Metastability, limit

The metastable limit can provide an empirical approach to modeling primary nucleation. This limit, which was first observed in 1951 (6), must be determined through experimentation, and nucleation rate is correlated with the following equation... 

Supersaturation reaches the metastable limit and nucleation is initiated supersaturation drops rapidly as crystals formed begin to grow... 

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. 

The time elapsed from the ereation of the initial supersaturation to the detee-tion of the first erystals formed in the system is known as the induetion period. The level of supersaturation attained is then akin to the metastable limit . Neither quantity (viz. the induetion time and metastable limit) is therefore a fundamental quantity. Both are useful measures, however, of the propensity of a solution to nueleate. Measurement of the induetion time as a funetion of supersaturation ean be used to help determine erystallization kineties and meehanism. Thus, the induetion time may be expressed by (Walton, 1967)... 

Now, if nueleation ean be eonsidered negligible by operation at a low level of supersaturation then the eooling eurve needed to maintain the supersaturation within the metastable limit may be expressed by (Mullin and Nyvlt, 1971)... 

FIG. 2 Growth rates as a function of the driving force A//. Comparison of theory and computer simulation for different values of the diffusion length and at temperatures above and below the roughening temperature. The spinodal value corresponds to the metastability limit A//, of the mean-field theory [49]. The Wilson-Frenkel rate WF is the upper limit of the growth rate. 

Hydrate nucleation and growth may have direct analogies in crystallization processes such as the precipitation of salt from solution. Metastability in salt crystallization was hypothesized to occur through supersaturation by Ostwald (1900). (A supersaturated solution is one in which the liquid [solvent] contains more dissolved solute than can be ordinarily accommodated at that temperature the greater the degree of supersaturation, the greater number of crystal nuclei that will form in solution.) Miers and Isaac (1907) experimentally proved metastability and postulated that for each solute-solvent pair, a concentration-temperature relationship exists that defines the metastable limit, formally called the thermodynamic spinodal. 

The classical nucleation theory embodied in Eq. (16) has a number of assumptions and physical properties that cannot be estimated accurately. Accordingly, empirical power-law relationships involving the concept of a metastable limit have been used to model primary nucleation kinetics ... 

Figure 1.32a. Schematic representation of the adsorption (1) - (3) and desorption (4) - (6) of a fluid from the gaseous phase in a cylindrical pore with radius a-l (1) stable adsorbed film with radius a - t (2) multilayer adsorbed film at the unstabillty limit r (3) completely filled capillary (4) unsymmetrical state of a partially filled pore at the metastability limit r (5) further desorption at the metastability limit r (6) stable film with radius r. " "...

Figure 1.33. Dependence of the unstable limit r /a and the metastable limit r /a on the dimensionless variable a/a according to the theory by Saam and Cole ...

The structural correlations are strongly enhanced in the under-cooled state as the temperature is reduced towaids the metastable limit of -40°C (to D2O) and various thermoph ical properties exhibit diverged behaviour [8]. The exact nature of this anomaly is still the subject of some controversy. However, the difiraction pattern indicates that the stmcture is evolving towards that of amorphous ice which is characterised as a continuous random networit of tetrahedral hydrogen-bonds [9]. Recent neutron measurements on amorphous ice [10] have re-infor the earlier conjectures tuid shown that the structure is similar to that of hyper-quenched glassy water produced by rapid cooling of micron-sized water droplets. It can now be realised that the CRN mo l for the disordered phase of ice is effectively the limiting stmcture of water at low temperatures. 

From a practical perspective, the initially slow cooling rate provides enough supersaturation to balance growth. Typically, this would help in producing a small number of larger crystals compared to the case when the cooling rate is initially high. In the latter case, the solution will tend to cross the metastable limit for the system. 

Fig. 7 Schematic of the metastable zone, which is the region surrounded by the solubility curve and the metastable limit. This is the operating region for most seeded batch crystallization.

Fig. 9 Steps for automated determination of metastable zone using ATR-FTIR and FBRM. While automatically collecting the IR spectra for calibration, the metastable limit is determined using FBRM. Then the model for relating the IR spectra to solution concentration is constructed using multivariate analysis such as principal component regression (PCR) or partial least squares (PLS). Using this model, the solubility curve can be obtained from the IR spectra of saturated slurry.

The use of ATR-FTIR spectroscopy is not a requirement for the determination of the metastable zone. If the goal is the determination of metastable zone or solubility curve alone, then there are less technically complicated methods, such as the gravimetric method for solubility measurement and observation by eye for detection of the metastable limit. Even for the automation of the system, ATR-FTIR spectroscopy is not a requirement. Automated determination... 

In the direct design approach, a desired supersaturation profile that falls between the solubility curve and the metastable limit of the system is followed based on feedback control of the concentration measurement. This is in contrast to the traditional first-principles approach, where a desired temperature profile or antisolvent addition rate profile is followed over time such as shown in Fig. 14. For a cooling crystallization, the direct design approach follows a setpoint profile that is solution concentration vs. temperature (or solvent-antisolvent ratio) as opposed to temperature (or addition rate) vs. time. Because the desired crystallizer temperature is determined from an in-situ solution concentration measurement, the batch time is not fixed. 

Recently Klein and Leyvraz considered nonclassical nucleation near a spinodal using a model with weak long-range repulsive potentials, and showed that it differed substantially from the classical picture, as expected from the considerations of Harrowell and Oxtoby. However, it is unclear whether real systems behave in such a way as to show the influence of a metastability limit for the undercooled liquid. 

A mechanism for oiling out can be postulated as follows When supersaturation is achieved rapidly such that the concentration is beyond the upper metastable limit—as can often be the case in a nucleation-based process—the substrate is forced to separate into a second phase by the creation of the resulting high solution concentration. However, crystallization is delayed by a slow crystallization rate. This combination may result in the creation of a nonstructured oil or possibly an amoiphous solid. The rates of phase separation and nucleation are relative to each other such that slow nucleation implies only that nucleation was not fast enough to create discrete particles before oil separation. 

Crystallization is thus a two-step process nucleation and growth. The first step is nucleation, which occurs spontaneously when the supersaturation attains the metastable limit at the spinodal curve. Knowledge of the spinodal curve is useful in understanding the mechanism of crystallization. However, the actual metastable limit may often exceed the spinodal curve in certain systems, and in such cases the phase transition mechanism is explained in terms of spinodal decomposition (9) or the solution history, impurities present, or rate of increase of supersaturation. 

Abstract The virtual terms binodal and spinodal are equivalent to the experimental terms eoexistence curve (CXC) and metastability limit (ML), respectively, within an inherent accuracy of any semi-empirical EOS at the description of a real fluid behavior. Any predicted location of mechanical spinodal at positive pressures Psp(T)>0 merits verification because the Maxwell rule is a model (EOS)-dependent method based on the non-measurable values of chemical potential for both phases. It is not a reliable tool of CXC- and ML-prediction especially at low temperatures between the triple and normal boiling ones [Tt,Ti where the actual vapor pressures Ps(T) > 0 are quite small while... 

Keywords disorder parameter, order parameter, metastability limit, asymmetry, quasi-binodal, particle-hole-type symmetiy... 

Temperature has a great influence on the initial reaction rate. Under conditions of low slurry density, where the concentration of magnesium bicarbonate does not reach its solubility limit, the rate of dissolution of MgO increases with increasing temperature. However, under conditions where the concentration of magnesium bicarbonate reaches its metastable limit, the maximum solution of Mg(OH)2 is reached at about 15°C, after which magnesium carbonate starts to precipitate. 

The metastable limit for slurry density is about 10 g/L MgO in solution. 

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