Size distributions

In Sections III and IV, the principles of nucleation and growth were discussed separately. Now the crystallization process as a whole will be considered. In any practical application of crystallization, a stable solid phase must first be formed from the metastable liquid phase, and then additional molecules are deposited on the nucleus to form the macroscopic crystalline solid. Since nucleation and growth are taking place simultaneously, the theoretical principles discussed earlier are difficult to apply quantitatively to crystallization practice. Consequently, empirical expressions are still generally used in the design of equipment and prediction of its operation. 

In the production of a crystalline material to be sold as a consumer product, crystal size, shape, and uniformity are often quite important. Sometimes a crystal product can be manufactured most economically by a particular process, but the product may not have the desired sales appeal. In such instances more costly manufacturing techniques may have to be employed in order to achieve the desired crystal characteristics. Factors such as these must be considered in economic calculations on a crystallization process. Often, as in the case of table salt, size and uniformity of size are important for practical reasons. Fines are undesirable, since they may be lost as an objectionable dust, while coarse crystals may not fit through small salt shaker openings. Uniform size also aids in 

Even when size and size distribution per se impose no restrictions on the process, the filtration, washing, or centrifugation steps which usually follow crystallization may call for special crystal characteristics. Since fine crystals have a large specific surface area, excessive loss of product during washing may be encountered. Time and cost of filtration or centrifugation are highly dependent on crystal size distribution. 

The ratio of growth rate to nucleation rate is a measure of the crystal size obtained from a given process the larger this ratio, the coarser the product. It is also clear that production rate falls off with decreasing supersaturation. The operation of a crystallizer is a compromise between these two factors. If size is of little importance, high supersaturations will result in high production rates and small crystals. Conversely, coarse product is obtained at lower supersaturation, but at the expense of production rate. Crystallizer operation will be discussed in greater detail in Section VI. 

Data obtained from a sieving operation are commonly presented as the weight per cent of crystals associated with each crystal size range. From these data a cumulative plot, showing the total weight per cent of crystals, finer or coarser than a given size, may be constructed. 

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For a single fluid flowing through a section of reservoir rock, Darcy showed that the superficial velocity of the fluid (u) is proportional to the pressure drop applied (the hydrodynamic pressure gradient), and inversely proportional to the viscosity of the fluid. The constant of proportionality is called the absolute permeability which is a rock property, and is dependent upon the pore size distribution. The superficial velocity is the average flowrate... 

Rowell and co-workers [62-64] have developed an electrophoretic fingerprint to uniquely characterize the properties of charged colloidal particles. They present contour diagrams of the electrophoretic mobility as a function of the suspension pH and specific conductance, pX. These fingerprints illustrate anomalies and specific characteristics of the charged colloidal surface. A more sophisticated electroacoustic measurement provides the particle size distribution and potential in a polydisperse suspension. Not limited to dilute suspensions, in this experiment, one characterizes the sonic waves generated by the motion of particles in an alternating electric field. O Brien and co-workers have an excellent review of this technique [65]. 

SANS Small-angle neutron scattering [175, 176] Thermal or cold neutrons are scattered elastically or inelastically Incident-Beam Spectroscopy Surface vibrational states, pore size distribution suspension structure... 

Emulsion A has a droplet size distribution that obeys the ordinary Gaussian error curve. The most probable droplet size is 5 iim. Make a plot of p/p(max), where p(max) is the maximum probability, versus size if the width at p/p(max) = j corresponds to... 

The method to be described determines the pore size distribution in a porous material or compacted powder surface areas may be inferred from the results. 

We have considered briefly the important macroscopic description of a solid adsorbent, namely, its speciflc surface area, its possible fractal nature, and if porous, its pore size distribution. In addition, it is important to know as much as possible about the microscopic structure of the surface, and contemporary surface spectroscopic and diffraction techniques, discussed in Chapter VIII, provide a good deal of such information (see also Refs. 55 and 56 for short general reviews, and the monograph by Somoijai [57]). Scanning tunneling microscopy (STM) and atomic force microscopy (AFT) are now widely used to obtain the structure of surfaces and of adsorbed layers on a molecular scale (see Chapter VIII, Section XVIII-2B, and Ref. 58). On a less informative and more statistical basis are site energy distributions (Section XVII-14) there is also the somewhat laige-scale type of structure due to surface imperfections and dislocations (Section VII-4D and Fig. XVIII-14). 

Brunauer and co-workers [211, 212] proposed a modelless method for obtaining pore size distributions no specific capillary shape is assumed. Use is made of the general thermodynamic relationship due to Kiselev [213]... 

Most microporous adsorbents have a range of micropore size, as evidenced, for example, by a variation in or in calorimetric heats of adsorption with amount adsorbed [227]. As may be expected, a considerable amount of effort has been spent in seeing how to extract a size distribution from adsorption data. 

Fig. XVII-31. (a) Nitrogen adsorption isotherms expressed as /-plots for various samples of a-FeOOH dispersed on carbon fibers, (h) Micropore size distributions as obtained by the MP method. [Reprinted with permission from K. Kaneko, Langmuir, 3, 357 (1987) (Ref. 231.) Copyright 1987, American Chemical Society.]...

Another important characteristic of the late stages of phase separation kinetics, for asynnnetric mixtures, is the cluster size distribution fimction of the minority phase clusters n(R,z)dR is the number of clusters of minority phase per unit volume with radii between R and + cW. Its zeroth moment gives the mean number of clusters at time r and the first moment is proportional to die mean cluster size. 

For a general dimension d, the cluster size distribution fiinction n(R, x) is defined such that n(R, x)dR equals the number of clusters per unit volume with a radius between andi + dR. Assuming no nucleation of new clusters and no coalescence, n(R, x) satisfies a continuity equation... 

Figure A3.3.11 The asymptotic cluster size distribution f(x) from LS analysis for

It has been shown that spherical particles with a distribution of sizes produce diffraction patterns that are indistingiushable from those produced by triaxial ellipsoids. It is therefore possible to assume a shape and detemiine a size distribution, or to assume a size distribution and detemiine a shape, but not both simultaneously. 

Morris K F and Johnson C S Jr 1993 Resolution of discrete and continuous molecular size distributions by means of diffusion-ordered 2D NMR spectroscopy J. Am. Chem. See. 115 4291-9... 

Figure Bl.14.6. J -maps of a sandstone reservoir eore whieh was soaked in brine, (a), (b) and (e), (d) represent two different positions in the eore. For J -eontrast a saturation pulse train was applied before a standard spin-eeho imaging pulse sequenee. A full -relaxation reeovery eiirve for eaeh voxel was obtained by inerementing the delay between pulse train and imaging sequenee. M - ((a) and (e)) and r -maps ((b) and (d)) were ealeulated from stretehed exponentials whieh are fitted to the magnetization reeovery eurves. The maps show the layered stnieture of the sample. Presumably -relaxation varies spatially due to inliomogeneous size distribution as well as surfaee relaxivity of the pores. (From [21].)...

As an example figure B 1.14.13 shows the droplet size distribution of oil drops in the cream layer of a decane-in-water emulsion as determined by PFG [45]. Each curve represents the distribution at a different height in the cream with large drops at the top of the cream. The inset shows the PFG echo decay trains as a fiinction of... 

Figure Bl.14.13. Derivation of the droplet size distribution in a cream layer of a decane/water emulsion from PGSE data. The inset shows the signal attenuation as a fiinction of the gradient strength for diflfiision weighting recorded at each position (top trace = bottom of cream). A Stokes-based velocity model (solid lines) was fitted to the experimental data (solid circles). The curious horizontal trace in the centre of the plot is due to partial volume filling at the water/cream interface. The droplet size distribution of the emulsion was calculated as a fiinction of height from these NMR data. The most intense narrowest distribution occurs at the base of the cream and the curves proceed logically up tlirough the cream in steps of 0.041 cm. It is concluded from these data that the biggest droplets are found at the top and the smallest at the bottom of tlie cream.

MoDonald P J, Ciampi E, Keddie J L, Fleidenreioh M and Kimmioh R, Magnetio resonanoe determination of the spatial dependenoe of the droplet size distribution in the oream layer of oil-in-water emulsions evidenoe for the effeots of depletion floooulation Rhys. Rev. E, submitted... 

Horvath G and Kawazoe K 1983 Method for oaloulation of effeotive pore size distribution in moleoular sieve oarbon J. Chem. Eng. Japan 16 470-5... 

Even when carefully prepared, model colloids are almost never perfectly monodisperse. The spread in particle sizes, or polydispersity, is usually expressed as the relative widtli of tire size distribution,... 

Physical properties affecting catalyst perfoniiance include tlie surface area, pore volume and pore size distribution (section B1.26). These properties regulate tlie tradeoff between tlie rate of tlie catalytic reaction on tlie internal surface and tlie rate of transport (e.g., by diffusion) of tlie reactant molecules into tlie pores and tlie product molecules out of tlie pores tlie higher tlie internal area of tlie catalytic material per unit volume, tlie higher the rate of tlie reaction... 

Madsen C and Jacobsen C J FI 1999 Nanosized zeolite crystals—convenient control of crystal size distribution by confined space synthesis Chem. Commun. 673-4... 

Wlrile size distribution is important, control over tire nanocrystal surface is equally important. The best nanocrystal syntlieses provide avenues for nanocrystals to be purified, collected as powders, and tlien redissolved in appropriate solvents. This requires control over tire surface chemistry, in order to control tire solubility of tire nanocrystals. 

Figure C2.17.4. Transmission electron micrograph of a field of Zr02 (tetragonal) nanocrystals. Lower-resolution electron microscopy is useful for characterizing tire size distribution of a collection of nanocrystals. This image is an example of a typical particle field used for sizing puriDoses. Here, tire nanocrystalline zirconia has an average diameter of 3.6 nm witli a polydispersity of only 5% 1801.

Peng Z G, Wickham J and Alivisatos A P 1998 Kinetics of ll-VI and lll-V colloidal semiconductor nanocrystal growth focusing of size distributions J. Am. Chem. Soc. 120 5343... 

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