MECHANISM OF CRYSTALLIZATION SUPERSATURATION Non equilibrium NUCLEATION CRYSTAL GROWTH Equilibrium CRYSTAL
How dos nucleation occur? NUCLEATION SPONTANEOUS ARTIFICIAL • Agitation • Mechanical shock • Friction • Electric/magnetic field • Sonic/ultrasonic irradiation • Etc.
HOMOGENEOUS NUCLEATION The classical theory of nucleation assumes that clusters are formed in solution by an addition mechanism: A + A A2 A2 + A A3 A3 + A A4 . . . . . . . . . . . An-1 + A An (critical cluster) that continues until a critical size is reached (n = tens to thousands)
CLASSICAL THEORY OF NECLEATION The overall excess free energy, G, between a small solid particle of solute (assumed to be a sphere of radius r) and the solute in the solution is defined as: (1) where GS :surface excess free energy, i.e. the excess free energy between the surface of the particles and the bulk of the particle) GV :volume excess free energy, i.e. the excess free energy between a very large particle, r = , and the solute in solution.
GS is a positive quantity and is proportional to r2. In a supersaturated solution GS is a negative quantity and is proportional to r3. (2) where Gv :free energy change of the transformation per unit volume (a negative quantity) :surface energy, i.e. interfacial tension between the developing crystalline surface and the supersaturated solution in which it is located The two terms on the right-hand side of eq. (2) are of opposite sign and depend differently on r, so G passes through a maximum.
This maximum value, Gcrit, corresponds to the critical nucleus, rc. For spherical cluster, it is obtained my maximizing eq. (2), setting d(G)/dr = 0: (3) Where Gv is a negative quantity. From eqs. (2) and (3): (4) Particles smaller than rc will dissolve, or evaporate if the particle is a liquid in a supersaturated vapor, because only in this way can the particle achieve a reduction in its free energy. Similarly, particles larger than rc will continue to grow.
The rate of nucleation, B0, e.g. the number of nuclei formed per unit time per unit volume, can be expressed in the form of the Arrhenius type of expression: (5) k : Boltzmann constant = 1.3805 10-23 J K-1 A : pre-exponential factor (theoretical value = 1030 nuclei/cm3 s) where The basic Gibbs-Thomson relationship for a non-electrolyte: (6) S : supersaturation ratio = c/c* v : molecular volume where
From eq. (6): Introducing the above equation into eq. (3) gives: (7) Introducing eq. (7) into eq. (4) gives: (8) And from eq. (5) gives: (9)
Induction Time for Nucleation of Water Vapor In the case of nucleation of water vapor, a ‘critical’ super-saturation could be said to exist in the region of S ~ 4.0, but it is also clear that nucleation would have occurred at any value of S > 1 if sufficient time had been allowed to elapse.
The rate equation predicts exponential growth once a critical supersaturation is attained. • In practice, an optimal temperature exists below which the liquid is too viscous to nucleate and above which molecular motions prevent crystal formation. • The viscous effects can be incorporated into the rate equation by taking into account the viscous free energy (10) • This was observed by Tamman (1925) for several organic salts, and Mullin and Leci (1969) for the spontaneous nucleation of citric acid solutions and is shown in the following figure.
Spontaneous nucleation in supercooled citric acid solution: (A) 4.6 kg of citric acid monohydrate/kg of "free" water (T= 62 °C) and (B) 7.0 kg/kg (T= 85 °C).
An empirical approach to the nucleation process is described by Nielsen (1964), expressing a relationship between the induction period, tind (the time interval between mixing two reacting solutions and the appearance of crystals) and the initial concentration, c, of the supersaturation solution: (11) where k : constant p : number of molecules in a critical nucleus
HETEROGENEOUS NUCLEATION • A foreign substance present in a supersaturated solution is generally known to reduce the energy required for nucleation. • Nucleation in a heterogeneous system generally occurs at a lower supersaturation than a homogeneous system. • Partial attraction is possible in a case where the foreign substance and the crystal have almost identical atomic arrangement. • It was shown (Preckshot and Brown 1952) that the energy for nucleus formation was reduced only if the difference in iso-morphismbetween the crystal and the foreign particle was <15%. • For differences >15%, the energy requirements were similar to that for a homogeneous system.
SECONDARY NUCLEATION • Secondary nucleation results from the presence of crystals in the supersaturated solution. • These parent crystals have a catalyzing effect on the nucleation phenomena, and thus, nucleation occurs at a lower supersaturation than needed for spontaneous nucleation. • Although several investigations of secondary nucleation exist, the mechanisms and kinetics are poorly understood.
Strickland-Constable (1968) described several possible mechanisms of secondary nucleation: • ‘Initial’ breeding (crystalline dust swept off a newly introduced seed crystal) • ‘Needle’ breeding (the detachment of weak out-growths) • ‘Polycrystalline’ breeding (the fragmentation of weak polycrystalline mass) • ‘Collision’ breeding ( a complex process resulting from the interaction of crystals with one another or with parts of the crystalline vessel)
Factors Affecting Secondary NucleatiON The rate of secondary nucleation is governed by three processes: the generation of secondary nuclei on or near a solid phase; removal of the clusters; and growth to form a new solid phase Several factors influence these processes: the supersaturation, the rate of cooling, the degree of agitation, and the presence of impurities.
Supersaturation • The degree of supersaturation is the critical parameter controlling the rate of nucleation. • The size of the critical nucleus decreases with increasing super-saturation, thus, the probability of the nuclei surviving to form crystals is higher. • In general, nucleation rates are enhanced with increasing super-saturation. However, the nucleation exponent is found to be lower than that for primary nucleation
Temperature • The role of temperature in the production of secondary nuclei is not fully understood. • For several systems, the nucleation rate declined with increasing temperature for a given supersaturation. • A few contradictory results exist - Genck and Larson (1972) found a decrease in nucleation rate with increasing temperature for a potassium nitrate system and increasing rates with increasing temperature for a potassium chloride system. • It was shown by Nyvlt (1981) and others that the nucleation order is not sensitive to temperature variations.
Stirring • Stirring the solution leads to lower nucleation rates. • However, Sikdar and Randolph (1976) found that the nucleation rate increased with the degree of agitation for smaller crystals of magnesium sulfate (8-10 m) — the nucleation rates were independent of the degree of agitation for larger crystals. • The results of Melia and Moffit (1964a, 1964b) on the secondary nucleation of potassium chloride are shown in the following figure; they found that the nucleation rate increases with supersaturation, and the degree of supercooling and agitation
Dependence of number of secondary nuclei produced on stirrer speed and supercooling in secondary nucleation of potassium chloride
Hardness of the contact material • In general, it was found that a harder material is more effective in enhancing the nucleation rates. • For example, it was found that a polyethylene stirrer reduced the nucleation rates by a factor of 4-10, depending on the agitation • Crystal hardness also affects nucleation behavior — a hard, smooth crystal is less effective. • Irregular crystals with some roughness are generally more active.
Effect of agitator speed on secondary nucleation rate for steel and plastic impellers.
Impurity • It is well known that a small amount of impurity can profoundly affect the nucleation rate, however, it is impossible to predict the effect prior. • The presence of additives can either enhance or inhibit the solubility of a substance. • Enhanced solubilities would lead to lower supersaturations and lower growth rates. • The effect of impurities is complex and unpredictable.
NUCLEATION KINETICS • A general theory for the prediction of nucleation rates does not exist. • Several correlations based on the power law model have been found to explain most of the experimental data satisfactorily. • The power law is given by: (no/m3.s) (12) • This form is valid if the adsorption layer mechanism is the source of nuclei. • The nucleation rate in this case is independent of the suspension concentration.
In the industrial crystallizer, most of the nuclei are generated by contact with the crystallizer environment. • The nucleation rate in this case is a function of the degree of agitation, the suspension density, and the supersaturation. (13) where W :agitation rate (rpm) MT:suspension density (mass of crystals per volume of solution). • In some situations an equation that does not include the effect of agitation is used (14) In this case may vary with the agitation rate.
The kinetics for secondary nucleation can be measured either by measuring the width of the metastable zone, the induction time, or by counting the number of nuclei formed. • One of the methods for the determination of nucleation rates is by measuring the maximum possible supercooling that can be obtained in a saturated solution when it is cooled at different rates (metastable zone width measurement). • The polythermal experiment (proposed by Nyvlt 1968) is carried out in a jacketed crystallizer cooled by a circulating water/ethylene-glycol bath accurate to ± 0.1 °C. • The temperature can be increased or decreased at a constant rate by a programmed controller. • The crystallizer is fitted with an accurate thermometer ±0.1 °C to read the solution temperatures. • A schematic of the apparatus is shown in the following figure.
Approximately 200 ml of saturated solution is placed in a crystallizer and allowed to equilibrate thermally. • The solution is stirred at a constant rate and cooled slowly until a number of small crystals are formed. • The temperature of the solution is then raised at a very slow rate until the last crystal disappears. • This temperature is denoted as the saturation temperature, Ts. • The solution is then heated to a temperature 1° above Ts and maintained for 30 min. • The solution is now cooled at a constant rate (r1) and the temperature at which the first crystal appears is noted (T1). • The difference between this temperature and the saturation temperature is denoted as AT1max for the cooling rate r1. • The experiment is repeated for two different cooling rates.
The nucleation rate at the metastable limit can be approximated as (15) The maximum supersaturation is given by (16) Combining Eqs. (15), (16), and (12), simplifying, and taking logarithms, yields (17) where m is used in place of n to signify an apparent nucleation order.
The dependence of cooling rate on the maximum attainable supercooling in aqueous sodium chromate solutions
The rate of nucleation can also be determined by observing the time elapsed between the creation of supersaturation and the formation of a new phase. • This time interval is defined as the induction time and is a function of the solution temperature and supersaturation. • The formation of a new phase can be detected in several different ways—for example, by the appearance of crystals or by changes in properties (turbidity, refractive index) of the solution. • The induction time, tind is the sum of the time needed for reaching steady-state nucleation, ttr;the nucleation time, tn;and the time required for the critical nucleus to grow to a detectable size, tg. (18)
It can be shown (Sohnel and Mullin 1988) that the transient period (ttr) is unimportant in aqueous solutions of moderate supersaturations and viscosities. • In certain special cases however, at very low supersaturations the transient period cannot be ignored (Packter 1974) and the following analysis is not applicable. • Recently, Kubota et al. (1986) has suggested a method to take into account the transient period. • If the transient period can be ignored, the induction time is a function of the nucleation and growth times. Three cases emerge: tn >> tg tn~tg tn << tg
If tn >> tg , the induction time is inversely proportional to the steady-state nucleation rate and (19) where is the wetting angle that is 1 for homogeneous nucleation and is <1 for heterogeneous nucleation, and F is the shape factor ratio. If tn~ tg , the induction time for nucleation followed by diffusion growth is given by (20) where x is the solute mole fraction
If tn << tg , the induction times for various cases are given by Sohnel and Mullin (1988). For mononuclear growth, the induction time in this case is given by (21) where Ds is the surface diffusion coefficient and rc is the critical radius of the nuclei. In general, the induction time is given by the expression (22)
Induction period as a function of supersaturation for CaCO3 precipitation at 25 °C showing regions of homogeneous and heterogeneous nucleation.