CHAPTER 22 PRECIPITATION AND CRYSTALLIZATION KINETICS : MODELS

22.1 Introduction………………………………………………………………………………1

22.2 Nucleation………………………………………………………………………………... 2

22.2.1 Supersaturation…………………………………………………………………. 222.2.2 Effect of Particle Size on Solubility……………………………………………..322.2.3 Nucleation Kinetics……………………………………………………………...6

22.3 Crystal Growth by Monomer Addition………………………………………………. 10

22.3.1 Growth Models………………………………………………………………... 1022.3.2 Monuclear Growth……………………………………………………………..1222.3.3 Polynuclear Growth…………………………………………………………… 1922.3.4 Spiral Growth…………………………………………………………………..19

22.4 Aggregative Growth ……………………………………………………………………25

22.5 Synthesis of Monodispersed Colloidal Particles……………………………………... 32

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22.1 Introduction

Crystallization and precipitation are solubility-related processes. That is, a solid crystal

or precipitate forms when a solute exceeds its solubility in the aqueous solution. In this respect the solubility product (Kso) represents a useful parameter for ascertaining the thermodynamic

feasibility of solid-formation reactions. Consider the general reaction:

MaAb(s) = a Mz+(aq) + bAx- (aq) (22.1)

From a thermodynamic standpoint, the transfer of a solute from the aqueous to the solid phase is feasible provided the condition log Q > log Kso is satisfied, where

Q = {Mz+ (aq)}a {Ax-(aq)}b (22.2)

and Q = K at equilibrium.

Two different approaches may be taken in the attempt to satisfy the condition of log Q > log Kso, i.e., (a) for a given log Q, decrease log Kso, or (b) for a given log Kso, increase logQ. In

industrial practice, the term crystallization typically refers to the first case, i.e. the situation where the condition log Q > log Kso is achieved by decreasing log Kso, e.g. via temperature

variation (heating or cooling). On the other hand, precipitation refers to the case where the condition log Q > log Kso is attained by altering log Q via the addition of a reagent (precipitant)

which changes {Mz+} or {Ax-}. This chapter is devoted to a discussion of the kinetics of

precipitation and crystallization processes.

22.2 Nucleation

22.2.1 Supersaturation

A precipitation or crystallization process involves three main steps, i.e., the development of

supersaturation, followed by nucleation, and then growth. Supersaturation may be defined in

several ways, e.g., as the supersaturation ratio (S), the concentration driving force (C), and the

relative supersaturation (). The relevant relations are given by Equations 22.3, 22.4, and 22.5:

S = C/Ceq (22.3)

C = C - Ceq (22.4)

= C/Ceq = S - 1 (22.5)

where C and Ceq respectively represent the solute concentrations in the supersaturated and

equilibrium saturated solutions.

The precipitation of relatively insoluble compounds exhibits many of the characteristics associated with the crystallization of relatively soluble salts (i.e., salts with Ceq > 0.1 to 1 kmol

m-3). However, precipitation differs from crystallization in that it involves a much higher degree of supersaturation (S = (C-Ceq)/Ceq ~ 1,000 for precipitation compared with ~ 0.001 to 0.20 for

crystallization systems). As a consequence of this comparatively high degree of supersaturation,

the tendency in precipitation processes is to form relatively small crystals.

The supersaturation ratio (S) represents the thermodynamic driving force for

crystallization. Consider the precipitation reaction,

Mz+ + Az- = MA(s) (22.6)

The corresponding Gibbs free energy of reaction is given by

Gr = -RT ln K + RT ln Q (22.7)

where K is the equilibrium constant:

K = 1/{Mz+}eq {Az-}eq (22.8)

and Q is the reaction quotient:

Q = 1/{Mz+} {Az-} (22.9)

It follows from Equations 22.7-22.9 that

Gr  = - RT ln (C/Ceq) (22.10)

= - RT lnS (22.11)

The precipitation reaction will be favored, i.e., Equation 22.6 will proceed in the forward direction if Gr < 0. Therefore on the basis of Equations 22.10 and 22.11, precipitation will

proceed so long as C > Ceq or S > 1

EXAMPLE 22.1. Thermodynamic Driving Force for Crystallization

22.2.2 Effect of Particle Size on Solubility

In preparation for our later discussion of nucleation kinetics, it is helpful to consider

particle size effects on solubility. Consider a solid particle bounded by an arbitrary

solid/aqueous interface. The Gibbs free energy is given by:

dG = -SdT + Vdp + dA + (22.12)

where S = entropy, T = absolute temperature, V = volume, p = pressure, = interfacial energy, A = interfacial area, i = chemical potential of component i, ni = moles of component i. The

chemical potential is given by

i = ( G/ ni)T,P,nj,A (22.13)

Let us focus on a solid consisting of only one type of compound, i.e., the solid contains

only one component. Then at constant temperature and pressure, Equation 22.13 becomes:

dG = dA + dn (22.14)

where is the bulk chemical potential of the crystal.Equation 22.13 assumes that ni can change at constant A. For very small particles,

however, n cannot be transferred without changing A. If dV corresponds to the volume change

associated with introduction of dn molecules, then it follows that

dV = mdn (22.15)

where m is the molar volume of the solid particle. But A = 4r2 and V = (4/3)r3 and thus

dA = 8rdr = 2dV/r (22.16)

Therefore it follows from Equations 22.15 and 22.16 that

dA = 2mdn/r (22.17)

Introducing Equation 22.17 into Equation 22.14,

dG = (2m/r + ) dn (22.18)

Thus ' the chemical potential of the microparticle is given by

' = ( G/ n)T,p,nj = 2m/r + (22.19)

Comparing Equations 22.13 and 22.19 it can be seen that in the case of the microparticle, the

constant A constraint has been removed.

Consider the case where a solute distributes between the aqueous phase and the solid

particle. If ' and are respectively the chemical potentials of the solute for a microparticle and

for macroparticle, then it follows from Equation 22.19 that

' - = 2m/r (22.20)

Now the chemical potential of the solute in the aqueous phase in contact with the macroparticle

is given by:

= ˚ + RT ln a (22.21)

where a is the aqueous phase activity of the solute. Similarly for the case of the microparticle,

' = ˚ + RT ln a' (22.22)

At equilibrium the chemical potentials of the solute in the contacting solid and aqueous

phases are equal. Therefore,

s = = ˚ + RT ln a (22.23)

= ' = ˚ + RT ln a' (22.24)

Therefore combining Equation 22.20, 22.23, and 22.24 gives

ln (a'/a) = 2m/rRT (22.25)

Under conditions where activity coefficient effects can be ignored, Equation 22.25 may be

expressed in terms of the solute concentrations as:

ln (C'/C) = 2m/rRT (22.26)

= 2M/rRT (22.27)

where M is the molecular weight of the solid compound and is the density of the solid.

______

EXAMPLE 22.2. Effects of particle size on solubility and crystal growth.

(a) Show that the particle size effect on solubility can be incorporated into the expression for the solubility product constant as:

log Kso(r) = log Kso() + (2/3) A/2.3RT (1)

where Kso(r) represents the solubility product for a small precipitate with a molar surface area

A(meter2/mole), Kso() is the solubility product constant for a large precipitate.

(b) On the basis of Equation 1 and the data provided below, derive expressions for the solubility product constants of Cu(OH)2(s) and CuO(s) which take surface effects into account.

Reaction log Kso( ) (erg cm -2)

CuO(s) + H2O = Cu2+ + 2OH- -20.0 690

Cu(OH)2(s) = Cu2+ + 2OH- -19.0 410

(c) Consider an aqueous solution at pH 7.0. As the Cu2+ concentration is increased, which of the two solids Cu(OH)2 and CuO will nucleate first? Assume that the precipitate has an initial molar surface area of

5x104 m2/mole.

(d) As crystal growth proceeds, the compound which first precipitates becomes unstable relative to the second compound. At what value of molar surface area does the first precipitate invert to give the

second solid? What is the equilibrium Cu2+ concentration at this stage?____________________________________________________________________________________________

22.2.3 Nucleation Kinetics

The establishment of supersaturation does not lead automatically to solid formation. For

this, it is necessary that crystal nuclei be formed. In general, the ions present in solution are in

constant motion and their mutual interaction results in the formation of temporary clusters. The

process leading to the formation of a permanent or critical cluster, i.e., a crystal nucleus, can be

viewed as a succession of bimolecular reactions:

M + M M2

M2 + M M3

Mn-1 + M Mn (critical cluster) (22.28)

In view of its extremely small size, it is practically impossible to determine the structure of this

critical cluster. However, it is apparent that clusters which contain less molecules than the

critical cluster are unstable and redissolve.

It can be shown from a thermodynamic analysis of the free energy changes associated with the nucleation process that a cluster with a radius greater than that of the critical cluster (rc)

is stable because if it grows, it can reduce its free energy. On the other hand, a molecular assemblage with a radius less than rc can only decrease its free energy by decreasing in size, i.e.,

dissolving.The free energy change, Gn, associated with nucleation has two main contributions.

When molecules or ions assemble into a cluster, energy is given off (e.g. latent heat of

condensation when gaseous molecules condense to liquid). The corresponding free energy change, Gv, is therefore a negative quantity. On the other hand, creation of the aggregate

results in the formation of a new surface. The corresponding interfacial energy, Gs, is a

positive quantity. Thus

Gn = Gv + Gs (22.29)

… … …

= (4/3)r3 G + 4r2 (22.30)

where is the interfacial energy and G is the free energy of aggregation per unit volume of

aggregate.

Gcrit

rcrit

r

Gs

Gn

Gv

G

0

Figure 22.1 The effect of the cluster radius (r) on the free energy of nucleation (Gn).

The radius of the critical nucleus is obtained when dGn/dr = 0, i.e.,

dGn/dr = 8r + 4r2 G = 0 (22.31)

Thus,

rc = -2/G (22.32)

Inserting Equation 22.32 into Equation 22.30 gives:

Gcrit = 163/3(G)2 = 4rc/3 (22.33)

Figure 22.1 presents a schematic illustration of the dependence of r on Gn. From the free

energy diagram it follows that a cluster with a radius greater than rc is stable because if it grows

it can reduce its free energy (i.e. |G| > Gs). On the other hand, a particle with a radius less

than rc can only decrease its free energy by decreasing in size, i.e. dissolving.

How is the amount of energy needed to overcome the energy barrier, Gcrit, obtained?

At constant temperature and pressure, the average energy in the system is constant. However the

energy fluctuates around this average value and it is therefore possible for some parts of the fluid

to momentarily acquire energy levels which will permit the corresponding clusters to overcome Gcrit.

The transition from a critical radius to a stable nucleus can be viewed in terms of the

transformation of a microparticle to a macroparticle, as discussed in Section 22.2.2.

Accordingly, the solute concentration (C') associated with the microparticle represents the solute

concentration in the supersaturated solution that surrounds the nucleus. On the other hand the

solute concentration (C) in the solution adjacent to the macroparticle may be linked to the solute

concentration in the saturated solution obtained at equilibrium around the fully grown particle.

Thus recalling Equations 22.3 and 22.26,

ln (C'/C) = ln S = 2m/rcRT (22.34)

Thereforerc = 2m/RT ln S (22.35)

Combining Equation 22.33 with Equation 22.35 gives

Gcrit = 163/3 (RT ln S)2 (22.36)

An Arrhenius type of expression can be used to relate the rate of nucleation and Gcrit:

Rn = ko exp(-Gcrit /RT) (22.37)

where ko is a constant. It follows from Equations 22.36 and 22.37 that

Rn = ko exp[-163/3R3T3(ln S)2] (22.38)

According to Equation 22.38, three key variables control the rate of nucleation:

temperature (T), interfacial energy (), and the degree of supersaturation (S). In particular, it can

be seen from Equation 22.38 that as the supersaturation increases, the nucleation rate also

increases.

When nucleation proceeds without assistance from foreign substances, including the

container walls, the process is termed homogeneous nucleation. On the other hand when external

agents such as seed crystals or container walls are used to initiate nucleation, the process is

termed heterogeneous nucleation. The ability of solid surfaces to catalyze nucleation reactions

stems from the fact that the activation energy for the formation of a two-dimensional nucleus is

smaller than that required for a three-dimensional nucleus.

___________________________________________________________________________

EXAMPLE 22.3 The number of moles in a critical nucleus

(a) Starting from Equation 22.35, show that nc, the number of moles in a critical nucleus is given by

nc = 323/3(RT ln S)3 (1)

(b) Under certain conditions the experimentally determined nucleation rate is found to follow the following power law:

R = k Cn (2)

Show that n = nc.

Solution

(a) According to Equation 22.35,

rc = 2m /RT ln S

where m is the molar volume of the solute. Let the volume of the critical nucleus be Vc. Then the

number of moles in the critical nucleus is given by

nc = Vc/m

= (4rc/3)/m = 323/3(RT ln S)3 (3)

(b) It follows from Equation 2 that:

ln R = lnk + n ln C (4)Thus

n = d ln R/d ln C (5)

But according to Equation 22.38,

R = ko exp[-163/3 R3 T3 (ln S)2] (22.38)

Thus

ln R = ln ko - 163/3 R3 T3 (ln S)2 (6)

Therefore

= (7)

Comparison of Equations 1 and 7 indicates that

d ln R/d ln S = nc (8)

But by definition, S = C/Ceq (see Equation 22.3). Therefore

d ln R/d ln S = d ln R/d ln C (9)

It follows from Equations 5, 8, and 9 that n = nc

22.3 Crystal Growth by Monomer Addition

22.3.1 Growth Models

Once a stable nucleus is formed, it can grow by two main pathways, i.e., monomer

addition, and nuclei aggregation, as illustrated in Figure 22.2.

In the past, the formation of primary particles was viewed mostly in terms of monomer

addition. However, there is increasing evidence that in some cases (e.g. oxide and sulfide

precipitations, formation of colloidal gold) aggregative growth constitutes a significant pathway

in the formation of primary particles.

PRECURSOR

INTERMEDIATE (MONOMER)

NUCLEI (SUBPARTICLES)

PRIMARY PARTICLES PRIMARY PARTICLES

AGGREGATES

+ PRIMARY PARTICLES

+ PRIMARY PARTICLES

+ MONOMERS + NUCLEI

AGGREGATIVE GROWTH

MONOMER ADDITION GROWTH

Figure 22.2 Stages in the particle growth process

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EXAMPLE 22.4 Aggregation of silica nanoparticles

(a) According to the Smoluchowski theory of rapid aggregation (i.e., where there is no energy barrier) the time (t1/2) taken for an initial particle number density of Co to decrease by one-half is given by

t1/2 = 3/4kBTCo (1)

where = viscosity of the solvent, kB = Boltzmann constant, T = absolute temperature. An experiment to

synthesize silica particles (by the ammonia-catalyzed hydrolysis of alkoxides) yields a number density of

1018 m-3. Determine the half-life for aggregation, given that the viscosity of the continuous phase is 10-3

Pa s.(b) In the presence of an energy barrier, Equation 1 may be modified as:

t1/2 = 3Wii/4kBTCo (2)

where Wii is termed the "stability ratio for two particles of size i". In the case of equal-sized particles

where the interaction forces are derived solely from van der Waals attraction and electrostatic repulsion, Wii is given approximately by

Wii = (1/2 a)exp [4oo2/kBT] (3)

where

= relative permitivity of the liquido = permitivity of free space

o = surface potential of the particle

a = particle radius = reciprocal of the double-layer thickness (see Chapter 5).

Assuming a surface potential of 25mV, determine the half-life for particles with a radius of (i) 25nm, (ii) 5nm.

Solution

(a) It can be shown that the relevant half-life is 0.1s.(b) It can be shown that for the 25nm particles, t1/2 > 12 h whereas for the 5 nm particles, t1/2 < 1s.

____________________________________________________________________________________________

Example 22.4a illustrates the fact that in the absence of an energy barrier, aggregation is

highly favored. Thus the formation of a colloidally stable precipitate implies the presence of a

sufficiently high energy barrier (e.g. electrostatic, solvation, steric). On the other hand Ex. 22.4b

illustrates the fact that even in the presence of electrostatic repulsion, extremely small particles

(approaching the size of a nucleus) are unstable towards aggregation.

22.3.2 Mononuclear Growth

As illustrated in Figure 22.3, the growth of an ionic crystal by monomer addition may be

viewed in simple terms as a two-step process involving (a) the diffusion of ions from the bulk

solution through a stagnant boundary layer to an adsorption layer:

Jd = (D/)(C - Ci) (22.39a)

= kd (C - Ci) (22.39b)

Figure 22.3 Crystal growth via combined film diffusion and surface reaction.

and (b) interfacial reaction, i.e. the incorporation of the adsorbed ions into the crystal lattice:

M(ads) M(lattice) (22.40)

Assuming a first-order reversible reaction at the surface, the flux (Jr) associated with the

reaction can be expressed as:

Jr = (1/A) dn/dt = k1 Ci - k-1 CL (22.41a)

= k1 Ci - k-1 (22.41b)

where n is the moles of the solid present at time t, A is the solid/solution interfacial area, and C L

is the concentration of the ion in the crystal lattice (which is a constant). At equilibrium dn/dt = 0 and Ci = Ceq and therefore it follows from Equation 22.41 that:

k1 Ceq = k-1 (22.42)

and

Jr = k1 (Ci - Ceq) (22.43)

At steady-state the rate of diffusion must equal the rate of the surface reaction. Thus it

follows from Equations 22.39 and 22.43 that

J = Jd = Jr = k(C - Ceq) (22.44)

where k is the overall mass transfer coefficient for crystal growth given by,

1/k = 1/kd + 1/k1 (22.45)

It can be seen from Equations 22.44 and 22.45 that when the surface reaction is very fast, k~k d

and crystal growth is diffusion controlled. On the other hand, when diffusion is relatively fast, k~kr and crystal growth is surface reaction controlled.

It can be seen from Equation 22.44, that for a first-order surface reaction, the crystal growth rate is proportional to the concentration difference (C-Ceq). It must be noted that here

Ceq refers to the solute concentration in equilibrium with the surface of the freshly precipitated

solid. Thus if the surface of the fresh precipitate eventually undergoes rearrangement into a more stable form, the solute concentration (C ) in equilibrium with this more stable form may

differ from Ceq. In general C < Ceq.

______________________________________________________________________________

EXAMPLE 22.4 Calcite growth kinetics

In the growth of calcite crystals from supersaturated solutions, it may be assumed that the "adsorbed lattice

ions combine in the surface layer to form an ion-pair, CaCO3o, that is subsequently incorporated into the surface

lattice:

Ca2+ + CO32- = CaCO3

o (1)

The equilibrium constant corresponding to the above reaction is given by:

K1 = {CaCO3o}/{Ca2+}{CO3

2-} (2)

(a) Show that if the growth rate is first order with respect to the growth unit,, then the corresponding rate law is:

-d[Ca2+]/dt = (kK1A22)([Ca2+][CO3

2-] - Kso/22/ (3)

where k is the first order rate constant, A is the "area of seed per unit volume of solution", 22 is the

divalent ion activity coefficient, K1 is the formation constant of CaCO3o, and Kso is the solubility

product of calcite.

(b) Verify that the data below are consistent with the above growth model

Solution

(a) Let us recall Equation 22.44:

J = Jd = Jr = k(C - Ceq) (22.44)

For calcite crystal growth, this equation can be re-expressed as:

J = Jd = Jr = k([CaCO3o] - CaCO3

o]eq) (4)

According to Equation 2,

{CaCO3o} = K1{Ca2+}{CO3

2-} = K122[Ca2+][CO3

2-] (2)

If it assumed that the neutral species, CaCO3o, has unit acivity coefficient, then it follows that

[CaCO3o] = K1{Ca2+}{CO3

2- } = K122[Ca2+][CO3

2-] (5)

Also, at equilibrium,

[CaCO3o]eq = K1{Ca2+}eq{CO3

2- }eq = K1Kso (6)

Substitution of Equations 5 and 6 into Equation 4 gives the desired expression.

EXAMPLE 22.5 Silica growth kinetics

The kinetics of silica growth were investigated by Fleming (J. Colloid Interface Sci., 110, 40-64 (1986)). Seeded growth experiments were conducted where monodisperse colloidal amorphous silica (Ludox) was introduced into a supersaturated solution of silicic acid (Si(OH)4). Under the experimental conditions (25°C, 0.1M

NaCl, pH 6-8), the equilibrium silica solubility (determined as silicic acid concentration) is 115 ppm.

It has been proposed that growth occurs via a condensation polymerization reaction between a silicic acid molecule and a surface hydroxyl:

-SiOH(s) + Si(OH)4(aq) -Si-O-Si(OH)3(s) + H2O (1)

(a) Figure E22.5a presents a set of experimental data plotted as 1n (C-Ceq) vs. time. Establish that this plot,

which assumes that Ceq corresponds to the equilibrium solubility of amorphous silica (115 ppm), does not

conform to the mechanism described by Equation 1.

(b) Fleming argues that a distinction should be made between a silica surface coated with chemisorbed silicic acid (i.e., -Si(OH)3) and the surface of stable amorphous silica. Thus the appropriate Ceq value to use in

analyzing the experimental data is not C = 115 ppm (the value used in plotting Figure E22.5a), but rather

Cx, the silicic acid concentration in equilibrium with chemisorbed silicic acid. Replot the Figure 1 data as -

dC/dt vs C and show that the rate is relatively low when silicic acid concentration is below ~200 ppm. Above this concentration, however, the rate is linearly dependent on the silicic acid concentration.

(c) The transitional concentration of ~200 ppm represents the silicic acid concentration that is in equilibrium with a silica surface coated with chemisorbed silicic acid. Show that replotting the experimental data with Ceq = 200 ppm leads to a trend that is consistent with the proposed growth mechanism (Equation 1).

Solution

(a) Let C represent silicic acid concentration. Then the rate law corresponding to the proposed growth mechanism (Equation 1) is:

-dC/dt = k1S[-SiOH]C - k-1S (2)

where S is the surface area of silica per volume of solution, and [-SiOH] is the surface concentration of hydroxyl groups.

When the solution is in equilibrium with the chemisorbed surface, dC/dt = 0 and C = Ceq.. Thus,

k-1 = k1S[-SiOH]Ceq (3)

Accordingly Equation 2 can be re-written as:

-dC/dt = k1S[-SiOH](C - Ceq) = k' (C - Ceq) (4)

where k' = k1S [-SiOH]. Equation 4 may be integrated with the initial condition C = Cwhen t = 0, to give

the following integrated rate law:

ln (C - Ceq) = -kt (5)

where k = k'C and C >> Ceq.

In view of Equation 5, the fact that the rate data plotted in Figure E22.5a do not give straight lines means that either the proposed mechanism is inappropriate or the pertinent Ceq is not the solubility of stable

amorphous silica.

(b) Figure E22.5b shows the required plot.

(c) Figure E22.5c shows the required plot.

Figure E22.5 Silica precipitation kinetic data.

Figure E22.5 (cont’d) Silica precipitation kinetic data.

ln (

C-C

e) (

ppm

)

700

pH = 8.017.316.33

0 100 200 300 400

Time, minutes

500 6003.6

3.8

4.0

4.2

4.4

4.6

4.8

5.0

5.2

5.4

5.6As = 4195 60 cm2/cm3

I = 0.1 M NaCl

(a)

Sil

icic

Aci

d D

epos

itio

n R

ate,

-C

/I

(ppm

/min

)

pH = 8.017.316.33

100 200 300

1

2

3

4

5 Region 2 Region 1

Cx

400

Ce

Silicic Acid Concentration, C (ppm)

Increasing Time

(b)

pH = 8.017.316.33

3

2

4

5

0 100 200 300 400Time, minutes

ln (

C-C

X)

(ppm

)

(c)

According to Equations 22.44, the crystal growth rate is proportional to the concentration difference (C - Ceq). However growth rates of inorganic salts in aqueous solution frequently

follow the form J ~ (C - Ceq)x where x = 1.5 - 2. Thus the actual situation is much more

complicated than the simple two-step process presented above. In fact, several physicochemical

processes occur simultaneously, e.g. dehydration of ions, counter-diffusion of rejected water

molecules, surface diffusion, surface nucleation, etc.

The reaction-controlled growth models may be divided into (a) mononuclear layer, (b)

polynuclear layer, and (c) surface dislocation models. The mononuclear layer mechanism views

crystal growth in terms of the build-up of one monolayer of the crystal lattice, followed by the

formation of another layer on top of the previous one, etc. A new layer can grow only when a

two-dimensional nucleus is formed. The energetics of this two-dimensional nucleus can be

developed by following an approach similar to that used above for two-dimensional (i.e.

homogeneous) nucleation. That is, recalling Equation 22.29, the free energy of two-dimensional

nucleation consists of two terms:

G = VG + A (22.46)

where V is the volume of the nucleus and A is its total surface area. If a disc-like nucleus is

considered, with radius r and thickness h, then Equation 22.46 becomes:

G = 2rh + r2hG (22.47)

It can be shown that the critical radius (rc) is given by

rc = -/G (22.48)

and that the corresponding critical energy barrier Gcrit is given by

Gcrit = hrc (22.49)

Comparing Equations 22.32 and 22.48, it can be seen that the radius of a critical two-

dimensional radius is half that of a critical three-dimensional radius. It follows from Equations

22.26 and 22.49 that:

Gcrit = h2m/RT ln S (22.50)

Therefore in analogy with the case of homogeneous nucleation (Equation 22.37), the rate

of two-dimensional nucleation may be written as:

Rn = ko exp (-h2m/R2 T2 ln S) (22.51)

Comparing Equations 22.36 and 22.50, it can be seen that

= (22.52)

___________________________________________________________________________

EXAMPLE 22.6 Comparison of energy barriers for three- and two-dimensional nucleation.

For (a) S = 1.1, and (b) S = 1.2, compare the energy barriers for three- and two-dimensional nucleation for

an inorganic salt for which = 10-1 Jm-2, m = 2x10-29 m3 mol-1, h = 5x10-10 m; also kT = 4x10-21 J.

Solution

Gcrit (3 dim)/Gcrit (2 dim) = 50/1 for S = 1.1

= 1.2/1 for S = 1.2

These results indicate that at low supersaturations the critical energy barrier for three-dimensional nucleation is significantly higher than that for two-dimensional nucleation. In other words low supersaturations will favor growth rather than nucleation.____________________________________________________________________________________________

22.3.3 Polynuclear Growth

22.3.4 Spiral Growth

It is frequently observed that for a given growth rate, the surface nucleus model of crystal

growth requires higher supersaturations than are actually observed experimentally. This

observation suggests that there may be more growth sites than indicated by the layer-by-layer

mechanism. The surface dislocation growth model explicitly takes account of the fact that

imperfections exist on crystal surfaces and that these crystallographic features can significantly

influence interfacial reactions.

Figure 22.5 presents a schematic illustration of the surface of a growing crystal. The

growth of a crystal involves several steps which can be summarized as follows:

(a) Transport of ions from the bulk solution to the solution immediately adjacent to

the crystal surface.

(b) Transfer of ions from the solution to an adsorption layer.

(c) Transfer of ions from the solution or adsorption layer to a growth step.

(d) Transfer of ions from the solution, adsorption layer or growth step to a growth

site, i.e., a lattice position at a kink.

Figure 22.5 Schematic illustration of the surface of a growing crystal: (a) Smooth surface,(b) edge or step, (c) kink site, (d) two dimensional nucleus.

Figure 22.6 Schematic illustration of a growth spiral showing the kink distance (xo), the step distance (yo), and the step height (a) (After Nielsen).

If the rate of crystal growth is determined by step (d), then in order to obtain the rate

equation it is necessary to have quantitative information on the number density of growth sites,

i.e., the number of kink sites per unit area. According to the Burton, Cabrera and Frank (BCF)

theory of crystal growth, the sites needed for growth are provided by screw dislocations that

terminate in the surface of the crystal. The screw dislocations are in the form of a spiral

staircase, as illustrated in Figure 22.6. The main characteristics of the surface structure needed for the crystal growth model are the kink distance (xo), the step distance (yo), and the step height

(a). It follows from Figure 22.6 that the density of kinks (n) in the crystal surface (i.e., the

number of kinks per unit area of crystal surface) is given by

n = (xo yo)-1 (22.53)

The BCF theory gives the following expressions for xo and yo:

xo = a S-1/2 exp (a2/kT) (22.54)

yo = 19a3/(kTlnS) (22.55)

where a is the step height, S is the saturation ratio (S = C/Co, where C and Co are respectively

the concentrations of the electrolyte in the supersaturated and the equilibrium saturated

solutions), and is the interfacial tension of the crystal/aqueous interface. Thus it follows from

Equation 22.53-22.55 that the density of kinks may be expressed as:

n = (22.56)

(22.57)

Equation 22.57 uses the approximation:

S1/2lnS (S-1) (22.58)

Consider the crystal growth of a symmetrical electrolyte MA. It shall be assumed that

step (d) (in particular, transfer of ions from the adsorption layer to the growth site) is rate-

determining. Thus the electrolyte in the solution is in equilibrium with the ions in the adsorption

layer, i.e.,

Mz+(aq) = Mz+(ad) (22.59)

K+ = +/CM (22.60)

Az-(aq) = Az-(ad) (22.61)

K- = - /CA (22.62)

where + and - are the adsorption densities of the cation and anion, and K+ and K- are the

respective adsorption constants.

Let us focus on the transfer of the growth units from the adsorption layer into a kink site:

Mz+ (ad) Mz+ (kink) (22.63)

Az- (ad) Az- (kink) (22.64)

It should be noted that Mz+ will deposit onto an Az- ion in the lattice. Similarly Az- will deposit onto an Mz+ ion in the lattice. Let be the fraction of the kink site with exposed Mz+ ions and

let + be the fraction of the kink site with exposed Az- ions. Then

+ + - = 1 (22.65)

It follows therefore that the rate equations for the deposition of Mz+ and Az- at the kink

sites are given by Equations 22.66 and 22.67 respectively:

r+ = k+ + - - k + (22.66)

r- = k- - + - k- (22.67)

where r+ and r- respectively, the rate of deposition per kink for the cation and anion.

The formation of a stoichiometric MA crystal requires that cation and anion are incorporated into the crystal structure at equal rates, i.e., r+ = r- = r. Therefore, it follows from

Equations 22.65-22.67 that:

+ = (22.68)

andr = r+ = r- =' (22.69)

At equilibrium, r = r+ = r- = 0. Accordingly Equations 22.66-22.67 respectively give

Equations 22.70-22.71:

k+' = k+ o+ o-/o+ (22.70)

k+

k+

k-

k-

k-' = k- o- o+/o- (22.71)

where the subscript ("o") refers to equilibrium conditions. Similarly under equilibrium

conditions, Equation 22.69 gives

k+' k-'/k+ k- = o+ o- (22.72)

Combination of Equations 22.69 and 22.72 gives:

r = (22.73)

If the adsorbed layer is approximately electroneutral, then + -. Therefore Equation

22.73 becomes:

r = (22.74)

Furthermore, if at equilibrium the fraction of a kink site occupied by the cation equals that occupied by the anion (i.e., o+ = o-), then recalling the assumed electroneutrality of the

adsorbed layer (i.e., o+ = o-), Equations 22.70 and 22.71 may be combined to give

o+ = (k+' + k-')/(k+ + k-) (22.75)

Combination of Equations 22.74 and 22.75 gives:

r = (22.76)

kk

k k ( o ) o

o

kk

k k

o (22.77)

According to Equation 22.60,

+ = K+CM (22.78)

o+ = K+ CMO (22.79)

Using Equations 22.78 and 22.79 in Equation 22.77 gives:

r = (22.80)

= (22.81)

where S is the saturation ratio (= CM/CMO)

Recalling that the density of kinks is given by Equation 22.57, the crystallization rate per

unit area will be given by

EXAMPLE 22.7 Parabolic crystal growth rate

Liu and Nancollas (J. Cryst. Growth, 6, 281-289(1970)) investigated the kinetics of crystal growth of

calcium sulfate dihydrate. The following represents a selection of their results:

t(min) 0 5 15 30 40 50 60

[Ca2+]x102(mol/L) 3.90 3.66 3.45 3.23 3.11 2.98 2.94t(min) 80 100 120 140 160 180

[Ca 2+ ]x10 2 (mol/L) 2 .81 2 .70 2 .62 2 .53 2 .50 2 .45

Show that these results obey the rate law:

-d[Ca2+]/dt = k([Ca2+] - [Ca2+]eq)2 (1)

Solution

Equation 1 may be integrated with the initial conditions [Ca2+] = [Ca2+]o when t = 0, to give

1/([Ca2+] - [Ca2+]eq) - 1/([Ca2+]o - [Ca2+]eq) = kt (2)

Figure E22.7a shows a plot of the original data and Figure E22.7b shows a plot of these data according to Equation

2. The straight lines observed in Figure E22.7b confirm that the crystal growth of calcium sulfate dihydrate follows

the parabolic rate law given by Equation 1.

40Time, minutes

4.0

3.0

2.080 120 160

Con

cent

rati

on x

102

40Time, minutes

240

160

80 120 160(m

– m

o)-1 –

(m

i – m

o)-1 x

10

80

(a) (b)Figure E22.7

22.4 Aggregative Growth

The evidence for aggregative growth includes the following: (a) TEM micrographs that

indicate that primary particles in the 10 to 1000 nm size range consist of aggregates of smaller

subparticles (e.g. TiO2 by Santacesaria et al., gold by Uyeda et al., iron hydroxides by Murphy et

al, van der Woude and deBruyn, Murphy et. al, zirconia by Bleier and Cannon, titania by

Dirksen and Ring, silica by Bogush and Zukoski), (b) observation that particle densities decrease

with increase in size (e.g. iron (III) hydroxide by von Gunten and Schneider), and (c) aggregative

growth models that are reasonably consistent with experimental data (e.g. silica from alkoxides,

Philipse; Bogush and Zukoski).

___________________________________________________________________________

EXAMPLE 22.8 Evidence for aggregative growth: Dissolution of colloidal Fe(OH)3

It has been suggested by von Gunten and Schneider (J. Colloid Interface Sci., 145, 127-139 (1991)) that the

aggregative nature of colloidal precipitates can be probed by conducting carefully designed dissolution experiments.

The overall reaction describing the acid decomposition of colloidal Fe(OH)3 can be expressed as:

(Fe(OH)3)p + 3pH+ = pFe3+ + 3pH2O (1)

The overall reaction consists of two main steps:

I. Disintegration (i.e. disaggregation):

(Fe(OH)3)p = (p/m)(Fe(OH)3)m (2)

II. Dissolution of primary particles:

(Fe(OH)3)m + 3mH+ = mFe3+ + 3m H2O (3)

Figure 22.8a

1.0 1.5 2.00

10

20

30

40

50

60120Å

120Å/HClO4/HCl

10Å/HCl

HCl Concentration, M

t 1/2 [

sec]

In synthesis experiments conducted by von Gunten and Schneider, Fe(II) solution was oxygenated in the

presence of tris (i.e., tris (hydroxymethyl)aminomethane). The size of the colloidal Fe(III) hydroxide particles

ranged from 10Å in 0.5M tris to 140Å in 0.1M tris.

Examine each of the following observations for its compatibility with an aggregative growth model.

(a) In a dissolution experiment conducted in 2M HCl, the half-life was found to be

independent of the hydrodynamic radii of the starting particles.

(b) Separate dissolution experiments were conducted with 10Å and 120Å particles. With both small and large

particles, the half-life decreased with increase in HCl concentration. However, while t1/2 was independent

of particle size for 2M HCl, as the acid concentration was decreased from this level, t1/2 (120Å) increased

relative to t1/2 (10Å).

(c) Both acid-and ligand-promoted dissolution occur when HCl is used as the lixiviant whereas with HClO4,

only proton-promoted dissolution is significant. Thus with 120Å particles, the half-life is respectively 1

min and 12 min for HCl and HClO4. A solution containing 120Å particles was mixed with an equal

volume of 2M HClO4. One minute after the addition of HClO4, the pH was raised to 3 by addition of 3M

tris (pH 8). Following this HCl was added and the dissolution rate was monitored. The results are

presented in Figure E22.8a, where the 10Å/HCl data are also shown for comparison.

Solution

(a) The reported observation is illustrated schematically in Figure E22.8b

r

[Tris], mol/L

t 1/2

0.1 0.5

Figure E22.8b

Since dissolution is a heterogeneous process, the rate is expected to be proportional to the external surface

area. The fact that the rate was found to be independent of the "external" surface area (as determined by

the hydrodynamic radius) indicates that this is not the relevant external surface area. This result implies

that if (i) the primary particles are aggregates of equal-size subparticles, and (ii) the disintegration step

(Equation 2) is extremely fast, then the external surface seen by the aqueous solution corresponds to that

provided by the subparticles. Therefore the half-life should be constant for all sizes of the primary

particles, since in all cases the actual dissolving particles are the subparticles (i.e.,(Fe(OH)3)m, see

Equation 3).

(b) A schematic illustration of this observation is shown in Figure E22.8c. Two effects are at work here. With

2M HCl the acid concentration is high enough to permit fast disaggregation of the 120Å particles. Thus in

this case the hydrodynamic radius of the starting particles will have no effect on the half-life. However, as

the acid concentration is decreased the disaggregation step becomes increasingly slow and the external

surface area increasingly has contributions from the outer surface delimited by the hydrodynamic radius.

(c) During the preliminary HClO4 treatment, the primary particles disaggregate to give the subparticles, but

because of the slow dissolution kinetics with this acid (compared with HCl), very little actual dissolution

(i.e., step II, Equation 3) occurs. In the subsequent reaction in HCl, the subparticles generated in the

pretreatment undergo dissolution. The fact that the dissolution behavior coincides with that of the 10Å/HCl

system strongly suggests that the subparticles (i.e. (Fe(OH)3)m) are of the order of 10Å in radius.

[HCl]

t1/2

2 M

10 Ao

120 Ao

Figure E22.8c____________________________________________________________________________________________

The sequence of steps involved in aggregative growth are illustrated in Figure 22.7.

Quantitative analysis of the overall process, taking into account all the separate steps, is a

formidable task. However, by means of carefully designed seeded growth experiments, it is

possible to obtain useful quantitative insight into the factors that control an aggregative growth

process.

Consider an experimental system where a supersaturated solution containing initially contains a number concentration no of subparticles (nuclei). At t = 0, a known amount of

primary particles is introduced into the solution. The primary particles are much larger than the subparticles. Let r1 = radius of a primary particle, r2 = radius of a subparticle. It is assumed that

(a) the primary particles do not aggregate (e.g. presence of significant electrostatic repulsion), (b)

no nucleation occurs during the seeded growth process, (c) particle growth may be viewed in

terms of the coagulation of small subparticles on a large stationary central primary particle.

Figure 22.7 Schematic illustration of the stages of aggregative growth (after Bagnall et.al.

1990).

Let us consider a spherical surface located at a distance r from a central primary particle.

Figure 22.8 illustrates this situation. The flux of subparticles towards the central primary particle

is given by

J = DCr (22.83)

The number of subparticles crossing this shell in a given time is given by

R = 4r2J (22.84)

At steady-state, the number of subparticles crossing the shell in a given time, must equal the

number of subparticles colliding with the central sphere in the same time interval, i.e.,

R = (dn/dt)r=r1 = (dn/dt)r = 4r2J = constant (22.85)

r

r2

r12

r1

SUBPARTICLE

PRIMARY PARTICLE

Figure 22.8 Schematic illustration of the subparticle (r2)-primary particle (r1) aggregation (after Philipse, 1988)

Recalling Equation 9.47, and the assumption that r12 = r1 + r2 r1, the flux may be

expressed as

J = Dr12n/r2 Dr1n/r2 (22.86)

where n is the concentration of subparticles at infinite distance from the central primary particle.

Substituting Equation 22.86 into 22.85,

R = (4r2)(Dr1n/r2) = 4Dnr1 (22.87)

Since the central absorbing sphere is also subject to Brownian motion, the appropriate

expression for the diffusion coefficient is

D = D1 + D2 (22.88)

where D1 and D2 respectively represent the diffusion coefficients of a primary particle and a

subparticle. However, since it is assumed that the primary particles are much larger than the

subparticles, the diffusion coefficient of the primary particles may be reasonably taken to be

relatively small, i.e.,

D D2 (22.89)

The central sphere grows by absorbing the subparticles. Let V1 and V2 respectively be

the volumes of a primary particle and a subparticle. Then it follows that

dV1/dt = V2R (22.90)

Recalling that V1 = 4r13/3, Equation 22.90 can be rewritten as:

4r12dr1/dt = V2R (22.91)

Combining Equations 22.87 and 22.91 gives:

4r12dr1/dt = 4Dnr1V2 (22.92)

That is,r1dr1 = DnV2dt (22.93)

Infinite Volume of Aqueous Phase. If the assumption is made that the aqueous phase is of

infinite volume, then the bulk concentration of subparticles is constant. Then Equation 22.93 may be integrated with the initial condition, r1 = ro at t = 0 to give:

r12 = ro

2 + 2V2nD2t (22.94)

Finite Volume of Aqueous Phase. When the aqueous phase has a finite volume, the

possibility exists that the subparticles may become exhausted in the course of the growth process. Let npo represent the initial number concentration of primary particles, each with an

initial radius of ro. At a given time, t, when each primary particle has attained a size of r1, the

prevailing bulk concentration of subparticles (n) will be

n = no - (4/3)npo(r13 - ro

3)/V2 (22.95)

where no is the initial number concentration of subparticles.

The particles attain their maximum size (rm) when the bulk concentration of subparticles

becomes exhausted. Therefore inserting n = 0 in Equation 22.95 gives:

no = (4/3)npo(rm3 - ro

3)/V2 (22.96)

Combining Equations 22.95 and 22.96,

n = (4/3)npo(rm3 - r1

3)/V2 (22.97)

The growth of each sphere may be described with Equation 22.93, where n is now given

by Equation 22.97. It can be shown (see Philipse, 1988) that Equations 22.93 and 22.97 give the

following result:

F(x) = F(xo) + 8rmnpoD2t (22.98)

wherex = r1/rm, xo = ro/rm (22.99)

and

F(x) 1n1 x x2

x 1 2

2 3 tan 1 2x 1

3

(22.100)

___________________________________________________________________________

EXAMPLE 22.9 Aggregative growth: Infinite vs finite volume of aqueous phase.

In a seeded growth experiment, silica particles grow from an initial radius of 60  nm to 80 nm in 1-2h. The

volume fraction of subparticles is initially 0.005. The diffusion coefficient for a subparticle is of the order of 10 -5

cm2 s-1. Determine whether the observed particle growth rate is consistent with an infinite volume of aqueous

phase.

Solution

For an infinite volume of aqueous phase, the aggregative growth obeys Equation 22.94:

r12 = ro

2 + 2V2nD2t (22.94)

That is,

t = (r12 - ro

2)/2V2nD2 (1)

In this expression the term V2n represents the volume fraction of subparticles, i.e., V2n = 0.005; also r1 = 80nm,

ro = 60nm, D2 = 10-5 cm2 s-1. Thus

t = [(80 nm)2 - (60 nm)2]/2 (0.005)(10-5 cm2 s-1) = 5.1 x 10-4 s

The observed time was 1-2h, which is much greater than the theoretical value of 5.1x10-4s based on an

infinite volume of aqueous phase. Thus it may be concluded that the silica growth did not occur in an infinite

volume of aqueous phase.

____________________________________________________________________________________________

22.5 Synthesis of Monodispersed Colloidal Particles

In the monomer-addition growth model, the formation of monodispersed colloidal

particles is rationalized with the aid of the LaMer diagram, as shown in Figure 22.9. Three

stages are identified. Stage I involves slow monomer generation (e.g. via decomposition of

precursor compounds). No particle formation occurs during this stage. Stage II represents the

nucleation stage. Nucleation commences when the monomer concentration exceeds C. At first

the monomer concentration continues to rise since monomers are produced more rapidly than

they are consumed by nucleation. However, eventually there is a depletion of monomers as

monomer generation is no longer able to keep up with its consumption by nucleation. When C falls below C , no more nucleation occurs. Stage III represents the growth stage; here the

predominant process is the addition of monomers to the available nuclei.

I II III

C*max

C*min

Cs

CONC'N (C)

TIME (t)

Figure 22.9 LaMer diagram illustrating the stages in the formation of monodispersed colloidal particles.

C represents the critical limiting supersaturation. In order to obtain monodisperse

particles, the time lapse between C and C must be as small as possible. That is, in order to

obtain monodispersed particles, there must be a clear temporal separation between the nucleation

and growth stages.EXAMPLE 22.15 Critical supersaturation during homogeneous precipitation of metal sulfides with thioacetamide

The precipitation of a divalent metal sulfide (MS) can be expressed as:

M2+ + S2- = MS (s) (1)

The sulfide ions needed for the precipitation reaction may be generated in-situ via the thermal decomposition of

thioacetamide (TA):

CH3C(S)NH2 + H2O CH3C(O)NH2 + H2S (2)

The corresponding rate law is given by,

-d[TA]/dt = k[H+] [TA] (3)

Taking into consideration the deprotonation reactions of H2S,

H2S = HS- + H+ K1 (4)

HS- = S2- + H+ K2 (5)

show that prior to precipitation, the time dependence of sulfide concentration is given by:

[S2-] = [TA]o{1 - exp(-k[H+]t)}/{[H+]2/K1K2 + [H+]/K2 + 1} (6)

Solution

It follows from the rate law (Equation 3) that,

[TA]o - [TA]t = [TA]o{1 - exp(-k[H+]t)} (7)

Also, it must be recognized that, according to the stoichiometric relationships in Equation 2, the total dissolved

sulfide corresponds to the amount of decomposed thioacetamide. Therefore,

[TA]o - [TA]t = [S]T (8a)

= [H2S] + [HS-] + [S2-] (8b)

Combining Equations 4 and 5, we get:

H2S = S2- + 2H+ K1K2 (9)

Thus,

[H2S] = ([H+]2/K1K2)[S2-] (10)

Also, from Equation 5,

[HS-] = ([H+]/K2) [S2-] (11)

It follows from Equations 10 and 11 that

[S]T = [H2S] + [HS-] + [S2-]

=( [H+]2/K1K2 + [H+]/K2 + 1 ) [S2-] (12)

That is,

[S2-] = [ST]/([H+]2/K1K2 + [H+]/K2 + 1) (13)

It follows from Equations 7, 8a and 13 that

[S2-] = [TA]o{1-exp(-k[H+]t}/{[H+]2/K1K2 + [H+]/K2 + 1}

(See Celikkaya and Akinc, JACerS, 73, 2360 (1990))

FURTHER READING

References on Crystallization and Precipitation

1. J. Nyvlt, O. Sohnel, M. Matuchova, and M. Broul, The Kinetics of Industrial Crystallization, Elsevier, New York, NY, 1985.

2. J. W. Mullin, Crystallization, 2nd Ed., CRC Press, Cleveland, OH, 1972.

3. W. L. McCabe and J. C. Smith, Unit Operations of Chemical Engineering, 3rd Ed., McGraw-Hill, New York, 1976.

4. L. Gordon, M. L. Salutsky and H. H. Willard, Precipitation from Homogeneous Solution, Wiley, New York, 1959.

5. G. H. Nancollas, "The Growth of Crystals in Solution", Adv. Colloid Interface Sci., 10, 215-252(1979).

6. G. H. Nancollas and N. Purdue, "The Kinetics of Crystal Growth", Quart. Rev., 18, 1-20(1964).

7. A. E. Nielsen, "Theory of Electrolyte Crystal Growth. The Parabolic Rate Law", Pure Appl. Chem., 53, 2025-2039(1981).

8. A. E. Nielsen, "Electrolyte Crystal Growth Mechanisms", J. Cryst-Growth, 67, 289-310(1984).

9. A. S. Myerson, ed., Crystallization as a Separations Process, ACS Symp. Ser., Vol. 438, 1990.

10. A. E. Nielsen, Kinetics of Precipitation, Macmillan, New York, 1964.

11. A. G. Walton, The Formation and Properties of Precipitates, Interscience, New York, 1967.

12. P. Hartman, ed., Crystal Growth: An Introduction, North-Holland, Amsterdam, 1973.

13. R. F. Strickland-Constable, Kinetics and Mechanism of Crystallization, Academic, London, 1968.

References on Hydrogen Reduction

1. F. A. Schaufelberger and T. K. Roy, "Separation of Copper, Nickel and Cobalt by Selective Reduction from Aqueous Solution", Trans. IMM, 64, 375-93 (1955).

2. F. A. Schaufelberger, "Precipitation of Metal from Salt Solution by Reduction with Hydrogen", Trans. AIME, , 539-548 (1956).

3. B. Meddings and V. N. Mackiw, "The Gaseous Reduction of Metals from Aqueous Solution", in Unit Processes in Hydrometallurgy, M. E. Wadsworth and F. T. Davis, eds., Gordon and Breach, New York, NY, 1964, pp. 345-384.

4. A. R. Burkin, "Production of Metal Powders and Coatings by Precipitation Techniques, and their Fabrication", Met. Revs.,12, 1-14 (1967).

5. D. J. I. Evans, "Production of Metals by Gaseous Reduction from Solution - Processes and Chemistry", in Advances in Extractive Metallurgy, IMM, London, 1968, paper 35.

6. J. Halpern, "Homogeneous Catalytic Activation of Molecular Hydrogen by Metal Ions and Complexes", J. Phys. Chem., 63, 398-403 (1959).

7. M. Findlay, "The Use of Hydrogen to Recover Precious Metals", in Precious Metals 1982, M. I. El Guindy, ed., Pergamon, New York, NY, 1983, pp. 477-501.

References on Sulfide Precipitation

1. C. S. Simons, "Hydrogen Sulfide as a Hydrometallurgical Reagent", in Unit Processes in Hydrometallurgy, M. E. Wadsworth and F. T. Davis, eds., Gordon and Breach, New York, NY,1964, pp. 592-616.

2. T. K. Roy, "Preparing Nickel and Cobalt Concentrates" Ind and Eng Chem., 53, 559-566 (1961).

3. S. Joris, "La Cinétique de précipitation des Sulfures de Cobalt et de Nickel par L'Hydrogéne Sulfuré," Bull. Soc. Chim. Belges., 78, 607-619 (1969).

4. H. A. Pohl, J. Am. Chem. Soc., 76, 2182-2184 (1954).

5. M. C. Jha and G. A. Meyer, "Physical Chemistry of Nickel Sulfide Precipitation from Acidic Sulfate Solutions", TMS Paper Selection, A80-51, 1980.

6. M. C. Jha, G. A. Meyer, and G. R. Wicker, "An Improved Process for Precipitating Nickel Sulfide from Acidic Laterite Leach Liquors", J. Metals, November, pp. 48-53, 1981.

7. B. Tougarinoff, "Dénickelage du Cobalt par Cémentation Sulfurante," Ind. Chim. Belge, Tome XX, Numéro Spécial, Vol. II, pp. 532-536 (1955).

References on Electroless Plating

1. F. Pearlstein, "Electroless Plating", in Modern Electroplating, F. A. Lowenheim, ed., 3rd Ed., Wiley, New York, 1974, pp. 710-746.

2. A. K. Graham, ed., Electroplating Engineering Handbook, 3rd Ed., Van Nostrand Reinhold, New York, 1971, Chapt 15, "Nonelectrolytic Metal Coating Processes", pp. 475-507.

3. G. O. Mallory and J. B. Haydu, eds., Electroless Plating - Fundamentals and Applications, American Electroplaters and Surface Finishers Society, Orlando, FL, 1990.

4. Y. Okinaka and T. Osaka, "Electroless Deposition Processes: Fundamentals and Applications," in Advances in Electrochemical Science and Engineering, Vol. 3, pp. 55-116 (1994).

References on Aggregative Growth

1. E. Santacesaria, M. Tonello, G. Storti, R. C. Pace, and S. Carra, "Kinetics of Titanium Dioxide Precipitation by Thermal Hydrolysis", J. Colloid Interface Sci., 111, 44-53 (1986).

2. A. P. Philipse, "Quantitative Aspects of the Growth of (Charged) Silica Spheres", Prog. Colloid Polym. Sci., 266, 1174-1180 (1988).

3. C. M. Bagnall, L. G. Howarth, and P. F. James, "Modelling of Aggregation Kinetics of Colloidal Silica Particles", J. Non-cryst. Solids, 121, 56-60 (1990).

4. G. H. Bogush and C. F. Zukoski, "Uniform Silica Particle Precipitation: An Aggregative Growth Model", J. Colloid Interface Sci., 142, 19-34 (1991).

5. J. Y. Bottero, M. Axelos, D. Tchoubar, J. M. Cases, J. J. Fripiat, and F. Fiessinger, "Mechanism of Formation of Aluminum Trihydroxide from Keggin Al13 Polymers", J. Colloid Interface Sci., 117, 47-57 (1987).

6. A. Bleier and R. M. Cannon, "Nucleation and Growth of Uniform m-ZrO2", in Better Ceramics Through Chemistry II, C. J. Brinker, D. E. Clark, and D. R. Ulrich, eds., MRS, Pittsburgh, PA, 1986, pp. 71-78.

7. J. A. Dirksen and T. A. Ring, "Production of Powders for High-tech Ceramics", in High-tech Ceramics: Viewpoints and Perspectives, G. Kostorz, ed., Academic, New York, 1989, pp. 29-39.

8. T. Sugimoto and E. Matijevic, "Formation of Uniform Spherical Magnetite Particles by Crystallization from Ferrous Hydroxide Gels", J. Colloid Interface Sci., 74, 227-243 (1980).

9. W. P. Hsu, L. Ronnguist, and E. Matijevic, "Preparation and Properties of Monodispersed Colloidal Particles of Lanthanide Compounds. 2. Cerium (IV)", Langmuir, 4, 31-37 (1988).

10. J. H. A. van der Woude, J. B. Rijnbout, and P. L. de Bruyn, "Formation of Colloidal Dispersions from Supersaturated Iron (III) Nitrate Solutions, IV. Analysis of Slow Flocculation of Goethite", Colloids Surf., 11, 391 - 400 (1984).

11. P. J. Murphy, A. M. Posner, and J. P. Quirk, "Characterization of Partially Neutralized Ferric Nitrate Solutions", J. Colloid Interface Sci., 56, 270-283 (1976); "Characterization of Partially Neutrallized Ferric Chloride Solutions", J. Colloid Interface Sci., 56, 284-297 (1976); "Characterization of Partially Neutralized Ferric Perchlorate Solutions", J. Colloid Interface Sci., 56, 298-311 (1976); "Characterization of Hydrolyzed Ferric Ion Solutions. A Comparison of the Effects of Various Anions on the Solutions", J. Colloid Interface Sci., 56, 312-320 (1976).

12. K. M. Towe and W. F. Bradley, "Mineralogical Constitution of Colloidal Hydrous Ferric Oxides", J. Colloid Interface Sci., 24, 384-392 (1967).

13. N. Uyeda, M. Nishino, and E. Suito, "Nucleus Interaction and Fine Structures of Colloidal Gold Particles", J. Colloid Interface Sci., 43, 264-276 (1973).

References on Synthesis of Monodispersed Particles

1. E. Matijevic, "Production of Monodispersed Colloidal Particles", Ann. Rev. Mater. Sci., 15, 483-516 (1985).

2. T. Sugimoto, "Preparation of Monodispersed Colloidal Particles", Adv. Colloid Interface Sci., 28, 65-108 (1987).

3. M. Haruta and B. Delmon, "Preparation of Homodisperse Solids", J. Chem. Phys., 83, 859-868 (1986).

4. J. Th. G. Overbeek, "Monodisperse Colloidal Systems, Fascinating and Useful", Adv. Colloid Interface Sci., 15, 251-277 (1982).

5. T. Sugimoto, "Preparation and Characterization of Monodispersed Colloidal Particles", MRS Bull., Dec., 23-28 (1989).

6. J. Livage, M. Henry, J. P. Jolivet, and C. Sanchez, "Chemical Synthesis of Fine Powders", MRS Bull., Jan., 18-25 (1990).

7. M. Ozaki, "Preparation and Properties of Well-defined Magnetic Particles", MRS Bull., Dec., 35-40 (1989).

8. V. K. LaMer and R. H. Dinegar, "Theory, Production and Mechanism of Formation of Monodispersed Hydrosols", J. Amer. Chem. Soc., 72, 4847-4854 (1950).

9. K. Osseo-Asare, “Microemulsion-Mediated Synthesis of Nanosize Oxide Materials”, in Handbook of Microemulsion Science and Technology, P. Kumar and K. L. Mittal, eds., Marcel Dekker, New York, NY, 1999, pp. 549-603.

10. K. Osseo-Asare and F. J. Arriagada, “Growth Kinetics of Nanosize Silica in a Nonionic Water-in-Oil Microemulsion: A Reverse Micellar Pseudophase Reaction Model”, J. Colloid Interface Sci., 218, 63-76 (1999).

11. T. Sugimoto, ed., Fine Particles. Synthesis, Characterization, and Mechanisms of Growth, Marcel Dekker, New York, NY, 2000.

References on Electrodeposition

1. J. O'M. Bockris and G. Razumney, Fundamental Aspects of Electrocrystallization, Plenum, New York, 1967.

2. A. Brenner, Electrodeposition of Alloys, Vols. I and II, Academic, New York, 1963.

3. M. Fleishmann and H. R. Thirsk, "Metal Deposition and Electrocrystallization", in P. Delahay and C. Tobias, eds., Advances in Electrochemistry and Electrochemical Engineering, Vol. 3, 1963, p. 123.

4. A. Damjanovic, "The Mechanism of the Electrodeposition of Metals", in J. O'M. Bockris and B. E. Conway, eds., Modern Aspects of Electrochemistry, Vol. III, 1964, p. 224.

5. J. O'M. Bockris and A. K. N. Reddy, Modern Electrochemistry, Vol. 2, Plenum New York, 1970, pp. 1173 - 1231.

6. L. Young, Anodic Oxide Films, Academic, New York, 1961.

7. A. Calusaru, Electrodeposition of Metal Powders, Elsevier, New York, 1979.

8. M. Paunovic and M. Schlesinger, Fundamentals of Electrochemical Deposition, Wiley,

New York, 1998.