Supersaturation dependence

To reduce or avoid agglomeration removal of certain impurities such as charged polymers could be effective - once identified, while to enhance agglomeration certain additives could be used as are commonly employed as flocculating agents to enhance solid-liquid separation in the water industry for example. These effects have to be determined empirically with care, however, since they can be pH and supersaturation dependent. 

Solutions can be unsaturated, saturated, or supersaturated, depending on how much solute is dissolved compared to the solubility of the solute in the solvent. 

At some level of calcite supersaturation, depending on the concentration and reactivity of inhibitors and other kinetic factors, calcite tends to nucleate and precipitate. Herman and Lorah (1987) noted previous field studies in which calcite precipitation did not occur until CO2 outgassing had raised calcite supersaturation levels to 5 or 10 times above equilibrium. In their own work on Falling... 

There are various models for the potential (i.e. supersaturation) dependence of the heterogeneous nucleation rate. According to the small cluster model developed by Walton (21) and Stoyanov (22), the formation of a cluster can be treated as a sequence of attachment and detachment steps. In equilibrium, the attachment and detachment rates are equal, whereas supersaturation leads to an increase in the attachment rate and growth of the cluster. The result of this theoretical analysis is the following expression for the nucleation rate, Jnuci (15) ... 

Commercial crystallizers may operate either continuously or batchwise. Except for special applications, continuous operation is preferred. The first requirement of any crystallizer is to create a supersaturated solution, because crystallization cannot occur without supersaturation. Three methods are used to produce supersaturation, depending primarily on the nature of the solubility curve of the solute. (1) Solutes like potassium nitrate and sodium sulfite are much less soluble at low temperatures than at high temperatures, so supersaturation can be produced simply by cooling. (2) When the solubility is almost independent of temperature, as with common salt, or diminishes as the temperature is raised, supersaturation is developed by evaporation. (3) In intermediate cases a combination of evaporation and cooling is effective. Sodium nitrate, for example, may be satisfactorily crystallized by cooling without evaporation, evaporation without cooling, or a combination of cooling and evaporation. 

Note that supersaturations depend both on the macroscale cloud dynamics represented by the cloud updraft velocities (larger updrafts result in higher supersaturations) and on the... 

Figure 3.23 Illustration of the variation of the face growth rale of a paraffin (C36H74) crystal in the presence of the impurity dioctadecylamine (Ci8H37)2NH. (From Supersaturation Dependence of Crystal Growth in the Presence of Impurity , J. Crysl. Growth 182, pp. 86 94. Used by permission of Elsevier Science, 1997.)...

The quantity that changes most in example 1 is Ac neither S nor soluble substances considerable changes can occur in all expressions of supersaturation depending on the concentration units used, as seen in example 2 where [Pg.126]

Direct observation of impact-induced microattrition at the surfaces of potash alum crystals immersed in supersaturated solution (Garside, Rush and Larson, 1979) indicated that the majority of the fragments produced were in the 1-10 pm size range and had a supersaturation-dependent size distribution. Impact energy and the frequency of impact also have an important influence on the number of crystals resulting from contact secondary nucleation (Larson, 1982). 

Furthermore, he gave evidence that silica powders put into the perchlorate solu tion first formed a supersaturated solution in 10 days. The degree of supersaturation depended on the amount of silica that had been added to the system. The supersaturation was higher with silica preheated to higher temperature (up to 700°C). However, silica that had previously been equilibrated with the solution did not cause supersaturation and followed the dissolution rate... 

The time dependency of the particle growth and nucleation rates can be converted to their rightful dependence on supersaturation when estimates of the latter are available from experimental measurements at different times. Among other interesting attributes of this technique, the use of inverse problem strategies to identify the characterization in (6.3.4) from estimated nucleation rates deserves special mention. Thus, based on the assumption that the supersaturation dependence can be described by a power law, identification of the optimum exponent and the function p l) have been possible. The method is under active investigation by Mahoney (2000). 

As higher the supersaturations are, as faster the crystal growth is and as more effective the crystalliser. Certainly, not any supersaturation can be chosen, because also the nucleation processes are supersaturation dependent. There is the spontaneous or primary nucleation which is caused by a critical height of supersaturation... 

Nucleation — Atomistic theory of nudeation — Figure 2. Supersaturation dependence of the stationary nucleation rate /o according to the atomistic theory of nucleus formation (a schematic representation)... 

Apart from the purely thermodynamic analysis, the description of the electrocrystallization phenomena requires special consideration of the kinetics of nucleus formation [i-v]. Accounting for the discrete character of the clusters size alteration at small dimensions the atomistic nucleation theory shows that the supersaturation dependence of the stationary nucleation rate Jo is a broken straight line (Figure 2) representing the intervals of A p within which different clusters play the role of critical nuclei. Thus, [A//, A/l"] is the supersaturation interval within which the -atomic cluster is the critical nucleus formed with a maximal thermodynamic work AG (nc). 

Figure 10.3 shows the schematic of solubility curve and metastable hmit. The metastable zone is shown between the solubility curve and metastable limit. Although supersaturation depends on solubility and supersaturation occurs when AC is greater than zero, nuclei may start forming before the supersaturation at any point in the metastable zone. The metastable zone width (MSZW) varies depending on the system being studied. It is usually quite narrow for small ionic crystals, such as NaCl, but can be much wider for organic molecules, such as citric acid. 

Description of the thermodynamic state of the adsorbate needs further consideration. Foremost it is necessary to stress that equation (1.19) represents a general expression for the supersaturation dependence of the activity of adatoms on the inert foreign substrate. In order to derive an explicit formula for the surface concentration 2 of adatoms, it is necessary to find out the interrelation between the activity adsorbed atoms, and the number N of adsorption sites on the foreign substrate (see e.g. [1.5-1.7] and the references cited therein). The result will depend on how the tree energy of the adsorbed metal phase is calculated, and here we present three examples for the functional relationship No). 

Finally, equation (1.32) shows that given the supersaturation the work of formation of any n-atomic cluster of the new phase can be found if ( ) is calculated taking into account the supersaturation dependence of the nucleus size n. In what follows we shall show how this is done in the framework of the classical nucleation theory. 

The general formula for the nucleation work AG(n) = -nAji +0 n) (equation (1.32)) provides the possibility to obtain explicit expressions for this quantity if the surface free energy 0(n) is evaluated accounting for the supersaturation dependence of the nucleus size MAji). To illustrate the thermodynamic method developed by Gibbs [1.2] and Volmer [1.11,1.16] we shall calculate the woik of formation of two-dimensional (2D) and three-dimensional (3D) liquid and crystalline nuclei on flat foreign substrates. For the sake of simplicity in these calculations we shall consider the idealized case of a stmctureless substrate thus neglecting any lattice mismatch between the electrode surface and the nuclei of the new phase. 

In Chapter 1 we considered the equilibrium properties of small clusters and derived explicit thermodynamic expressions for the work AG(r ) of nucleus formation. We hould emphasize, however, that giving us the value of the energy barrier AG n) thermodynamics do not say anything about the rate J of appearance of nuclei within the supersaturated parent phase. The reason is that this important physical quantity depends on the mechanism of nucleus formation and can be determined only by means of kinetic considerations. It is the purpose of this Chapter to present the fundamentals of the nucleation kinetics, to derive explicit expressions for the nucleation rate J and to reveal the supersaturation dependence of this quantity. In doing this we consider the nucleus formation on the assumption that the process is a set of consecutive bimolecular reactions of the type ... 

The supersaturation dependence seems to be the most frequently and the most extensively studied feature of the stationary nucleation rate ever since the time of Volmer [2.29]. For that reason in this Chapter we shall derive some explicit expressions for the Is/(Ajx) relationship assuming that the number No of the nucleation sites is not a potential dependent quantity. 

Concerning the supersaturation dependence of the Zeldovich factor/ the classical relation 2AG(n jp)/Afi (equation (1.64)) combined with... 

What one should do next is to find out the supersaturation dependence of the frequency in equation (2.19). This quantity depends on the... 

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