Crystallization Kinetics
The time necessary to initiate nucleation of crystals from a supersaturated solution is called "induction time" (Reddy, 1995). It is a function of the solution supersaturation, that is, the ratio of the ion activity product to the solubility product of the precipitating crystalline matter as demonstrated in this article by Reddy (1995) for calcium carbonate nucleation in the presence of magnesium ions. This article indicates that the induction time is lower for higher supersaturation.
Below the supersaturation value of about 10, the induction time for calcium carbonate nucleation is very long. In this region, the solution is at a "metastable" condition and, therefore, calcium carbonate crystals cannot be formed without the aid of a matching growth surface or substrate (Reddy, 1995). It can also be observed that the presence of magnesium ions increases the induction time for calcium carbonate nucleation and therefore has a retardation and/or inhibition effect.
Reddy (1995) explains the magnesium ion inhibition of calcium carbonate nucleation by adsorption of the magnesium ions and thus, the occupation of some crystal growth sites on the calcium carbonate crystal surface. For a quantitative interpretation of this phenomenon, Reddy (1986; 1995) resorts to a growth rate analysis and a Langmuir adsorption isotherm model using experimental data obtained by a seeded growth method. He expressed the crystal growth rate as being proportional to the surface available for crystal growth and the square of the driving force for precipitation:
where N represents the calcium carbonate crystal concentration in the solution in moles/liter, t denotes the time measured from the time of
initiation of the crystallization by seeding, s is the concentration of the seed added to provide the surface area for growth in mg/liter, and k is the crystal growth-rate constant. If N0 denotes the initial theoretical crystal concentration that would be produced by precipitation from a stable supersaturated solution at the time of seeding, the integration of Eq. 9-20 yields (Reddy, 1986):
The plot of the calcium carbonate growth data given by Reddy (1995) in this page confirms the validity of Eq. 9-21 and indicates that the presence of magnesium ions reduces the slopes of the straight lines and, thus the crystallization rate constant and inhibits the calcite formation. Reddy (1995) shows a rapid decline of the crystallization rate constant by the increasing magnesium ions presence. The plot of data according to the Langmuir model
given in this article by Reddy (1995) clearly indicates that the mechanism of the inhibition of the calcite crystal growth is the magnesium ion adsorption on the growth sites, where ka and kd denote the rate constants for adsorption and desorption of the magnesium ions at the growth sites, k0 and k are the crystallization growth-rate constants without and with the presence of magnesium ions, and TM 2+ is the total concentration of the magnesium ions present in the system.
Particle Growth and Dissolution in Solution
The particle growth is assumed to occur at a rate proportional to the surface, Ac, available for growth and the deviation of the saturation ratio from unity (Chang and Civan, 1992; Civan, 1996):
for which the initial amount of crystals present per unit bulk medium is given by
Relating the crystal shape to spherical shape, the mass and surface area of the crystalline particle is given, respectively, by:
in which C{ and C2 are the shape factors, pc, is the density, and Dc is the diameter. kc is a crystallization rate constant. Thus, Eqs. 9-23 through 9-26 lead to the following model in which the shape factors and the constant ( y 2 ) have been incorporated into the constant k'c:
For example, Dc = 5 urn, and k = k'c(Fs-l)/pc is equal to 1.4 and 10.3 um/s for calcium° carbonate crystal growth at 25 and 50°C, respectively, using the Dawe and Zhang (1997) data.
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