Supersaturation rates

Among the various methods for preparing different polymorphs are sublimation, crystallization from the melt, crystallization from supercritical fluids, and crystallization from liquid solutions. In the pharmaceutical sciences, different polymorphs are usually prepared by crystallization from solution employing various solvents and various temperature regimes, such as initial supersaturation, rate of de-super-saturation, or final supersaturation. The supersaturation of the solution... 

Shen et al. [50] carried out a detailed investigation in the MIV mixer, studying the incorporation of (5-carotene and polyethyleneimine (PEI) selected as model drug and cationic macromolecule various copolymers in different physical states were selected an amorphous one, poly(ethylene oxide)-l>-poly(styrene) (PEO-b-PS), a semicrystalline one, poly(ethylene oxide)-b-poly(e-caprolactone) (PEG-fc-PCL), and an ionic copolymer, poly(ethylene oxide)-b-poly(acrylic acid) (PEG-b-PAA) this also allowed the investigation of the influence of the type of interaction forces, in addition to hydrodynamics and supersaturation rate. 

In principle, nucleation should occur for any supersaturation given enough time. The critical supersaturation ratio is often defined in terms of the condition needed to observe nucleation on a convenient time scale. As illustrated in Table IX-1, the nucleation rate changes so rapidly with degree of supersaturation that, fortunately, even a few powers of 10 error in the preexponential term make little difference. There has been some controversy surrounding the preexponential term and some detailed analyses are available [33-35]. 

Because of the large surface tension of liquid mercury, extremely large supersaturation ratios are needed for nucleation to occur at a measurable rate. Calculate rc and ric at 400 K assuming that the critical supersaturation is x = 40,000. Take the surface tension of mercury to be 486.5 ergs/cm. ... 

The central quantity of interest in homogeneous nucleation is the nucleation rate J, which gives the number of droplets nucleated per unit volume per unit time for a given supersaturation. The free energy barrier is the dommant factor in detenuining J J depends on it exponentially. Thus, a small difference in the different model predictions for the barrier can lead to orders of magnitude differences in J. Similarly, experimental measurements of J are sensitive to the purity of the sample and to experimental conditions such as temperature. In modem field theories, J has a general fonu... 

Aluminum hydroxide gel may be prepared by a number of methods. The products vary widely in viscosity, particle size, and rate of solution. Such factors as degree of supersaturation, pH during precipitation, temperature, and nature and concentration of by-products present affect the physical properties of the gel. 

Over 50 acidic, basic, and neutral aluminum sulfate hydrates have been reported. Only a few of these are well characterized because the exact compositions depend on conditions of precipitation from solution. Variables such as supersaturation, nucleation and crystal growth rates, occlusion, nonequilihrium conditions, and hydrolysis can each play a role ia the final composition. Commercial dry alum is likely not a single crystalline hydrate, but rather it contains significant amounts of amorphous material. 

Crystallization. Acidified aluminum sulfate solutions can be supercooled 10 °C or more below the saturation point. However, once nucleation begins, the crystallization rate is rapid and the supersaturated solution sets up. The onset of nucleation in a gentiy stirred supersaturated solution is marked by the appearance of silky, curling streamers of microscopic nuclei resulting from orientation effects of hydraulic currents on the thin, platelike crystals. Without agitation, nucleation in an acidified solution, in glass tubes, can yield extended crystalline membranes of such thinness to exhibit colors resulting from optical interference. 

Ammonium tetraborate tetrahydrate is prepared by crystallization from an aqueous solution of boric acid and ammonia having a B202 (NH4)20 ratio of 1.8 2.1. Ammonium pentaborate is similarly produced from an aqueous solution of boric acid and ammonia having a B202 (NH4)20 ratio of 5. Supersaturated solutions are easily formed and the rate of crystallization is proportional to the extent of supersaturation (130). A process for the production... 

The most important variables affecting nucleation rate are shown by equations 10 and 11 to be iaterfacial eaergy, temperature, and supersaturation. 

Models used to describe the growth of crystals by layers call for a two-step process (/) formation of a two-dimensional nucleus on the surface and (2) spreading of the solute from the two-dimensional nucleus across the surface. The relative rates at which these two steps occur give rise to the mononuclear two-dimensional nucleation theory and the polynuclear two-dimensional nucleation theory. In the mononuclear two-dimensional nucleation theory, the surface nucleation step occurs at a finite rate, whereas the spreading across the surface is assumed to occur at an infinite rate. The reverse is tme for the polynuclear two-dimensional nucleation theory. Erom the mononuclear two-dimensional nucleation theory, growth is related to supersaturation by the equation. 

The screw dislocation theory (27), often referred to as the BCE theory (after its formulators), shows that the dependence of growth rate on supersaturation can vary from a paraboHc relationship at low supersaturation to a linear relationship at high supersaturation. In the BCE theory, growth rate is given by... 

Both supersaturation and temperature can have different effects on the growth rates of different faces of the same crystal. Such occurrences have implications with respect to crystal habit, and these are dealt with in a later section. 

Batch Crystallization. Crystal size distributions obtained from batch crystallizers are affected by the mode used to generate supersaturation and the rate at which supersaturation is generated. For example, in a cooling mode there are several avenues that can be followed in reducing the temperature of the batch system, and the same can be said for the generation of supersaturation by evaporation or by addition of a nonsolvent or precipitant. The complexity of a batch operation can be ihustrated by considering the summaries of seeded and unseeded operations shown in Figure 19. 

Control of supersaturation is an important factor in obtaining crystal size distributions of desired characteristics, and it would be useful to have a model relating rate of cooling or evaporation or addition of diluent required to maintain a specified supersaturation in the crystallizer. Contrast this to the uncontrolled situation of natural cooling in which the heat transfer rate is given by... 

Better product characteristics are obtained through control of the rate at which supersaturation (cooling, evaporation, and addition of a nonsolvent or precipitant) is generated. An objective of the operation may be to maintain the supersaturation at some constant prescribed value, usually below the metastable limit associated with primary nucleation. For example, the batch may be cooled slowly at the beginning of the cycle and more rapidly at the end. 

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