CRYSTAL GROWTH & DESIGN
The Evolution of Crystal Shape During Dissolution: Predictions and Experiments
2008 VOL. 8, NO. 4 1100–1101
Ryan C. Snyder,† Stéphane Veesler,‡ and Michael F. Doherty*,† Department of Chemical Engineering, UniVersity of California, Santa Barbara, California 93106, and Centre de Recherche sur les Mécanismes de la Croissance Cristalline CRMCN-CNRS, Campus de Luminy, Case 913, F-13288 Marseille Cedex 09, France ReceiVed September 5, 2007; ReVised Manuscript ReceiVed January 25, 2008 嘷 w This paper contains enhanced objects available on the Internet at http://pubs.acs.org/crystal.
ABSTRACT: A comparison between a priori model predictions and experimental results for crystal shapes of succinic acid crystals dissolving in aqueous solution is presented. The model is the simplest possible model which neglects all effects other than a determination of which faces to include on the crystal habit and their perpendicular growth rates. The experiments are carried out in a thermostatted peltier cell and are imaged in situ using a microscope with digital imaging capabilities. The predicted and experimentally measured crystal shapes are very similar throughout the dissolution process. Thus, the experiments provide support for the model’s predictive power. The change in crystal shape as a result of dissolution is a topic of interest for a wide range of materials including organics such as pharmaceuticals, proteins and specialty chemicals as well as inorganics in geology and semiconductors. For inorganic and organic materials, dissolution not only affects the crystal shape but also plays a vital role in polymorphic phase transformations. The possible end-shapes in crystal dissolution have been discussed in many contexts.1–4 For example, Moore suggested that vertices on growth shapes would end up as faces on dissolution shapes,2 and Gibbs suggested that shapes in dissolution will differ from the equilibrium form in a direction “opposite” to that of growth.3 Despite these qualitative descriptions, a model that can quantitatively predict the evolution of crystal shapes during dissolution has only recently been developed.5 This communication presents recently produced in situ experimental confirmation of the model’s predictions for the dynamics of dissolving crystal shapes, along with the related modeling information necessary for comparison. We have chosen β-succinic acid crystals dissolving in water as the system to demonstrate the effective predictions of the model. The crystallization and subsequent dissolution and visualization were performed in a thermostatted peltier cell equipped with a Nikon microscope and digital imaging capabilities, and has been previously described in detail.6 A saturated solution of β-succinic acid7 in 2 mL of water was prepared at a saturation temperature of 36 °C. Nucleation was induced at a lower temperature (25 °C). Once a crystal could be observed in the solution, the temperature was raised to within a degree of saturation, and the crystal was allowed to grow to a size suitable for dissolution visualization (∼100 µm). Then, the temperature was increased to one degree above saturation, leading to dissolution during which the crystal shape evolution was recorded. Since the mass of the dissolving crystal is very small (∼0.001 mg compared to the 2 mL volume of the crystallizer) the undersaturation is approximately constant throughout the dissolution process. The model for shape evolution can predict and track each of the faces of a fully three-dimensional crystal during growth and dissolution. It requires a minimal set of physical properties for implementation (face dissolution rates and crystallographic informa* To whom correspondence should be addressed. Phone: 805-893-5309; fax: 805-893-4731; e-mail:
[email protected]. † University of California. ‡ Centre de Recherche sur les Mécanismes de la Croissance Cristalline CRMCN-CNRS.
Table 1. Relative Dissolution Rates for Each of the Families of Planes That Are Required for a Dissolution Shape Evolution Prediction of β-Succinic Acid in Water plane family
relative dissolution rate
{100} {020} {011} {11j1} {111j} {002} {120} {031} {102} {102j} {131}
-1.00 -4.19 -2.98 -10.00 -10.00 -10.00 -10.00 -10.00 -10.00 -10.00 -10.00
tion). The rate of change of the distance of each face from the center of a crystal is dHi ) Gi, i ) 1, . . . , N (1) dt where Hi and Gi are the perpendicular distance and normal velocity of face i, respectively, and N is the number of real crystal faces, which may change as the crystal evolves. To determine the shape at any point in time, a convex hull is calculated.8 A convenient choice of variables for tracking the evolution are the relative perpendicular distance and relative growth or dissolution rate. This formulation results in a set of differential equations for dissolution given by dxi ) xi - RD,i, i ) 1, . . . , N - 1 dξD dξD ) -
GD,ref dt Href
(2) (3)
where the dimensionless perpendicular distance of face i is xi ) Hi/Href, and the (dimensionless) relative growth velocity of face i is RD,i ) GD,i/GD,ref. GD,ref is the dissolution rate of the reference face and Href is the perpendicular distance of the reference face. Similar equations exist for growth; however, the sign of the righthand side of eq 2 is reversed.8 Equation 2 is written in matrix notation as dx ) Ax - RD, i ) 1, . . ., N - 1 dξ
10.1021/cg7008495 CCC: $40.75 2008 American Chemical Society Published on Web 03/05/2008
(4)
Communications
Crystal Growth & Design, Vol. 8, No. 4, 2008 1101
Figure 1. Comparison of the model predictions (top) and experimental results (bottom) for β-succinic acid dissolving in water. The initial shape (a) is similar to the steady-state growth shape.
where A ) I is the identity matrix (N - 1 × N - 1), x is a vector of N - 1 relative perpendicular distances, and RD is a vector of N - 1 relative dissolution rates. When RD is constant as in our experiments, eq 4 is a linear system of ordinary differential equations with N - 1 eigenvalues whose values are all +1. Since all of the eigenvalues are real and positive, the steady state of the system (xi ) RD,i) is unique and unstable. Thus, as crystals dissolve, this model predicts they will evolve away from the unique and unstable steady-state shape. (The model for growth predicts that crystals evolve toward a unique and stable steady-state shape.8) In addition, a set of algebraic conditions that determine when faces appear at edges or vertices as the shape of the crystal evolves in time are coupled to the differential equations. The model also contains a methodology to determine the overall set of planes that must be considered for a dissolution evolution.5 Many of the planes that can appear at edges or vertices are stepped (S) or kinked (K) faces and thus will be roughened. While roughened faces often have curvature in directions without a strong bond chain, our first-order approximation to the crystal shape during dissolution treats these faces as being flat, providing for a faceted crystal shape. A more detailed model would include roughening and curvature to provide for predictions of even higher fidelity. The model inputs are the crystallography as well as the relative dissolution rates of each face, which can be taken from either experimental data or models such as the kinetic growth models of Burton, Cabrerra and Frank9 and Chernov.10 The purely predictive model has been implemented for β-succinic acid dissolving in water to compare to the experiments on the same system. The predictions are generated using the relative dissolution rates in Table 1, which have been predicted based upon a spiral dissolution model for each face.5 The experiments were performed after the predictions were submitted for publication and are reported here for the first time. The model given by 4 predicts the shapes that can be obtained in dissolution and not the amount of time required to achieve those shapes. Thus, in addition to the relative rates of each crystal face, two further pieces of information are required to appropriately compare the model with the experiment. The initial condition for the dissolution experiment is a succinic acid crystal with H020 ) 75 µm and a shape that is consistent with the predicted steady-state growth shape. The total duration of the experiment is 22.50 min from the beginning of the dissolution until the crystal has completely dissolved (images of the crystal shape were recorded every 0.25 min). The predictions then used these two pieces of data for the initial and final conditions of the shape evolution in order to compare the predictions to the experiments throughout the dissolution. In other words, the absolute dissolution rate of the crystal was chosen such that a simulated crystal with an initial H020 ) 75 µm and initial shape corresponding to the steadystate growth shape would dissolve in 22.50 min. Figure 1 shows both the experimental results (bottom) and the model predictions (top) at several points in time. The first pair of
images (a: t ) 0 min) is similar to the hexagonal plate-like steadystate growth shape of β-succinic acid in water and serves as the initial shape for the dissolution evolution. The next set of images (b: t ) 2 min) is at a time shortly after dissolution begins when several new faces have appeared along the edges. As the dissolution continues (c: t ) 5 min, d: t ) 12 min and e: t ) 16 min), the shape progresses from six to four sided. Finally, (f: t ) 20 min) the crystal becomes an equant rounded shape. A video in .avi format that presents the dynamic evolution of the experiment side by side with the predictions is available. The model predictions are very similar to the experimental results at each point in time. However, while the model is fully faceted and does not account for roughening the experimental shapes are more rounded due to this effect. Thus, the accuracy decreases as the crystal dissolves and rounding becomes evermore important. Despite the faceted assumption, the model is adept at predicting dissolution shapes and has the advantage of requiring the least possible amount of calculation and physical property data. On the basis of this support of the model’s predictive power, it is anticipated that it can be used to further develop methodologies for providing predictive capabilities in areas such as solution-mediated polymorphic phase transformations and the cycling of growth and dissolution to enhance crystal shapes.
Acknowledgment. We are grateful for financial support provided by Merck and Co. and the National Science Foundation (Award No. CBET-0651711).
References (1) Lacmann, R.; Franke, W.; Heimann, R. J. Cryst. Growth 1974, 26, 107. (2) Moore, M. Mineral. Mag. 1986, 50, 331. (3) Gibbs, J. W. The Scientific Papers of J. Willard Gibbs; Dover Publications Inc.: New York, 1961. (4) Frank, F. C. On the Kinematic Theory of Crystal Growth and Dissolution Processes. In Growth and Perfection of Crystals; Doremus, R. H., Roberts, B. W., Turnbull, D., Eds.; John Wiley & Sons, Inc.: New York, 1958; p 411. (5) Snyder, R. C.; Doherty, M. F. AIChE J. 2007, 53, 1337. (6) Veesler, S.; Lafferrere, L.; Garcia, E.; Hoff, C. Org. Proc. Res. DeV. 2003, 7, 983. (7) Crystal Structure Data: space group P21/c, a ) 5.519, b ) 8.862, c ) 5.101, β ) 91.59°. Leviel, J. L.; Auvert, G.; Savariault, J. M. Acta Crystallogr. 1981, B37, 2185. (8) Zhang, Y.; Sizemore, J.; Doherty, M. F. AIChE J. 2006, 52, 1906. (9) Burton, W. K.; Cabrera, N.; Frank, F. C. Phil. Trans. R. Soc. 1951, 243, 299. (10) Chernov, A. A. Modern Crystallography III: Crystal Growth; SpringerVerlag: Berlin, 1984.
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