beyond which it is positive, and either has a maximum or does not depend on the relative values of b1 and b2. If, on the other hand, a1 e a2, i.e., γ e 0 and b* e 1, then three intervals for b have to be considered (see Figure A-1b). If b < b*, then eq A-7 shows that the function f(m) has two roots, between which it is positive. If b* < b < 1, then f(m) has no roots because its maximum is below zero; if b g 1, it has no roots because it has no maximum in the domain of interest. In both cases, i.e., for b > b*, the function f(m) is always negative for m > σ.

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Thus, in summary, there are three possible situations in terms of the relative values of J1 and J2 as a function of m, i.e., as a function of supersaturation: I. a1 > a2 and either b1 g b2 or b1 < b2. The less stable polymorph nucleates faster than the more stable polymorph only beyond a certain supersaturation threshold. II. a1 e a2 and b1 > b2b*. The more stable polymorph nucleates always faster than the less stable one. III. a1 e a2 and b1 < b2b*. The less stable polymorph nucleates faster than the more stable one in an intermediate range of supersaturation values, whereas the opposite holds for both low and very large supersaturation values. These are exactly the conditions determined by Cardew and Davey, as expected.20,21 Acknowledgment The authors thank Roger Davey (University of Manchester) and Joop ter Horst (Delft University of Technology) for stimulating discussions. Literature Cited (1) Bernstein, J.; Davey, R. J.; Henck, J. O. Concomitant polymorphs. Angew. Chem., Int. Ed. 1999, 38 (23), 3441. (2) Davey, R. J.; Blagden, N.; Potts, G. D.; Docherty, R. Polymorphism in molecular crystals: Stabilization of a metastable form by conformational mimicry. J. Am. Chem. Soc. 1997, 119 (7), 1767. (3) Hilfiker, R. Polymorphism in the Pharmaceutical Industry, 1st ed.; Wiley-VCH: Weinheim, Germany, 2006. (4) Scho¨ll, J.; Bonalumi, D.; Vicum, L.; Mazzotti, M.; Muller, M. In situ monitoring and modeling of the solvent-mediated polymorphic transformation of L-glutamic acid. Cryst. Growth Des. 2006, 6 (4), 881. (5) Cornel, J.; Lindenberg, C.; Mazzotti, M. Experimental Characterization and Population Balance Modeling of the Polymorph Transformation of L-Glutamic Acid. Cryst. Growth Des. 2009, 9 (1), 243. (6) Kogermann, K.; Aaltonen, J.; Strachan, C. J.; Pollanen, K.; Heinamaki, J.; Yliruusi, J.; Rantanen, J. Establishing Quantitative In-Line Analysis of Multiple Solid-State Transformations during Dehydration. J. Pharm. Sci. 2008, 97 (11), 4983. (7) Tian, F.; Zeitler, J. A.; Strachan, C. J.; Saville, D. J.; Gordon, K. C.; Rades, T. Characterizing the conversion kinetics of carbamazepine polymorphs to the dihydrate in aqueous suspension using Raman spectroscopy. J. Pharm. Biomed. Anal. 2006, 40 (2), 271. (8) Burger, A.; Henck, J. O.; Hetz, S.; Rollinger, J. M.; Weissnicht, A. A.; Stottner, H. Energy/temperature diagram and compression behavior of the polymorphs of D-mannitol. J. Pharm. Sci. 2000, 89 (4), 457.

(9) Fronczek, F. R.; Kamel, H. N.; Slattery, M. Three polymorphs (alpha, beta and delta) of D-mannitol at 100 K. Acta Crystallogr., Sect. C: Cryst. Struct. Commun. 2003, 59, O567. (10) O’Sullivan, B.; Glennon, B. Application of in situ FBRM and ATRFTIR to the monitoring of the polymorphic transformation of D-mannitol. Org. Process Res. DeV. 2005, 9 (6), 884. (11) De Beer, T. R. M.; Alleso, M.; Goethals, F.; Coppens, A.; Heyden, Y. V.; De Diego, H. L.; Rantanen, J.; Verpoort, F.; Vervaet, C.; Remon, J. P.; Baeyens, W. R. G. Implementation of a process analytical technology system in a freeze-drying process using Raman Spectroscopy for in-line process monitoring. Anal. Chem. 2007, 79 (21), 7992. (12) Tao, J.; Jones, K. J.; Yu, L. Cross-nucleation between D-Mannitol polymorphs in seeded crystallization. Cryst. Growth Des. 2007, 7 (12), 2410. (13) Xie, Y.; Cao, W. J.; Krishnan, S.; Lin, H.; Cauchon, N. Characterization of mannitol polymorphic forms in lyophilized protein formulations using a multivariate curve resolution (MCR)-based Raman spectroscopic method. Pharm. Res. 2008, 25 (10), 2292. (14) Bruni, G.; Berbenni, V.; Milanese, C.; Girella, A.; Cofrancesco, P.; Bellazzi, G.; Marini, A. Physico-chemical characterization of anhydrous D-mannitol. 15th Int. Conf. Biol. Calorim. 2008, 871. (15) Cornel, J.; Lindenberg, C.; Mazzotti, M. Quantitative application of in situ ATR-FTIR and Raman spectroscopy in crystallization processes. Ind. Eng. Chem. Res. 2008, 47 (14), 4870. (16) Cornel, J.; Mazzotti, M. Calibration-Free Quantitative Application of in Situ Raman Spectroscopy to a Crystallization Process. Anal. Chem. 2008, 80 (23), 9240. (17) Qamar, S.; Warnecke, G.; Elsner, M. P.; Seidel-Morgenstern, A. A Laplace transformation based technique for reconstructing crystal size distributions regarding size independent growth. Chem. Eng. Sci. 2008, 63 (8), 2233. (18) John, V.; Angelov, I.; Oncul, A. A.; Thevenin, D. Techniques for the reconstruction of a distribution from a finite number of its moments. Chem. Eng. Sci. 2007, 62 (11), 2890. (19) Press, W.; Teukolsky, S. A.; Vetterling, W. Numerical recipes in Fortran; Cambridge University Press: Cambridge, U.K., 1992. (20) Cardew, P. T.; Davey, R. J. Kinetic factors in the appearance and transformation of metastable phases. Tailoring Cryst. Growth 1982, 2, 1 (Institute of Chemical Engineers, North West Branch Paper No. 1, Manchester, U.K.). (21) Cardew, P. T.; Davey, R. J. General Discussion. Faraday Discuss. 1993, 95, 160. (22) Young, S. W.; Burke, W. E. Further studies on the hydrates of sodium thiosulphate. J. Am. Chem. Soc. 1906, 28 (3), 315. (23) Roelands, C. P. M.; ter Horst, J. H.; Kramer, H. J. M.; Jansens, P. J. Precipitation mechanism of stable and metastable polymorphs of L-glutamic acid. AIChE J. 2007, 53 (2), 354. (24) Kitamura, M. Strategy for control of crystallization of polymorphs. CrystEngComm 2009, 11 (6), 949.

ReceiVed for reView December 10, 2009 ReVised manuscript receiVed April 24, 2010 Accepted April 26, 2010 IE9019616

Precipitation and Transformation of the Three Polymorphs of d

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