Modeling the crystallization of proteins and small organic molecules in nanoliter drops

Drop‐based crystallization techniques are used to achieve a high degree of control over crystallization conditions in order to grow high‐quality protein crystals for X‐ray diffraction or to produce organic crystals with well‐controlled size distributions. Simultaneous crystal growth and stochastic nucleation makes it difficult to predict the number and size of crystals that will be produced in a drop‐based crystallization process. A mathematical model of crystallization in drops is developed using a Monte Carlo method. The model incorporates key phenomena in drop‐based crystallization, including stochastic primary nucleation and growth rate dispersion (GRD) and can predict distributions of the number of crystals per drop and full crystal size distributions (CSD). Key dimensionless parameters are identified to quickly screen for crystallization conditions that are expected to yield a high fraction of drops containing one crystal and a narrow CSD. Using literature correlations for the solubilities, growth, and nucleation rates of lactose and lysozyme, the model is able to predict the experimentally observed crystallization behavior over a wide range of conditions. Model‐based strategies for use in the design and optimization of a drop‐based crystallization process for producing crystals of well‐controlled CSD are identified. © 2009 American Institute of Chemical Engineers AIChE J, 2010     

Growth regimes and the time development of lateral habits in polyethylene crystals

On the assumption that in regime I the growth rate of the lateral face of a lamellar crystal is proportional to the length of the face, and that in regime II the growth rate is independent of the length of the face, the time development of the size and aspect ratio of a polyethylene lamellar crystal is calculated. The aspect ratio is defined as the ratio of the length of the crystal in the b crystallographic direction to that in the a direction. It is assumed that steady state growth obtains, i.e. the solution concentration is constant. When both {110} and (200) faces are in regime I, the dimensions of the crystal increase exponentially in time, and lozenge shape (i.e. a crystal bounded by {110} faces) cannot be obtained under experimentally realizable conditions. When one of the faces is in regime I and the other in regime II, novel time dependencies of the crystal size and shape are derived, none of which has heretofore been observed. Despite the fact that the experimental conditions have never been realized in growth from solution, the now well-known fact that in regime I the growth rate is not proportional to the length of the growing face is re-established. Some new ways by which this assumption might be removed are suggested, and qualitative values for the upper bound for the ‘substrate length’ over which growth takes place from a nucleation event are discussed.     

Roughness of growth faces of polymer crystals: Evidence from morphology and implications for growth mechanisms and types of folding

It is proposed that the growth faces of lamellar polymer crystals can have an equilibrium roughness (or crenellation). This can explain why some polymer crystals show no evidence of faceting. Support for this idea comes from the extensive theoretical developments on the nature of crystal surfaces. The characteristic habits of polyethylene are analysed in terms of a roughness which, on the {110} faces, increases progressively over a temperature range of about 100°C. At a temperature near 110°C the roughness becomes sufficient for there to be no free energy penalty for arbitrary crystal shapes (e.g. rounded) compared with one bounded by {110} faces. Above this temperature of crystallization most of the habits which are observed are leaf-shaped, with an apex along 〈010〉. Below 110°C{110} facets (or microfacets) are normally seen. There is no positive evidence that faces approximately parallel to (100) planes, observed for crystallization temperatures in the range 80°–110°C, are ever other than rough. The relative rates of growth on the {110} compared with the (100) increase with temperature, since {110} faces predominate at low but not high temperatures of crystallization. These changes are attributed to the increase in roughness with temperature on the {110} faces. The existence of surface disorder (roughness) requires that the binding energy between units in the crystal is comparable with KT. Hence this unit is probably several monomer units of polyethylene (rather than, for example, a complete stem which contains a hundred or more monomers). There is therefore a surface lattice on the growth faces with twenty or more units in the direction perpendicular to the lamellae. Monte Carlo calculations are cited for lattices of 20 by 50 units. These show that a cooperative increase in surface roughness with temperature and a transition between faceted and non-faceted growth can be expected for lattices of such limited extents. No explicit allowance has been made as yet for the consequences of the units being linked into chains and, for that reason, not being able to arrive or leave the surface independently. It is noted that changes in the alkane lattice with temperature indicate a possible evolution in binding energy, and in mobility, and hence may influence surface roughness. Theories of crystallization in polymers have normally assumed a growth surface which is molecularly smooth in equilibrium, and have emphasised nucleation events. Since this paper shows that the equilibrium structure may often be rough, it may be necessary to re-examine the basis of these theories. A brief review is included of the experimental evidence for surface nucleation events: nucleation may be a more important barrier at low temperatures than at high. The type of folding will be influenced by equilibrium roughness just as it will be by kinetic roughness, and some comparisons are made with neutron scattering results on this topic. The degree of adjacent folding is higher in the faceted regime as expected. Brief comments are made on the applicability of this idea to polymers other than polyethylene.     

Rapid synthesis of CNTs@MIL-101(Cr) using multi-walled carbon nanotubes (MWCNTs) as crystal growth accelerator☆

In this work,hybrid material CNTs@MIL-101(Cr) was synthesized in 2 h using multi-walled carbon nanotubes(MWCNTs) as the crystal growth accelerator with hydrothermal method.The characteristic differences between the crystals of CNTs@MIL-101(Cr) and MIL-101 were investigated by N_2 adsorption–desorption isotherms,X-ray diffraction(XRD),scanning electron microscope(SEM) and thermogravimetric analyzer(TGA).The results showed that MWCNTs embedding in the hybrid material provide more mesoporous volumes than that of MIL-101.Moreover,the fast synthesized crystals of CNTs@MIL-101(Cr) still preserve the octahedral shape like MIL-101 and have a larger size ranging from 1.5 to 2.0 μm which were approximately three times larger than that of MIL-101.In the proposed mechanism,the roles of MWCNTs played in the crystallization were discussed where MWCNTs can be seen as coaxial cylindrical tubes composed of multi-layer graphenes and the place where nucleation and crystal growth processes occur at the tubes' out surface.Then,a crystal seeding layer bonding with the MWCNTs may be easily formed which accelerates the growth rate of MIL-101 crystals.Thus,larger crystals of CNTs@MIL-101(Cr) were formed due to the faster crystal growth rate of MIL-101.     

On the prediction of crystal shape distributions in a steady-state continuous crystallizer

Zhang and Doherty [2004. Simultaneous prediction of crystal shape and size for solution crystallization. A.I.Ch.E. Journal 50, 2101-2112] have provided a one-dimensional analysis of crystallization based on the assumption that the relative face-specific growth rates of a (2-D) crystal are independent of supersaturation and hence invariant with time. Subsequent work by these authors [Zhang, Y., Sizemore, J.P., Doherty, M.F., 2006. Shape evolution of 3-dimensional faceted crystals. A.I.Ch.E. Journal 52, 1906-1915) consider shape evolution of single three-dimensional crystals with morphological changes. In this work, we present a multidimensional population balance approach accounting for dependence of the relative face-specific growth rates on supersaturation, a situation more commonly encountered. For example, Joshi and Paul [1974. Effect of supersaturation and fluid shear on habit and homogeneity of potassium dihydrogen phosphate crystals. Journal of Crystal Growth 22, 321-327] and Mullin and Whiting [1980. Succinic acid crystal-growth rates in aqueous solution. Industrial & Engineering Chemistry Fundamentals 19, 117-121] report face-specific growth rates with different dependence on the supersaturation. Thus it has been observed that there exists significantly different crystal shapes in a crystallizer [Yang, G., Kubota, N., Sha, Z., Louhi-Kultanen, M. Wang, J., 2006. Crystal shape control by manipulating supersaturation in batch cooling crystallization. Crystal Growth and Design 6, 2799-2803]. Consequently, the population of crystals at any instant will have widely varying crystal shapes and sizes depending upon the initial crystal shape and size distribution. Computations are presented for the shape distributions of the crystal population emerging from a steady-state continuous crystallizer for two cases: (1) feed without crystals including nucleation for the formation of new crystals, and (2) feed with seed crystals of known shape, with suppressed nucleation. In the range of mean residence times investigated, the calculated crystal volume distributions for the first case show geometrically dissimilar shapes without morphological variations. However, in the second case, because the feed crystals of the chosen shape were susceptible to morphological changes, the volume distributions display this feature with shape and size distributions for each of a number of different morphologies. By varying operating conditions such as the flow rate, the inlet supersaturation, and the shapes of feed crystals, the proposed model can clearly be used to manipulate the crystal shape and size distributions and their morphologies.     

Reformulating multidimensional population balances for predicting crystal size and shape

There is a growing interest in predicting and controlling the size and shape of crystalline particles. Multidimensional population balances have been developed to accomplish this task but they suffer from the drawback of needing rate laws for the absolute growth rate for every family of faces that may appear on the crystal surface. Such growth rates are known for only a handful of crystalline materials and prospects are bleak for extending the library of growth rate data. This raises the question of where the surface growth rates for all the families of faces will come from to drive multidimensional population balance engineering technology. One answer is “from first principles.” We reformulate multidimensional population balances in terms of relative growth rates and show how to create first principles mechanistic models to calculate these quantities for real molecular crystals as a function of supersaturation. © 2013 American Institute of Chemical Engineers AIChE J, 59: 3468–3474, 2013     

Solution of the growth equation for asymmetric crystal faces

Most melt-grown and many solution-grown lamellar polymer crystals have curved lateral faces. Mathematical treatments by Mansfield, Point and Villers, and Toda, have provided a satisfactory interpretation of the shape of such crystal faces in terms of nucleation and relatively slow propagation rates of layers of attaching stems. The treatments by these authors, which start with the Frank-Seto growth model, assume that the propagation rates of growth steps to the right (v_r) and to the left of the secondary nucleus (v_l) are equal. However, for many crystal growth faces this is not the case; faces which lack a mirror plane perpendicular to the lamella have v_r≠v_l, resulting in asymmetric curvature. Here, we set up and solve the differential equations and reconstruct the shape of the growth front for the case of asymmetric spreading of steps. The solution is presented for the simple square lattice model. The asymmetric growth front is still described as part of an ellipse, as in the symmetric case, except that the centre of the ellipse is translated parallel to the underlying crystallographic plane in the direction of fast v. In forthcoming publications we will adapt the solution to other 2D Bravais lattices, appropriate to the crystal structures of specific polymers. Thus we will analyze complete habits of polymers such as polyethylene, poly(ethylene oxide), and poly(vinylidene fluoride), whose {110}, {120} and {110} growth faces, respectively, are asymmetric. The results of the present work allow a detail kinetic analysis of any well-developed polymer growth face in terms of the step initiation rate i and the propagation rates v_r and v_l. The present work also quantifies explicitly the deviations from elliptic shape and the substrate edge effects, and discusses when these can be ignored.     

Unusual Crystal Growth Kinetics of (RS)‐Ibuprofen from Ethanolic Solutions

The solubility, secondary nucleation threshold, and growth kinetics of (RS)‐ibuprofen have been studied in an aqueous ethanol solvent. The metastable zone for secondary nucleation is very narrow at lower temperatures in this range, but greatly enlarged at higher temperatures. The crystal growth kinetics not only display significant dispersion of growth rates, but also a dead zone that is dependent on the growth rates of the crystals. Faster growing crystals display almost no dead zone, whereas the smallest crystals have a large dead zone. The size of the dead zone is largely responsible for the dispersion of crystal growth rates, perhaps due to differences in the thermodynamic stability of the different crystals. The mechanism of growth rate dispersion relates to that of the dead zone.     

Control of crystal morphology via molecular imprinting

Imprinted polymers were prepared from divinylbenzene and 6‐methacrylamidohexanoic acid as the functional monomer, using calcite crystals of three different morphologies as templates. After template removal voids in the polymer remained which reflected the size and shape of the template crystals. The use of these imprinted polymers in aqueous supersaturated calcium carbonate solution resulted in the formation of some crystal objects of unusual morphology. In the case of spheroidal imprints, the nucleated crystals grow via the {104} faces in many directions creating clusters whose surface consists of many small rhombohedral crystallites. However, in the case of rhombohedral imprints, no conflict between crystal growth and the constraints of nucleation at the polymer surface arises and the new crystals very closely resemble the templates. © 2001 Society of Chemical Industry     

Crystallization of high purity ammonium meta-tungstate for production of ultrapure tungsten metal

The growth mechanism of Ammonium Meta-Tungstate (AMT) crystal was interpreted as two-step model. Growth rates of AMT crystals were measured in a fluidized bed crystallizer. The effects of temperature, supersaturation and crystal size on the crystal growth were investigated. The contribution of the diffusion step increased with the increase of temperature, crystal size and supersaturation. The nucleation kinetics from measurements on the width of the metastable region of Ammonium Meta-Tungstate (AMT) was also evaluated. The crystal size distribution from a programmed cooling crystallization system was predicted by the numerical simulation of a mathematical model using the kinetics of nucleation and crystal growth. It was also observed that the shape of AMT crystals was changed during the growth period. This paper was presented at The 5th International Symposium on Separation Technology-Korea and Japan held at Seoul between August 19 and 21, 1999.     

CL‐20 Evaporative Crystallization under Reduced Pressure

The effect of parameters of the CL‐20 crystallization process carried out by solvent removal by evaporation in vacuo on shape, polymorph type, crystal size, and on their shock sensitivity was studied. The CL‐20 crystallization process by this technique was shown to allow a precise control of the crystallization process parameters and of the process run. The o‐xylene/ ethyl acetate system proved to be highly effective. Selecting suitable values of the parameters such as: pressure, process time, temperature, stirring rate, CL‐20 crystals were obtained in the ε form (even with no need for inoculation of the crystallization system with polymorph ε seeds) and of the shape close to a spherical one. The crystal growth modifiers added allowed to additionally control the shape and size of the CL‐20 crystals formed and to produce crystals of reduced impact and friction sensitivity.     

Two-dimensional population balance modeling for shape dependent crystal attrition

Many suspension crystallization processes can be described by growth and secondary nucleation. The prevailing mechanism for secondary nucleation in industrial processes is attrition caused by crystal-impeller collisions. The understanding and prediction of attrition rates is of fundamental importance for product and process design.Attrition of a crystal is affected by the size of the crystals and, as experimental evidence reveals, by their shape. For crystals with the same mass, the attrition rate is significantly larger if crystals have distinct and sharp corners, than if they were spherical. Because of the associated modeling and computational effort, shape dependent behavior has mostly been disregarded in crystallization process modeling. In this work, a two-dimensional population balance model is formulated. The inner variables are the size and the shape of the crystals. Consequently, the model accounts for size and shape dependent process behavior. In order to close the model equation shape modification function are introduced. The reinforcement function specifies the increase in attrition resistance by rounding the sharp corners and increasing material strength. The face attrition ratio represents the differences of material removal from sharp corners and flat faces.The sensitivity of the results with respect to these shape modification functions is investigated. The results show that the model is capable of reflecting shape dependent attrition behavior in a physically meaningful way. To fit experimental data, mainly the parameters of the shape modification function need to be tuned.     

Needle Crystal Morphology Explained

The morphology of needle shaped crystals is, usually, ill‐predicted when using the common attachment energy approach. Here we explain the needle shape of triacylglycerol crystals on the basis of a two‐dimensional nucleation growth mechanism. For that the edge energies of 2D nuclei on the surfaces of the various crystal faces is determined and turns out to be much lower than expected when applying the attachment energy criteria. The edge energies are found by determining the connected nets of the various faces that follow from the crystal structure and the interaction energies between the molecules. The results are confirmed by Monte Carlo simulations.     

Modelling protein crystallisation using morphological population balance models

Protein crystallisation is known to be affected by many factors and inherently difficult to control. Being able to model the crystal growth behaviour, especially at process scale for the population of particles in a crystalliser will no doubt greatly help the formulation and controlled manufacture of protein crystals. In this paper, a morphological population balance model for crystallisation of tetragonal Hen-Egg-White (HEW) lysozyme is presented. Since the population balance model has incorporated crystal shape information, it is able to simulate the dynamic evolution of the shape distribution as well as size distribution. The morphological population balance model requires faceted growth kinetics data, which was obtained from published data in literature for the two identified independent crystallographic faces, {1 0 1} and {1 1 0}, of HEW lysozyme crystals.     

Kinetic roughening transition and missing regime transition of melt crystallized polybutene-1 tetragonal phase: growth kinetics analysis

The morphology and lateral growth rate of isotactic polybutene-1 (it-PB1) have been investigated for crystallization from the melt over a wide range of crystallization temperatures from 50 to 110°C. The morphology of it-PB1 crystals is a rounded shape at crystallization temperatures lower than 85°C, while lamellar single crystals possess faceted morphology at higher crystallization temperatures. The kinetic roughening transition occurs around 85°C. The nucleation and growth mechanism for crystallization does not work below 85°C, since the growth face is rough. However, the growth rate shows the supercooling dependence derived from the nucleation and growth mechanism. The nucleation theory seems still to work even for rough surface growth. Possible mechanisms for the crystal growth of this polymer are discussed.     

Analysis of crystal habits bounded by asymmetrically curved faces: Polyethylene oligomers and poly(vinylidene fluoride)

Curved lateral faces of lamellar polymer crystals have previously been described mathematically using a model of initiation and spreading of molecular steps on the growth surface. Previously it has been assumed that the steps spread with equal velocity v in both directions, and hence the model only applied to faces that have a two-fold axis or are bisected by a mirror plane normal to the lamella. Many lateral faces in polymer crystals do not have such symmetry. We recently solved the growth equation and reconstructed the growth profile for the case where the velocities in the left and right directions (v_l and v_r) are different, using a square lattice model. Here we adapt the model to oblique lattices suitable for {110} growth faces of polyethylene oligomers (ultralong alkanes) and PVDF. Very good fits are obtained with the observed unusual habits of crystals with curved {110} faces. It is shown that the shape of an asymmetric lateral crystal face is defined by two kinetic parameters, v_r/v_l and il₀²/(v_r + v_l), where i is the nucleation rate and l₀ the interchain distance in the growth plane. The value of v_r/v_l is found to be furthest from 1 at the smallest supercooling, which is consistent with the general principles of crystallization kinetics. The observed ten-fold difference between right- and left-moving steps on α-crystals of PVDF is attributed to the difference between edge-on and flat-on chain attachment at these steps, respectively. The high value of il₀²/(v_r + v_l), the cause of high curvature, gives a picture of a crystallization process that is only marginally nucleation controlled. Specific combinations of the above two parameters are predicted to produce exotic single crystals with re-entrant corners, so far unobserved.     

Analysis of Surface Features on Single Crystals of Synthetic Garnets

Surface features observed on the faces of synthetic garnets grown from lead oxide-lead fluoride melts are described and interpreted in terms of growth processes. The growth processes involve (1) nucleation of new growth layers at corners and edges between crystal faces and (2) propagation of layers associated with growth systems located on the crystal faces. Growth systems on the faces of crystals that terminated growth at about 1000°C appear as growth hillocks. Continuing growth to lower temperatures produces large vicinal faces, which result from the propagation of polygonized growth steps associated with a single, dominant growth system. Growth steps on the vicinal faces in turn possess faces that are parallel to the nearest adjacent crystal faces. Growth spirals have been observed on the faces of crystals that were grown slowly by evaporation of flux. Several observations, including the presence of growth spirals, the nature and location of line imperfections and hollow tubes, and the etching behavior of crystal faces and polished sections, are interpreted as evidence in favor of a latter-stage growth mechanism that involves the propagation of steps associated with screw dislocations.     

Novel Faceted α-SiAlON Micro-Crystals Prepared by Combustion Synthesis

Novel faceted α-SiAlON micro-crystals were prepared by combustion synthesis with proper additives. These crystals exhibited various end shapes, including smooth flat hexagonal faces, pyramids bounded by six small isosceles triangles, and partial pyramids with the sharp roofs removed. The formation mechanisms of different crystal shapes were discussed, and a terraced epitaxial nucleation mode for α-SiAlON was proposed. By this nucleation mode, two interesting crystal morphologies were formed: a terraced tower structure and a T-like shape. Transmission electron microscopy results showed that the fast growth direction of rod-like α-SiAlON crystals was parallel to the c -axis.     

Habit Modifications of SnO2 Crystals in SnO2–Cu2O Flux System in the Presence of Trivalent Impurity Cations

SnO₂ single crystals have short prismatic habits bounded by well-developed {110} and {111} faces in a pure SnO₂–Cu₂O flux system. When trivalent cations are added to the system, the habits drastically change to needle, acicular, or whisker forms with large aspect ratios. The addition of trivalent cations also greatly increases the nucleation rate and drastically decreases the crystal size. SEM observations and EPMA investigations reveal that the flat {111} faces transform to rounded or rough { hkl } faces by the addition of trivalent cations. This roughening transition of {111} faces, keeping {110} faces unchanged, is the cause of drastic habit modification that is attributed to the breaking of the periodic bond chain in {111} faces by impurity cations.