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Reconciliation of Kinetic Data for Organic Crystal Growth in Supercritical-CO2: Exploiting a Molecular Collision Theory Viewpoint, with Implications for Precipitator Process Design Daniel E. Rosner* Department of Chemical and Environmental Engineering, Yale University, New Haven, Connecticut 06520-8286, United States ABSTRACT: Exploiting a molecular “collision-theory” viewpoint, we reformulate and correct for systematic effects of fluid-phase solute diffusion, reported growth-rate data for {001} naphthalene single crystal surfaces under supercritical CO2-conditions at low supersaturations, S. By considering dimensionless incorporation probabilities, ε(S, T, ...; {hkl}) for this prototypical organic crystal, we initiate the process of quantifying the environment-dependence of ε as a rational route to crystal growth rate predictions at molecular volume fractions, ϕ, ranging all the way from ideal vapors (ϕ ≪ 1) to liquidlike densities (ϕ ≃ 0.64). Our rational “ansatz” for predicting growth species collision fluxes enables a two-stage data-reduction process. First we infer “apparent” incorporation probabilities, εapp, ignoring fluid-phase solute diffusion effects. Second, we recover their intrinsic counterparts, using εapp and invoking rational transport estimates for the prevailing crystal size/flow conditions. Treating available data near 318 K at 77−91 bar (Tai, C. Y.; Cheng, C.-S. AIChE J. 1995, 41, 2227−2236) and 150−200 bar (Uchida et al. Cryst. Growth Des. 2004, 4, 937−942), we conclude that ε decreases significantly with (CO2-) pressurewith important mechanistic and anti-solvent precipitation (ASP) process modeling implications.

1. INTRODUCTION, MOTIVATION, AND STRATEGY 1.1. Estimating Growth Rates of Crystalline Active Pharmaceutical Ingredients (API) Under Transcritical Conditions (e.g., GASP, ...), Difficulty of Intercomparing Crystal Growth (CG) Data Obtained in Different Growth Environments. The present study was motivated by the challenge of designing and optimizing compressed gas antisolvent precipitation (GASP) (“gentle” size-reduction) of APIs (see, e.g., Martin and Cocero (2008)3 and Rosner and AriasZugasti (2014)4,5). Following homogeneous nucleation, one of the many participating rate processes is the growth of individual organic solid precipitate particles in the prevailing gasexpanded, now supersaturated (S > 1), solvent. Published kinetic data are often unavailable or scant. Indeed, often growth kinetic data are available only for “similar” organic substances (possible “surrogates”?) under environmental conditions (melt? liquid solvent?, ideal vapor?, S, T, p) rather different from the conditions of current or future industrial interest. How then to rationally estimate the growth rates to be expected under industrial precipitation conditions? This task is further complicated by the present manner in which crystal surface {hkl} growth-rates are reported. Most often one finds partial information about the linear growth rate function: Ġ (S, T, ...;{hkl}) (with units m/s) or rate-parameters derived by forcing Ġ to be equal to, say, KG(T,...)·(S−1)g in the environment actually studied (see, e.g., the review of Mullin (2001)6). As demonstrated below, this type of parametrization makes it quite difficult to intercompare rate data obtained in © 2014 American Chemical Society

rather different CG environments, mainly because rate laws of this simple “phenomenological” form make no explicit reference to the growth species molecular arrival flux on the surface {hkl}. 1.2. Exploiting the Perspective of Molecular Collision Theory for More Appropriate Parameterizations of CG Kinetics; Attributes of Such a Reformulation. Whether one is attempting to predict CG rates in unusual environments or, conversely, cast one’s own CG rate data in the most “universal” (environment-insensitive-) form, we recommend adopting a molecular collision theory perspective which explicitly invokes the growth species impingement flux Ż ″A in the relevant rate law, thereby defining a dimensionless overall incorporation probability, written below as ε(S, T, ...;{hkl}). On this basis (which has deep roots in the earliest published treatments of ideal vapor CG rates;7,8 see, also, Part 3) we propose writing the general CG rate law in the factorable form: ln S Ġ = vm ·ZA″̇ ·ε(S , T , ...; {hkl})· S

(1)

where vm is the volume per molecule in the crystal phase, Ż ″A is the prevailing molecular impingement flux (molec/m2 s), and S the supersaturation ratio prevailing at the fluid side of the fluid/ crystal interface. In effect, eq 1 defines our overall incorporation probability, ε, which, in the limit of small S − 1, is equivalent to Received: February 13, 2014 Revised: May 30, 2014 Published: July 23, 2014 3783

dx.doi.org/10.1021/cg500223q | Cryst. Growth Des. 2014, 14, 3783−3790

Crystal Growth & Design

Article

the overall “condensation coefficients” that were routinely calculated/quoted in the earliest “ideal gas” physical vapor growth (PVG) literature (Volmer7,8). Indeed, motivated by the present study, this writer has initiated a review of earlier CG data in the classical (liquid) environments from this same “collision theory” perspective.9−11 However, in our view this law/parametrization is appropriate and useful for any environment (especially including supercritical solvents and gasexpanded liquids; see below!) provided one can supply a rational but convenient estimate for the relevant molecular impingement flux: Ż A″ in that environment. Considering the above-mentioned limiting cases that are known, as well as computational convenience, we provisionally suggest an interpolation based on molecular volume fraction ϕ, i.e., ZA″̇ ≃

ϕ/ϕ ⎛1 ⎞1 − (ϕ / ϕL) ⎛ 2DAL‐ mix nA ⎞ L ⎜ n c ⎟ ⎟ ⎜ · ⎝ 4 A A̅ ⎠ σ ⎠ ⎝

As a corollary, our present description can be applied even at very high supersaturatons (i.e., even when (S − 1) ≫ 6(1); however, our quantitative treatment of fluid phase solute diffusion limitations (section 2) presumes the absence of appreciable homogeneous nucleation in the immediate vicinity of the growing crystal surface. (For the analogous case of simple liquid condensation, see, e.g., Rosner and Epstein (1968).14) 1.3. Initiation of Systematic Studies of the Environment-Dependence of ε(S, T, p;{hkl}). This study can be viewed as the first step in this apparently long-overdue systematic “program” of investigating and interpreting the environment-dependence of the overall incorporation probability: ε(S, T, p;{hkl}) for the growth of crystals of industrial importance. To fully exploit its potential benefits, this ambitious program would not only include simple polycyclic hydrocarbons in familiar environments9−11 and section 1.4 below but also previously discovered and future APIs in transcritical environments of emerging interest. 1.4. Illustrative Case of {001} Naphthalene CG Kinetics Near 320 K. While the nucleation and crystal growth of more complex organics are of immediate interest to pharmaceutical producers, we exploit below the more readily available CG kinetic data on homologous polycyclic organics such as phenanthrene, anthracene, and naphthaleneoften considered API “surrogates” by process researchers.4,15,16 In this regard, for the purposes of this particular paper we will focus on growth rate data for the {001} surface of naphthalene (C10H8), which has been studied at CO2 pressures up to 200 bar (Uchida et al. (2004)2) at surface temperatures near 320 K and at effective growth species “A”-vapor supersaturations less than 1.1. In this paper we exploit the above-mentioned molecular “collision theory” perspective and critically examine/interrelate recently published CG kinetic data obtained on single crystals of the familiar organic crystal: naphthalene actually carried out under transcritical (GASP-type) conditions in the presence of a convective flow (both imposed and gravity-induced) of CO2 + solute A(g). We also reinterpret some “desublimation” kinetics for polycrystalline naphthalene and its three-ring homologue anthracene under ideal gas conditions. We provide (section 2.2) a tractable methodology to correct available data under supercritical (SC)-CO2 conditions for the intrusive role of solute diffusion limitations and demonstrate that, even after these (presently modest) corrections are made, there will generally be a nontrivial “environment-dependence” of the intrinsic overall incorporation probability ε(S, T, ...;{hkl}) on “carrier vapor” densityhere described by the CO2 pressure, p (section 2.4.2). Possible mechanistic causes of this “residual” dependence of ε on CO2 pressure will be briefly discussed in section 4.2. While an unambiguous “assignment” will probably have to await further experimental data, the practical implications of such a dependence cannot be overemphasized. This methodology/study should be viewed as the first step in gaining a mechanistic understanding of the possible “environment-dependence” of intrinsic overall incorporation probabilitiesquantities rarely, if ever, explicitly mentioned in the recent or current CG and ChE processing literature. However, this approach (illustrated for conventional CG environments in background papers9−11) is expected to simplify the task of predicting API crystal growth rates in SC-CO2 environments of emerging pharmaceutical interest. Applications in other commercially significant precipitation environments are also anticipated.

(2)

where ϕL is the molecular volume fraction in the “liquid-like” state, chosen here to be ca. 0.636 (random close packing of uniform hard spheres of diameter σ), and DLA‑mix is the growth species A Fick-diffusion coefficient in this same limit (known, or, say, estimated via a Stokes−Einstein relation12). Significantly and quite plausibly, Ż A″ remains simply proportional to the prevailing growth species number density, nA, over the entire range of molecular volume fractions: 0 < ϕ < ϕL. As an immediate corollary, eq 2 can be conveniently written in terms of Ż ″A,sat (proportional to nA,sat), provided one simply eliminates the prefactor (1/S) of the term ln S in eq 1. The numerous advantages of this reformulation/statement of CG kinetic data24 may be summarized as follows: A1: ε-values are dimensionless and expected to be bounded (from above) by 1/Z (where Z is the number of growth molecules per unit cell of the crystal structure) A2: The dependence of ε on the local state variables S, T (and possibly p, see section 2.4) are of greater fundamental interest than the corresponding dependencies of Ġ itself. Moreover, these sensitivities will be decreased; i.e., we expect: −9

∂ ln ε ∂ ln Ġ ∂ ln ε < −9 , and ∂(1/T ) ∂(1/T ) ∂ ln(S − 1) ∂ ln Ġ < ∂ ln(S − 1)

(3)

A3: ε-values are least likely to be “environment-sensitive” (compared to alternative parametrizations) A4: the sufficient condition for “interface-control” (i.e., negligibility of fluid-phase solute diffusional limitations) is dramatically simplifiedoften to the dimensionless inequality: ε ≪ 4lA/dp (where l is the growth species mean-free-path25 and the ratio lA/dp may be considered an appropriate Knudsen number) Clearly, attribute A3 will facilitate the prediction of growth rates in less familiar processing environments of emerging industrial interest, including crystal growth from supercritical CO2 (used as our example in sections 2.3, 3.1, and 4). Even in the absence of fluid phase diffusional limitations, no claim is made that ε is an “elementary step” rate “constant” hence the present name: “overall” incorporation probability. We return to this issue in section 3.1 (based on the data of Matsuoka (1982)13) and section 2.4.2, where we briefly address the underlying causes of our finding that ε for {001}-naphthalene at 318 K, S = 1.1 decreases with increasing CO2 pressure. 3784



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which facilitates deriving the following useful approximation28, eq 10, to the relation between apparent (measured) activation energies, Eεapp and the intrinsic value Eε associated with ε itself, utilized in sections 2.3 and 2.4.1 below:

2. REINTERPRETATION OF AVAILABLE CG RATE DATA UNDER SC-CO2 CONDITIONS 2.1. Conversion of Ġ Data to Apparent Incorporation Probabilities, εapp. When the growth species “A” is dilute in a carrier fluid and crystal growth is occurring, the local supersaturation established at the growing {hkl} surface, Sw, may be significantly lower that maintained in the bulk of the “solution”, Sb. This is due to the need for fluid-phase molecular diffusion to supply growth component A to the surface. We develop below a simple two-stage method for correcting for the effects of fluid phase diffusion, making no restriction on either the extent of supersaturation or the effective kinetic “order” of the growth process. Our principal assumption is that the quasi-steady (QS) crystal growth rate can also be expressed in the form: Ġ = vm ·Nu m(Re, Sc, ...)·

DA∞nA,sat dp

· (S b − Sw )

Eε ≃

+

Ġ (observed) ″̇ vmZA,sat ln S b

(4)

(5)

2.2. Conversion of Apparent Incorporation Probabilities to Intrinsic Values; Diffusional “Falsification” of both ε and Activation Energy, E. Making no further assumptions it is then easy to show that ε εapp

=

ln S ln(S − . ln S)

⎫ EμL ⎞⎤ ⎛ E ⎞⎪ ϕ ⎛⎜ ⎥ − ⎜1 + μ ⎟⎬ ⎟ + 1 ϕL ⎜⎝ 9T ⎟⎠⎥⎦ ⎝ 9T ⎠⎪ ⎭

(10)

and, where, in our notation: Eμ ≡ 9 d ln μ/d(1/T) (which may be regarded as the local “activation energy” for carrier “fluidity”12). In many cases, the first termi.e., (ε/εapp) Eεappdominates. The relation between Eεapp and EG is provided in section 2.4.1 below. 2.3. Treatment of Naphthalene {001} CG Rate Data in SC-CO2. 2.3.1. Data of Tai and Cheng (1995)1 near 84 bar. These authors reported growth rate data for {001} naphthalene at ca. 318 K employing a crystal size of ca. 1.5 mm at an imposed CO2-rich vapor velocity of only ca. 0.24 mm/s. Under these p, T conditions, we estimated a molecular volume fraction of ca. 0.44 corresponding to ϕ/ϕL = 0.69 and a collision flux reduction factor (compared to the formal IG-limit value) of ca. 0.78. Even though this crystal face grew much slower than the {110} face, and buoyancy-induced convection must have been helpful, using the equations above we estimate that these CG rate data probably require a ca. 2.2-fold correction (to the formally calculated εapp near 4 ppm) because of an estimated value of . near 0.55. This leads to the estimate of ca. 8.4 ppm for the overall incorporation probability (per naphthalene molecule impact), as plotted in Figure 1a value significantly

which (in the dilute solute limit) is linear in the Fickian diffusion “driving force” (Sb − Sw). Here dp is the characteristic effective (area-equivalent) diameter of the seed crystal,26 and Num(Re, Sc, ...) is the dimensionless local mass transfer coefficient (Nusselt number for mass transfer; Rosner12) at the prevailing Reynolds27 and Schmidt numbers. Our two-stage data treatment starts with defining and calculating the “apparent” value of ε, written εapp, calculated as though the feedstream (“bulk”) supersaturation Sb actually prevailed at the fluid/solid surfacei.e.: εapp ≡

⎧ ⎡ ⎪ ϕ⎞ . ⎢1⎛ ⎜⎜1 − ⎟⎟ + 9T ⎨ ⎪1 − . ⎢ 2 ϕL ⎠ 1−. ⎣ ⎝ ⎩ Eεapp

(6)

where the estimable dimensionless parameter . is a characteristic molecular flux ratio, bounded by unity: .≡

″̇ εapp·ZA,sat Nu m(Re, Sc, ...)·DA∞‐ mix nA,sat /d p

(7)

and we have dropped the (now implicit) subscript b. Note that ̇ ″/nA,sat, eq 6, along with rational estimates of Num, D∞ A‑mix, and ZA enable calculation of the “intrinsic” value of the overall incorporation probability ε pertaining to the (reduced) supersaturation: S−. ln S. However, if the growth rate “order”, n, is defined by n≡

∂ ln Ġ ∂ ln(S − 1)

T , p = const

(8)

Figure 1. Summary of available data on the growth kinetics of {001} naphthalene in SC-CO2 near 318 K at pressures between 77 and 200 bar; linear growth rate data recast as intrinsic overall incorporation probabilities, ε(1.1, 318 K, p(bar);{001}) (see eqs 1 and 2).

then ε will be S-independent for n = 1, a property mentioned in attribute A2 above and exploited in section 2.3 below (when we treat available CG rate data for the {001} surface of naphthalene growing from a dilute solution of naphthalene vapor in SC-CO2). In the important limit S−1 ≪ 1 eq 6 simplifies to ε = (1 − .)−1 εapp (9)

higher than ε(1.1, 318 K, pCO2,{001}) estimates made below for the higher CO2-pressure data of Uchida et al. (2004).2 Tai and Cheng (1995)1 also reported data for the {001} surface of naphthalene at 318 K over the pressure range from 3785



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Under supercritical temperature conditions, we can make use of the Joshi-Thodos corresponding states correlation (see, e.g., Poling et al. (2001)18) to estimate the prevailing value of Eμ, in which case we find the simple result:

76.9 to 90.7 bar bracketing 83.8 bar. As discussed briefly in section 2.4.2 below, these data also point to a reduction in εvalue at higher CO2 pressures. 2.3.2. Data of Uchida et al. (2004)2 (150−200 bar). Uchida et al. (2004)2 reported growth rate data for {001} naphthalene in SC-CO2 in the range of pressures between 150 and 200 bar, and at temperatures between ca. 308 and 318 K, using stationary seed crystals of ca. 5 mm diameter at imposed fluid velocities on the order of 1 cm/s. Because of the lower intrinsic incorporation probability at 200 bar (estimated below to be only ca. 0.5 ppm), naphthalene vapor diffusion corrections were found to be noticeably more important (again ca. 2-fold) for the lower pressure (150 bar) data, which led to an ε-value (plotted in Figure 1) of ca. 2 ppm. For the 200 bar, 308 K condition, the diffusion correction to εapp was only ca. 11 pct. At this higher pressure condition, we estimated that ϕ/ϕL was ca. 0.89, corresponding to a reduction in the IG collision flux of naphthalene vapor of ca. 46 pct. While details of the crystal/cell geometry and the (turbulence-generating?) role of a mentioned magnetic stirrer introduce inevitable uncertainties, we believe that the temperature and pressure trends inferred here (and in section 2.4.1) for the overall incorporation probabilities ε provide a much clearer picture of the role of supercritical CO2 environments than achievable by limiting oneʼs considerations to directly observable linear growth rates, Ġ (which ranged from 4.09 to 13.9 nm/s in these particular experiments). 2.4. Dependence of ε(intrinsic) on Temperature and Pressure at const S. 2.4.1. Activation Energy Associated with ε(S, T, ...;{hkl}). Fundamentally, the temperature (and pressure) dependence of ε(intrinsic) are of greater interest than the experimentally observable corresponding dependencies of the dimensional growth rate, Ġ , itself. Having presumably extracted ε(intrinsic) from Ġ data by “correcting for growth species diffusion limitations” in the fluid phase (section 2.2), we are now in a position to examine the value of the activation energy Eε defined by Eε ≡ −9

∂ ln ε(intrinsic) ∂1/T

EμL 9T

∂ ln yA,sat ∂ ln T

⎤ ⎛ ϕ 8DL ⎞⎟⎥ − βTT ⎜⎜1 + ln ϕL c ̅σCO2 ⎟⎠⎥⎦ ⎝

(11)

L

quantity

(units)

value

molecular weight heat of sublimation @Ttr triple point temperature dimensionless entropy change upon sublim. molecules per unit cell in crystal unit cell volume molecular volume molecular diameter

kg/kmol MJ/kmol K

128.2 72.6 353.4 24.7 2 0.360 0.186 0.61

(nm)3 (nm)3 nm

For CO2 we adopted: σCO2 = 0.45 nm, ρc = 467.6 kg/m3 (Anwar and Carroll (2011)23), and ρ/ρc = 2.08 when ϕ = ϕL = 0.636.

a

2.4.2. Pressure-Dependence Associated with ε(S, T, ...; {hkl}). After fluid-phase solute diffusion effects (section 2.2) were corrected for, it is of interest to inquire whether the overall incorporation probability ε(S, T, ...;{hkl}) exhibits any pressure dependence, to study the possible causes of any such dependence and to consider the practical implications. The available data for naphthalene crystal growth rates are such that we can only initiate this effort here by comparing available results near 318 K at carrier gas pressures of near 77−91 bar (Tai and Cheng (1995)1) and at 150−200 bar (Uchida et al. (2004)2). Our ansatz for calculating the growth species collision flux at saturation (eq 2) enables arriving at the following useful relation between the frequently reported exponent: ∂ ln Ġ /∂ ln p and the intermediate quantity: ∂ ln εapp/∂ ln p, holding S and T constant. Chain rule differentiation then provides the interrelation:

(12)

where, in our present notation: ⎛ ∂ ln μ ⎞ ⎟ EμL = 9⎜ ⎝ ∂1/T ⎠ϕ = ϕ

(14)

Table 1. Selected Properties of Naphthalene (C10H8)a

⎡ ⎛ EμL ⎞ ϕ⎞ ϕ ⎛⎜ 1 ⎟ ⎟⎟ + ≃ EĠ − 9T ⎢ ⎜⎜1 − + 1 ⎢⎣ 2 ⎝ ϕL ⎠ ϕL ⎜⎝ 9T ⎟⎠ +

⎛ μ0 μ0 ⎞ ⎟⎟β T + 3.05⎜⎜1 − μL μL ⎠ T ⎝

where ω is the exponent appearing in the IG limit T-dependence of μ0(T)close to 0.92 for CO2 near 320 K, βT is the thermal expansion coefficient (units K−1)and μL/μ0 is the prevailing p-induced increase in CO2 viscosity (typically less than a factor of 6.6 under our present conditions). The indicated “L” state for supercritical CO2 is considered to be that corresponding to ϕ = 0.636; i.e., when ρr = ρ/ρc = Vc/V = 2.08 at the prevailing temperature. Keeping in mind that 9T is only about 2.6 MJ/kmol at 308 K and evaluating ϕ/ϕL and μL/μ0 under the 200 bar conditions of Uchida et al. (2004),2 we find from eq 12 that E(0) would be smaller than EĠ by ca. 13.2 MJ/kmolnot ε insignificant compared to the reported apparent EĠ value of ca. 70.9 MJ/kmol. But we estimated (section 2.3.2) that if fluid phase solute diffusion effects were eliminated EĠ would have been higher than 70.9 MJ/kmol. On this basis, we conclude that the Uchida et al. data imply that Eε for the {001} surface of naphthalene may be as high as ca. 74 MJ/kmol under these conditionsnot very different from the heat of sublimation of naphthalene (Table 1).

Using our current estimate for the collision-flux under supercritical conditions (eq 2), there is necessarily a systematic difference between Eε and the frequently reported EĠ . For example, if conditions are such that εapp and ε are nearly the same (i.e., if . ≪ 1) then, invoking the definition eq 11 at constant S we find Eε(0)

≃ω

(13)

which will be recognized as the prevailing value of the activation energy for fluidity of CO2 when its molecular volume fraction is “liquid-like”. If the relative flux parameter . is not negligible then eq 12 in effect provides Eεapp and eq 10 above will then provide Eε, making use of the calculated value of . .

∂ ln εapp ∂ ln p

=

⎛ ∂ ln yA,sat ϕ 8DL ⎞⎟ ∂ ln Ġ ln − κ pp⎜⎜1 + − ⎟ c ̅σCO2 ⎠ ∂ ln p ∂ ln p ϕL ⎝ (15)

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where κpp = −∂ ln V/∂ ln p is near 0.24 for the 150 bar condition of Uchida et al. (2004)2 and 2.9 for the conditions of Tai and Cheng (1995).1 In the latter case, it was reported that even though the pressure level was changed by ca. 18 pct (i.e., from 76.9 to 90.7 bar) the value of Ġ /[ωN/(1−ωN)] remained about the same at constant S and T. However, we conclude from eq 15 that this itself implies that ∂ ln εapp/∂ ln p should be about −κpp[1 + (ϕ/ϕL) ln(8DL/ccσ ̅ CO2)] or ca. −1.3 under these particular conditions. After correcting for the effects of diffusion, this implies that the sensitivity ∂ ln ε/∂ ln p will be larger by the factor of ca. (1 − .)−1, putting the estimated local logarithmic slope in the vicinity of ca. −2.5. More direct evidence that the overall incorporation probability decreases appreciably with increasing CO2 pressure comes by simply comparing the inferred values of ε(1.1, 318 K, p;{001}) from the above-mentioned (section 2.3.2) studies carried out at 150 and 200 baras plotted in Figure 1 and discussed further in section 4 below. The mechanistic origins of such a CO2 pressure dependence are briefly considered in section 4.2; however, in view of the sparsity of the presently available data, an unambiguous assignment cannot be made at this time. Nevertheless, the practical importance of this CO2 carrier vapor pressure dependence is now quite evident. 3. Availability of Relevant Data on the PVG of Naphthalene and Anthracene under Ideal-Gas Conditions? Normally, “physical” organic crystal growth rate data from ideal gas environments are relatively abundant and among the best characterized/documented (see Myerson et al. (1996)19). However, the only relevant kinetic data we have found for solid naphthalene growth under these conditions (Matsuoka (1982)13 actually pertains to the growth of polycrystalline films in the presence of the background gas N2. We also consider below the possible relevance of the early PVG kinetic data of Sloan (1967)20 for the growth of polycrystalline anthracenethe next higher (three-ring) “polycene”. Unfortunately, as indicated in sections 3.1 and 3.2 below, neither of these particular data sets enables an unambiguous assignment of the above-mentioned value of ε0(1.1, 318 K, “0”, {001})which seems most likely to be at least 6(100 ppm). 3.1. Reinterpretation of the CG Kinetic Data of Matsuoka (1982)13 on Polycrystalline Naphthalene. To study polycrystalline naphthalene crystal growth rates under ideal gas mixture conditions, Matsuoka (1982)13 employed an axisymmetric jet of N2 carrier gas dilute in the growth species directed normal to a disc target cooled from behind. Naphthalene vapor diffusion limitations were not completely eliminated in these experiments, but, with the help of supplementary sublimation rate experiments, his data-reduction procedure evidently corrected for this by calculating the S-values prevailing one mean-free path from the growth surface. Using the summary of this earlier polycrystalline data appearing in Uchida et al. (2004),2 we formally estimated that ε0(1.1, 318 K, “0”, {poly}) would only be ca. 3 ppm, with Tables 3 and 4 of ref 2 reporting second-order behavior (in conflict with subsequent {001} data in SC-CO2) and EĠ = 90.9 MJ/kmol, respectively. But this ε0 value is ca. 1.5 decades below the above-mentioned “expected low-(CO2-)pressure asymptote” of the available SC-CO2 data for {001} naphthalene. While a nonmonotonic transition to this asymptote cannot be ruled out, a monotonic increase to the (presently unknown) ε0(1.1, 318 K, “0″, {001}) value appears more likely (see Figure 1). 3.2. Relevance of PVG Rate Data of Sloan (1967)20 for Polycrystalline Anthracene? Another possibly relevant

source of PVG kinetic data under ideal gas conditions is that for polycrystalline anthracene (C14H10), as reported in Sloan (1967).20 When these high temperature (393 K) growth-rate data for the three-ring homologue of naphthalene are converted to an overall incorporation probability in accord with eq 1, one finds (Rosner (2013)10 the impressive value 0.08. What ε would be at a much lower surface temperature (e.g., 318 K) is difficult to say, but it almost certainly would far exceed the above-mentioned ε 0 ≈ 6(100 ppm) asymptote anticipated for naphthalene {001} in the limit pCO2↓0. Indeed, for ϕ ≪ 1, and based on Matsuokaʼs early estimate of EĠ = 1.36ΔHsublim, one might expect: Eε = 0.36ΔHsublim +

1 9T 2

(16)

so Eε = 34 MJ/kmol (for anthracene near 393 K). This would imply a value of ε0 at 318 K of ca. 0.7 pctwhich remains surprisingly high.

4. IMPLICATIONS FOR THE PRECIPITATION OF ORGANIC CRYSTALS OF PHARMACEUTICAL INTEREST IN TRANSCRITICAL ENVIRONMENTS; GENERALIZATIONS 4.1. Temperature Dependence of ε (intrinsic). While the temperature changes associated with many GASP processes using CO2 as antisolvent are often modest (see, e.g., Martin and Cocero (2008)3), the temperature sensitivity of the intrinsic incorporation probability is likely to be high enough to be significant under conditions in which ε is many orders of magnitude smaller than 1/Z. In the case of the best-studied polycyclic model hydrocarbon compound: naphthalene, Eε, is evidently close to the heat of sublimation (73 MJ/kmol), implying that near 320 K a change of only some 10 K in surface temperature can bring about a change in incorporation probability ε of more than a factor of 2. The procedures outlined in sections 2 and 3 will enable such effects to be included in precipitation process simulations, thereby contributing to the reliability and predictive power of GASP-process models. 4.2. Carrier-Fluid Pressure Dependence of ε (intrinsic); Mechanistic Origins? We have assembled/reinterpreted the limited amount of available experimental data and conclude that it suggests that the intrinsic overall incorporation probability for {001} naphthalene CG near 320 K falls off with an increase in CO2 pressure (see Figure 1). Considering the “elementary steps” underlying crystal growth (cf. Burton et al. (1951)21)i.e., adsorption, surface-diffusion, and molecular incorporation at kink sitesthis behavior could be the result of one or more of the following impediments: (a) CO2 clustering around the growth species in the fluid phase, reducing its adsorption probability (b) competitive adsorption, with CO2 blocking terrace sites that would have been available to incident growth molecules (see, e.g., Ghez (1974)16), as well as modifying surface energies that help determine in situ surface morphology and N-admolecule desorption (c) reduction in the surface diffusivity of the growth species due to a and/or b above (d) reduction in the kink site incorporation rate constant associated with CO2 “solvation baggage” Identifying the dominant cause(s) in this particular case, and understanding the relative importance of these possibilities in 3787



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200 bar, and Tai and Cheng (1995)1 @ ca. 84 bar for modest anticipated solute diffusion limitations (i.e., calculating ε via εapp and . ), we find that ε(1.1, 318 K, p(CO2);{001}) betrays a systematic decrease with increased CO2 pressure (see Figure 1). While relevant ideal gas data for ε0 for the {001} surface of naphthalene are evidently still not available (!), this inferred (CO2-) pressure trend for ε would be quite consistent with, say, competitive CO2 adsorption on the {001} surface impeding either surface diffusion (of C10H8) and/or, perhaps, the incorporation step itself. However, even before this level of mechanistic understanding is achieved, this overall CO2 pressure effect on ε(S, T, pCO2,eff;{hkl}) can/should now be built into more realistic predictions/simulations of crystal growth for, say, semibatch GASP processes in which the pressure varies over a wide range (ca. 60-fold)as in the phenanthrene (C14H10) precipitation experiments of Muhrer et al. (2002).22 The molecular collision theory viewpoint and procedures we have developed/suggested/illustrated in this paper for interpreting such {hkl}-surface-specific linear growth rate data are rather general and should now enable: (a) more reliable estimates of API crystal growth rates under transcritical environments (GASP) of current/future pharmaceutical interest, as well as (b) a more mechanistic understanding of how carrier vapor pressure (here CO2) can influence the intrinsic growth kinetics of organic crystals. Reinstating and generalizing this collision-theory-based approach (as also illustrated in the authorʼs preprints I, II, and III (loc. cit.) for melt growth, PVG, and solution growth, respectively), together with our convenient ansatz for estimating Ż ″A for environments with any molecular volume fraction 0 < ϕ < ϕL, eq 2, should also facilitate more reasonable “parameterizations” of crystal growth rate laws applicable to emerging gas-expanded liquid environments and more meaningful judgments about mechanistic similarities (or differences) based on inferred ε(S, T, p;{hkl}) values. It is hoped that this study, and its ultimate companions (Parts I, II, III),7−11 will encourage/facilitate this alternative description of previous and future experimental crystal growth data, ultimately leading to improved performance predictions for crystallizers operating under nontraditional conditions of emerging importance in a number of chemical processing industries. We anticipate that future improvements can and will be made in our present procedurese.g., the evaluation of growth species collision flux at intermediate molecular volume fractions, and the evaluation of Fick diffusion coefficients under “expanded-liquid” fluid density conditions. However, the tools/approximations already suggested here seem adequate to advance this important yet nontraditional branch of crystal growth science, to the immediate benefit of ChE process designers/modelers.

general, are worthwhile research goals. But, even before this level of understanding is achieved, this presently revealed overall effect on ε(S, T, pCO2,eff;{hkl}) should definitely be built into predictions/simulations of crystal growth, especially for semibatch GASP processes in which the pressure varies over a wide rangeas in the case of Muhrer et al. (2002),22 in which the pressure increased by nearly 60-fold at nearly constant temperature. 4.3. Diffusional Falsification of CG Kinetics; e.g., {110} Naphthalene at 84 bar. Growth rates of the {110} face of naphthalene have also been reported by Tai and Cheng (1995),1 with their raw Ġ data suggesting much higher incorporation probabilities which are not S-independent. Indeed, we estimate that for this crystal face, which grew about one decade faster than the {001} surface when S = 1.1, diffusional corrections to εapp(S, T, p;{110}) are likely to be excessive. This despite the fact that natural convection effects probably dominated those associated with the (small) imposed CO2 velocity in their high pressure celli.e., Ug ≫ U (see, also, section 2.1). While geometric complexities preclude an accurate evaluation of the relevant dimensionless mass transfer coefficient Num, even εapp(1.1, 318 K, 84 bar;{110}) appears to be ca. 40 ppm, suggesting that ε itself could very well exceed 100 ppm.

5. CONCLUSIONS AND RECOMMENDATIONS From our present perspective, earlier attempts to interrelate/ understand organic crystal growth kinetics data obtained over a wide range of environmental conditions, including growth from supercritical CO2 from ca. 77−200 bar, have been hampered by a reliance on prescribed “phenomenological” rate laws/ parameters that do not explicitly take into account the prevailing molecular collision fluxes. However, once one: (a) restates the growth law in terms of overall incorporation probabilities: ε(S, T, p;{hkl}), making use of eq 1 and our proposed approach to calculating molecular arrival flux (in environments with an arbitrary molecular volume fraction ϕ) via eq 2 and (b) makes systematic corrections for the intervention of solute diffusion limitations (especially for data obtained on large single crystal “seeds”), it becomes possible to perceive systematic trends likely to be of mechanistic significance. An example of this is the reduction in incorporation probability for {001} naphthalene as the carrier vapor pressure is increased (Figure 1). Accordingly, our present approach promises to improve the prediction of many evolving precipitation processes, including gas antisolvent precipitation (ASP) process performance for the “micronization” of sparingly soluble pharmaceuticals (to increase their bioavailability). Regarding temperature sensitivity, it is interesting that the activation energy associated with the overall incorporation probability ε suggested/estimated here will inevitably be smaller than that associated with observed growth rates, Ġ , themselves. However, this difference will be smaller under SC-CO2 conditions than under ideal gas conditions. This is because the saturation number density of the growth species, nA,sat(T), and, hence, the molecular arrival flux Ż ″A,sat(T), is much less sensitive to T under SC-CO2 conditions than in the ideal vapor (ϕ ≪ 1) limit. As a timely prototypical example, we have focused here on the reinterpretation of available data for the growth rate of {001} naphthalene in dense CO2 environments near 320 K, S = 1.1. After correcting the data of Uchida et al. (2004)2 @150 and

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This study was supported by Industrial Affiliates of the Sol Reaction Engineering (SRE) Research Group at Yale, Yale ChE graduate alumni, and the Yale School of Engineering and Applied Science. It is a pleasure to acknowledge the helpful comments and assistance of Dr. Manuel Arias-Zugasti in preparing this paper for publication. An overview of this 3788



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σ

program was presented at Brookhaven National Laboratories on July 30, 2013 and was presented as Paper 300f of the AIChE Annual Mtg, November 5, 2013, in San Francisco, CA.

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Subscripts and superscripts

app b

apparent pertaining to bulk (mainstream) fluid entering CG cell c pertaining to thermodynamic critical point, or carrier fluid diff pertaining to vapor phase diffusion control eff effective interface pertaining to interface kinetic control L pertaining to the liquid state (T ≪ Tc) m pertaining to the crystallizing molecule mix pertaining to (carrier-species dominated-) mixture r “reduced” (ratiod to value at the critical state) s pertaining to crystalline solid, or sublimation sat pertaining to vapor equilibrium with condensed phase vap pertaining to vaporization (of liquid) ∞ evaluated at infinite dilution 0 evaluated in the limit nCO2 ↓ 0 (ideal vapor limit)

DEDICATION The author wishes to dedicate this paper to the memory of Prof. Michel Boudart [1924−2012], who “catalyzed” his early movement away from “purely phenomenological” descriptions of heterogeneous kinetics.

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molecular diameter (m) for component A or carrier fluid (CO2)

NOMENCLATURE

Symbol definition

A ci̅ C D

growth species (e.g., API surrogate) mean thermal speed of molecules i (m/s), (8kBT/πmi))1/2 carrier gas or fluid Fick diffusivity (for growth species i in prevailing carrier fluid) (m2/s) dp particle diameter (m) Eε activation energy (based on T-dependence of ε) EĠ activation energy (based on T-dependence of Ġ ) Ġ growth rate of crystal face {hkl} (m/s) characteristic ratio of molecular fluxes: growth/diffusion . (eq 7) Grh Grashof number (ref 25) ΔHs heat of sublimation (at Ttr), MJ/kmol {hkl} Miller indices defining crystal plane KG phenomenological growth rate constant (assumed T-dependent) kB Boltzmann constant SA mean-free-path for solute molecule A; [π (σA + σC)2 nC]−1 M molecular weight (kg/kmol) mi molecular mass of component i n kinetic “order” parameter in phenomenological growth rate law, d ln Ġ /d ln S NAvog Avogadroʼs number (0.6023 × 1027 molec/kmol) ni number density of component i (molec/m3) Num Nusselt number (mass transfer) (ref 25) universal gas constant (kBNAvog) 9 Re Reynolds number (ref 25) S supersaturation ratio (= 1 at interfacial equilibrium on a flat surface; footnote (ii); eq 2 Sc Schmidt number, v/DA∞ (momentum/species mass diffusivity ratio) T temperature (K) V molar volume of mixture (m3/kmol) vm molecular volume in the crystal phase (m3/molec); UCV/Z yi mole fraction of component i in prevailing mixture Ż ″i molecular impingement flux of species i (molec/m2 s) Z number of growth molecules in a “unit cell” of the crystal

Acronyms, Abbreviations, Operators

API ASP BCF CG GASP GK IG N 6 () polycr PVG QS SC UCV

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active pharmaceutical ingredient antisolvent precipitation Burton−Cabrera−Frank (CG theory) crystal growth gas antisolvent precipitation (process) Gibbs−Kelvin ideal gas (kinetic theory) naphthalene order-of-magnitude “operator” polycrystalline physical vapor growth quasi-steady supercritical unit cell volume

REFERENCES

(1) Tai, C. Y.; Cheng, C.-S. AIChE J. 1995, 41, 2227−2236. (2) Uchida, H.; Manaka, A.; Matsuoka, M.; Takiyama, H. Cryst. Growth Des. 2004, 4, 937−942. (3) Martin, A.; Cocero, M. J. Adv. Drug Delivery Rev. 2008, 60, 339− 350. (4) Rosner, D. E.; Arias-Zugasti, M. Ind. Eng. Chem. Res. 2014, 53, 4489−4498. (5) Rosner, D. E.; Arias-Zugasti, M.Theory of Pharmaceutical Powder ‘Micronization’ Using Compressed Gas Anti-Solvent (Re-) Precipitation. Chem. Eng. Sci. 2014, in preparation. (6) Mullin, J. W. Crystallization, 4th ed.; Elsevier/ButterworthHeinemann: Amsterdam, 2001; Chapter 6 Crystal Growth, pp 224− 296. (7) Volmer, M.; Schultz, W. Z. Phys. Chem. 1931, 156, 1−22. (8) Volmer, M. Kinetik der Phasenbildung; Mark, H., Polanyi, M., Eds., T. Steinkopff Verlag: Dresden, Germany, 1939; Vol 4. (9) Rosner, D. E. Collision Theory Re-Interpretation of Kinetic Data for the Growth of Organic Crystal Surfaces. Part I: Melt-Growth of Small Organics, to be submitted to Cryst. Growth Des., 2014. See ref 24. (10) Rosner, D. E. Collision Theory Re-Interpretation of Kinetic Data for the Growth of Organic Crystal Surfaces. Part II: Physical Vapor-Growth, to be submitted to Cryst. Growth Des., 2014. See ref 24. (11) Rosner, D. E. Collision Theory Re-Interpretation of Kinetic Data for the Growth of Organic Crystal Surfaces. Part III: SolutionGrowth, to be submitted to Cryst. Growth Des., 2014. See ref 24.

Greek letters

βT thermal expansion coefficient, 1/V(∂V/∂T)p ε overall incorporation probability (defined by eq 1) at stated S, T, {hkl} ϕ molecular volume fraction; nC (π/6)σC3 κp −1/V(∂V/∂p)T μ Newtonian viscosity of fluid (Pa s) ν “kinematic” viscosity, μ/ρ (momentum diffusivity12) ω ∂ ln μ0/∂ ln T (for CO2) ωN mass fraction of naphthalene ρ mass density (kg/m3) 3789



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(12) Rosner, D. E. Transport Processes in Chemically Reacting Flow Systems; Dover: Mineola, NY, 2000. (13) Matsuoka, M. J. Chem. Eng. Jpn. 1982, 15, 194−199. (14) Rosner, D. E.; Epstein, M. J. Colloid Interface Sci. 1968, 28, 60− 65. (15) Bakhbakhi, Y.; Charpentier, P. A.; Rohani, S. Can. J. Chem. Eng. 2005, 83, 267−273. (16) Ghez, R. J. Cryst. Growth 1974, 22, 333−334. (17) Churchill, S. W. Ind. Eng. Chem. Res. 2014, 53, 4104−4118. (18) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids, 5th ed.; McGraw-Hill: New York, 2001. (19) Myerson, A. S.; Green, D. A.; Meenan, P. Crystal Growth of Organic Materials; American Chemical Society: Washington, D.C., 1996. (20) Sloan, G. J. Mol. Cryst. 1967, 2, 323−331. (21) Burton, W. K.; Cabrera, N.; Frank, F. C. Philos. Transact. R. Soc. London, Ser. A 1950, 243, 299−358. (22) Muhrer, G.; Lin, C.; Mazzotti, M. Ind. Eng. Chem. Res. 2002, 41, 3566−3579. (23) Anwar, S.; Carroll, J. J. Carbon Dioxide Thermodynamic Properties Handbook; Wiley-Scrivener: Beverly, MA, 2011. (24) In preparation for the present paper, the author first explored this approach based on previously available Ġ data for organic crystals in the three “classical” crystal growth environments:6 pure melts, ideal vapors, and liquid solvents, with preliminary ε-results reported in the preprints I, II, III9−11available on request. (25) As seen in section 2, this particular sufficient condition will follow from the necessary condition: . ≪ 1 in the ideal gas limit (when Ż ″A ≃ (1/4)nAcA̅ and D∞ A‑mix ≃ (1/2)S AcA̅ when the dimensionless mass transfer coefficient Num is not very different from its quiescentisolated target limit value: 2). (26) We consider here the domain in which dp is small enough to preclude excessive diffusion “corrections” however not so small as to incur significant Gibbs−Kelvin−Ostwald curvature-modifications to yA,sat,w(T, p). For the more general case see, e.g., Rosner and AriasZugasti (2014).4,5 (27) In the absence of buoyancy-induced (“natural”-) convection Re ≡ Udp/ν would simply be based on the imposed fluid velocity U (often explicitly quoted). However, in the presence of a fluid density nonuniformity associated with even small imposed temperature differences, there will also be a “buoyancy”-driven velocity, Ug, associated with the gravitational body force/mass g. This velocity can be estimated via. 1.18Grh1/2ν/dp where the dimensionless group: Grh (named after the German engineer: Franz Grashof (1826−1893)) is proportional to the relative change in fluid density associated with ΔT at the prevailing pressure. For our present estimates, we simply employed a forced-convection relation for Num (see, e.g., Rosner12) but evaluated the effective fluid velocity as Ueff = (U2 + U2g )1/2a procedure which has several successful precedents (see, e.g., Churchill (2014).17 In the 84 bar experiments of Tai and Cheng (1995),1 despite the fact that ΔT was only about 0.2 K, we estimate that Ug actually dominated the small imposed velocity U (= 0.24 mm/s). (28) These explicit relations follow from the above-mentioned definitions, our suggested molecular flux law (eq 2) for arbitrary molecular volume fraction ϕ, and the chain rule for differentiation.

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