Nucleation order

Dateralination of i usually is conducted in small-scale laboratory equipment. Typically, a 4-8 L MSMPR crystallizer is used for this purpose. Other techniques for obtaining nucleation order indnde carefully conducted pilot or foil-scale tests, mininuclealor tests (sse Randolph and Larson in the General Bibliography), and repotted deta in the literature. 

Thus, a plot of the cooling rate and the maximum possible supercooling will give the apparent order m and the rate constant k - If the nuclei are visible immediately after they are formed, the true order (n) and the apparent order (m) are identical. In all other cases, the true nucleation order can be found by knowing the exponent for crystal growth (g). The relationship is given by... 

Figure 2.29 Nomograph for the estimation of the apparent nucleation order. The ratio of the cooling rates and the observed widths of the metastable zone are denoted as points on the left and right axes, respectively. The apparent nucleation order is given by the value at which the straight line joining those points intersects the diagonal. (Reproduced with permission from Nyvlt et al. 1985.)...

During nucleation ordered nuclei are formed as a result of matter fluctuations. If these are formed in the volume of ideally pure solutions, for example, containing only the protein and solvent molecules, the process is called homogeneous nucleation. If monomers nucleate on the surface of seeds added to the solution or on substrates that provide active centers for nucleation, it is referred as heterogeneous nucleation (Figure 3.2). 

The stability of a crystallizing system tends to increase with increasing crystal growth rate and magma density, and with decreasing nucleation order , minimum product size, and the relative amount of crystals withdrawn. Continuous seeding has the same effect as decreasing the effective nucleation order, and should lead to a stabilization of the system. 

The entropically driven disorder-order transition in hard-sphere fluids was originally discovered in computer simulations [58, 59]. The development of colloidal suspensions behaving as hard spheres (i.e., having negligible Hamaker constants, see Section VI-3) provided the means to experimentally verify the transition. Experimental data on the nucleation of hard-sphere colloidal crystals [60] allows one to extract the hard-sphere solid-liquid interfacial tension, 7 = 0.55 0.02k T/o, where a is the hard-sphere diameter [61]. This value agrees well with that found from density functional theory, 7 = 0.6 0.02k r/a 2 [21] (Section IX-2A). 

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... 

The growth of a well ordered fullerene monolayer, by means of molecular beam epitaxy, has been used for the controlled nucleation of single crystalline thin films. The quality and stability of molecular thin films has been shown... 

However, in the case of mini- and microemulsions, processing methods reduce the size of the monomer droplets close to the size of the micelle, leading to significant particle nucleation in the monomer droplets (17). Intense agitation, cosurfactant, and dilution are used to reduce monomer droplet size. Additives like cetyl alcohol are used to retard the diffusion of monomer from the droplets to the micelles, in order to further promote monomer droplet nucleation (18). The benefits of miniemulsions include faster reaction rates (19), improved shear stabiHty, and the control of particle size distributions to produce high soHds latices (20). 

Precipitation. The precipitation of aluminum tribydroxide ia the recovery step of the Bayer process is achieved either by loweting the temperature or by diluting the pregnant Hquor and reduciag its pH. Both methods reverse the direction of equation 35, but seeding with previously precipitated crystals is required ia order to initiate nucleation. 

Polypropylene molecules repeatedly fold upon themselves to form lamellae, the sizes of which ate a function of the crystallisa tion conditions. Higher degrees of order are obtained upon formation of crystalline aggregates, or spheruHtes. The presence of a central crystallisation nucleus from which the lamellae radiate is clearly evident in these stmctures. Observations using cross-polarized light illustrates the characteristic Maltese cross model (Fig. 2b). The optical and mechanical properties ate a function of the size and number of spheruHtes and can be modified by nucleating agents. Crystallinity can also be inferred from thermal analysis (28) and density measurements (29). 

Nucleation. Crystal nucleation is the formation of an ordered soHd phase from a Hquid or amorphous phase. Nucleation sets the character of the crystallization process, and it is, therefore, the most critical component ia relating crystallizer design and operation to crystal size distributions. 

Both homogeneous and heterogeneous mechanisms requite relatively high supersaturation, and they exhibit a high order dependence on supersaturation. These factors often lead to production of excessive fines ia systems where primary aucleatioa mechanisms are important. The classical theoretical treatment of primary nucleation results ia the expressioa (5) ... 

In order to treat crystallization systems both dynamically and continuously, a mathematical model has been developed which can correlate the nucleation rate to the level of supersaturation and/or the growth rate. Because the growth rate is more easily determined and because nucleation is sharply nonlinear in the regions normally encountered in industrial crystallization, it has been common to... 

Crystal growth is a layer-by-layer process, and the retention time required in most commercial equipment to produce crystals of the size normally desired is on the order of 2 to 6 h. On the other hand, nucleation in a supersaturated solution can be generated in a fraction... 

In processes where new powder feed has a much smaller particle size than the smallest granular product, the feed powder can be considered as a continuous phase which can nucleate to form new granules [Sastry Fuerstenau, Powder Tech., 7, 97 (1975)]. The size of the nuclei is then related to nucleation mechanism. In the case of nucleation by spray, the size of the nuclei is of the order of the droplet size and proportional to cos0, where 0 is binder fluid-particle contact angle (see Fig. 20-67 of Wetting section). 

The importance of inherent flaws as sites of weakness for the nucleation of internal fracture seems almost intuitive. There is no need to dwell on theories of the strength of solids to recognize that material tensile strengths are orders of magnitude below theoretical limits. The Griffith theory of fracture in brittle material (Griflfith, 1920) is now a well-accepted part of linear-elastic fracture mechanics, and these concepts are readily extended to other material response laws. 

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