Crystal growth overall

The fundamental basis of the sonoelectrochemical technique to form nanoparticles is massive nucleation using a high current density electrodeposition pulse (ca. 150-300 mA cm ), followed by removal of the deposit from the sonoelectrode by the sonic pulse. Removal of the particles from the electrode before the next current pulse prevents crystal growth. Overall there are many experimental variables involved in sonoelectrochemical deposition electrolyte composition and temperature, electrodeposition conditions including current density (le), pulse-on time (te(on)) and ratio between pulse-on time and pulse-off time (te(off)) (the duty cyde) sonic probe conditions sonic power (Is), sonic pulse parameters, fs(on) and ts(off). 

Crystal growth rate may be expressed either as a rate of linear inerease of eharaeteristie dimension (i.e. veloeity) or as a mass deposition rate (i.e. mass flux). Expressed as a veloeity, the overall linear erystal growth rate, G (=dL/dt where L is the eharaeteristie dimension that is inereasing). The rate of ehange of... 

A comparative study [10] is made for crystal-growth kinetics of Na2HP04 in SCISR and a fluidized bed crystallizer (FBC). The details of the latter cem be found in [11]. Experiments are carried out at rigorously controlled super-saturations without nucleation. The overall growth rate coefficient, K, are determined from the measured values for the initial mean diameter, t/po, masses of seed crystals before and after growth. The results show that the values for K measured in ISC are systematically greater than those in FBC by 15 to 20%, as can be seen in Table 2. On the other hand, the values for the overall active energy measured in ISC and FBC are essentially the same. 

Secondary nucleation is essentially a crystal growth process. Secondary nucleation occurs by the deposition of a stem of the polymer molecule on a preexisting crystal-face as shown in Figure 15. The overall rate of this process is given by the following expression [58],... 

It has been reported that the overall rate of crystallization of pure PHB is relatively low compared to that of common synthetic polymers, showing a maximum in the temperature range of 55-60°C [23]. The spherulite growth rate kinetics have been evaluated [59] in terms of the theory by Hoffmann et al. [63], At about 90 °C, the spherulite growth rate displayed a maximum, which is not excessively low compared to that of common synthetic polymers. Therefore it was stated that the low overall crystallization rate of PHB centers on the nuclea-tion process rather than the subsequent crystal growth. Indeed, it has been shown that PHB has an exceptionally low level of heterogeneous nuclei [18]. 

The SHG active compounds found in Tables II-VIII are just a small fraction of the hundreds of substances we have screened by the powder technique. Of the active compounds only another small fraction will provide crystals of size and quality as is required for optical application. A program in crystal growth as thorough as the effort which has gone into synthesis must be instituted in order to take full advantage of these materials. Overall, it will suffice to say that the promising nonlinear properties observed from such a small fraction of all the possible organic compounds bodes very well for future endeavor. 

The overall rate of crystallization is determined by both the rate of nuclei formation and by the crystal growth rate. The maximum crystal growth rate lies at temperatures of between 170 and 190 °C [71, 72], as does the overall crystallization rate [51, 61, 75], The former is measured using hot stage optical microscopy while the latter is quantified by the half-time of crystallization. Both are influenced by the rate of nucleation on the crystal surface and the rate of diffusion of polymer chains to this surface. It has been shown that the spherulite growth rate decreases with increasing molecular weight due to the decrease in the rate of diffusion of molecules to this surface [46, 50, 55, 71, 74],... 

As with nucleation, classical theories of crystal growth 3 20 2135 40-421 have not led to working relationships, and rates of crystallisation are usually expressed in terms of the supersaturation by empirical relationships. In essence, overall mass deposition rates, which can be measured in laboratory fluidised beds or agitated vessels, are needed for crystalliser design, and growth rates of individual crystal faces under different conditions are required for the specification of operating conditions. 

Eliminating c and introducing an overall crystal growth coefficient, KG gives the approximate relation ... 

The exponents i and s in equations 15.13 and 15.14, referred to as the order of integration and overall crystal growth process, should not be confused with their more conventional use in chemical kinetics where they always refer to the power to which a concentration should be raised to give a factor proportional to the rate of an elementary reaction. As Mullin(3) points out, in crystallisation work, the exponent has no fundamental significance and cannot give any indication of the elemental species involved in the growth process. If i = 1 and s = 1, c, may be eliminated from equation 15.13 to give ... 

Methods used for the measurement of crystal growth rates are either a) direct measurement of the linear growth rate of a chosen crystal face or b) indirect estimation of an overall linear growth rate from mass deposition rates measured on individual crystals or on groups of freely suspended crystals 35,41,47,48). 

Because the rate of growth depends, in a complex way, on temperature, supersaturation, size, habit, system turbulence and so on, there is no simple was of expressing the rate of crystal growth, although, under carefully defined conditions, growth may be expressed as an overall mass deposition rate, RG (kg/m2 s), an overall linear growth rate, Gd(= Ad./At) (m/s) or as a mean linear velocity, // (= Ar/At) (m/s). Here d is some characteristic size of the crystal such as the equivalent aperture size, and r is the radius corresponding to the... 

The overall rate of growth depends upon the slowest step in this sequence. Crystal growth may be controlled either by the transport processes or by the chemical reaction at the surface. The mechanism of growth and the resulting crystal morphology... 

The overall process of metal deposition and crystal growth involves several steps. One is the diffusion of ions in the solution to the metal surface. Another is the cathodic deposition step, i.e., the removal of the ion across the interfacial region to land somewhere on a terrace on the metal surface. 

It was possible for two of the systems chosen that the nucleation and crystallization activation energies could be determined separately by distinguishing the induction period and crystal growth period in the overall crystallization process. Of the two hypotheses proposed for zeolite crystallization, in the gel phase or from the solution phase, the data support the latter hypothesis for crystal growth with the crystal-liquid surface enhancing the nucleation process in seeded systems. The precise mechanism of nucleation in unseeded systems remains to be determined. 

As part of the overall scale-inhibiting threshold effect, the mechanism of crystal growth retardation is complemented by crystal distortion, whereby the crystal fails to successfully agglomerate into a larger crystalline scale. 

In order to avoid other complicated factors concealing the major problems being examined, the overall crystal-growth rate equation, Eq. (12.3), is assumed to be valid for the system under consideration and is used as the basis for the interpretation of data. 

Further, substituting Eq. (12-8) into Eq. (12-3), integrating the resulting equation between t - 0 and t = tf and rearranging yields the expression for the overall crystal-growth rate coefficient as... 

To ensure that all the overall crystal-growth rate coefficients are measured under the conditions without nucleation, the metastable region of the solution has to be determined first and therefore the solubility and super solubility need to be measured. 

For comparison, the experiments for measuring the overall crystal-growth rate coefficient are carried out in an impinging stream crystallizer (ISC) and a fluidized bed crystallizer (FBC). 

The values measured in the ISC for the overall crystal-growth rate coefficient of Na2HP04 are listed in Table 12.2. As can be seen, the reproducibility of the data is in a reasonable range and that of most data is very good. 

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