Nanocrystals from Solutions and Gels 1588830624

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Encyclopedia of Nanoscience and Nanotechnology

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Nanocrystals from Solutions and Gels Marc Henry Université Louis Pasteur, Institut Le Bel, Strasbourg, France

CONTENTS 1. Introduction 2. Nucleation 3. Crystalline Growth 4. Limited and Secondary Growth 5. Aggregation 6. Conclusion Glossary References

1. INTRODUCTION The interest in nanocrystalline materials comes from the occurrence of a large interface with a surrounding liquid or gaseous medium. A very elementary calculation is helpful here to grasp this importance of size. Any solid matter may be characterized by its molar volume Mv (cm3 mol−1
and its molar surface Ms (cm2 mol−1
. For a sphere of radius R, the number of bulk atoms Nv should be such that Nv × Mv = NA 4/3
R3 , whereas the number of surface atoms would be given by Ns × Ms = NA × 4R2 (with NA the Avogadro number). Consequently, we may define the atomic surface to volume ratio f = Ns /Nv = 3/R
× Mv /Ms
. As typical values for condensed solid matter would be Mv ∼ 20 cm3 mol−1 and Ms ∼ 6 × 108 cm2 mol−1 , it follows the very simple rule of thumb that f ∼ 1/R (nm). This analysis shows that for R = 1 nm there could exist no core atoms (the nanocrystal is a pure surface as f ∼ 100%), while for R = 1000 nm only one atom over 1000 remains accessible. The nanometer range 2–10 nm is thus particularly interesting as in this domain we have enough atoms (at least 50%) into the bulk to insure the existence of some useful mechanical, electrical, optical, magnetic, or chemical property. On the other hand we also have enough atoms (at most 50%) exposed to the outer space, creating an active interface able to deeply modulate the bulk properties. In fact, one may even hope for the appearance of a new property. A very nice example is provided by a quite old but very instructive paper ISBN: 1-58883-062-4/$35.00 Copyright © 2004 by American Scientific Publishers All rights of reproduction in any form reserved.

by Gortner concerning the state of water in colloidal and living systems [1]. The author was interested in the difference in mass between a fresh opaque jellyfish (weighting 500 g) and the dried transparent animal (weighting 0.45 g). As the volatile matter at room temperature is water, the conclusion of this simple experience is that the jellyfish was made of 99.9 wt% of water and only 0.1 wt% of solid matter. This solid matter includes all proteins, lipids, carbohydrates, and mineral salts needed to get a living animal. However, just take a glass containing 499.55 g of water and add 0.45 g of a well-dosed amount of organic and inorganic matter. You do not get a living animal, but just an inert more or less stable colloidal solution. Life is here really a new property not shared by a glass of water nor by a piece of sugar nor by a drop of oil nor by a chain of amino acids nor by sodium chloride crystals. The key to life apparition in this system is to create a nanosized solid/water interface, where the bulk, the surface, and the solution interact, leading to a system qualitatively different from the sum of its constituents. This simple example shows that making nanometric sized objects may be quite rewarding. However, in order to remain quite concise we will limit ourselves to the possible mechanisms allowing growing nanocrystalline dispersions and to their stability toward aging and/or aggregation processes. Even if several well-documented reviews already exist covering all these “well-known” aspects, it appears that none take into explicit account the chemical dimension of the problem. This complete absence of chemistry from nucleation, growth, and aggregation equations has the consequence that it is very difficult to establish a causal link between the final equilibrium size of the crystals and the growth conditions. In fact every crystal maker knows that playing with concentration, temperature, pressure, or ionic strength is not enough to fully control crystalline growth. The key to the success of controlled crystal growing lies in the wise use of chemical additives or suitable complexing reagents. It is the aim of this chapter to fully review an often quite old literature and bring to light all the hidden points where chemistry has a role to play. For each important step (nucleation, growth, aging, aggregation) we will put down for the first time a new analytical expression, linking the equilibrium final sizes to growth conditions. With these equations in hand crystal

Encyclopedia of Nanoscience and Nanotechnology Edited by H. S. Nalwa Volume 6: Pages (555–586)

556 scientists may then understand, on a quantitative basis, why the change in such physical or chemical parameter alters the final equilibrium size.

2. NUCLEATION 2.1. Physicochemical Parameters Nanocrystal formation from solutions and gels is ruled by several factors gathered in Figure 1 [2–4]. From the classical nucleation theory (CNT) viewpoint, five parameters are needed to fully characterize the state of the solution. These are the concentration of solute species c (mol l−1
, assumed to bear an electrical charge eq and moving with diffusion coefficient D, the dielectric constant of the solvent , and the temperature T . Recall that for typical molecular solute species we should have a constant value D ∼ 10−5 cm2 s−1 . For the growing solid phase, three parameters may be identified: its solubility cs (mol l−1
, its density  (g cm−3
, or molecular volume v = M/NA (with M molar mass of one structural motif and NA the Avogadro number). Notice that another useful way to express the molecular volume is to think to the characteristic molecular size of the structural repeating unit: d ∼ v1/3 . For typical binary compounds this value is roughly constant and may be taken approximately equal to v ∼ d 3 ∼ 10−285 m3 .

2.2. Interfacial Energy The interfacial energy  (J m−2
is finally the last critical parameter that remains to be introduced. Its physical meaning is perfectly clear if one refers to the top left diagram of Figure 1. In a growing crystal, two kinds of atoms may be identified. At the bottom, buried inside the core of crystal

Nanocrystals from Solutions and Gels

are atoms that interact with other similar atoms. The net force acting on these atoms is thus just zero. At the top, exposed to the surface are atoms displaying a clear force unbalance. On one side they are attracted by atoms buried within the bulk of the crystal, but on the other side they are involved in usually much weaker interactions (van der Waals or hydrogen bonds) with solvent molecules. In fact, the highest asymmetric situation is encountered when there is no solvent at all (i.e., crystals are placed under vacuum), leading to a maximum interfacial or surface energy. Consequently, in the presence of a given solvent, this interfacial energy may be strongly reduced. The stronger the interaction with the solvent, the lower the interfacial energy should be. This is just what is expressed by the Gibbs equation: d = − × d
P T

(1)

Here,  stands for the density in adsorption sites (sites nm−2
and refers to the electrochemical potential of adsorbed species. Table 1 gives some typical values of interfacial energies for aqueous solutions [3, 5–8]. This shows that expected values for this parameter should be in the range 50 ≤  ≤ 400 mJ m−2 . Let us notice that it has been very recently shown that vacuum surface energies may also be derived directly from the knowledge of the crystal structure [9]. In this last study, it was also proved that the surface energy concept remains valid even at a molecular scale. This last observation is important in a relation with the justified criticism that may be lodged against CNT that assumes constant surface tension for nucleation [10]. The first principles approach in [9] clearly shows that this criticism is not justified. For each crystal, one may define a characteristic vacuum surface energy averaged over a sphere whose value may be fixed (for most simple structures of course) by the relative spatial disposition of less than five atoms!!! Nevertheless, the most serious criticism remains as nucleation is an irreversible process that just cannot be handled by equilibrium thermodynamics. A more correct approach for this review would have then been to use entropy-based nucleation theory [10] instead of CNT. This was, however, not possible as analytical solutions exists only for liquid–vapor interfaces and not for solid–liquid or solid– solid interfaces, the subject of this chapter.

2.3. Nucleation Rate J From a macroscopic point of view, the most important quantity governing the final size and number of nanocrystals is the nucleation rate J (number of nuclei formed per unit volume and per unit time). A detailed analysis of the nucleation phenomenon [6] leads to the following expression for this nucleation rate:
 D GN J = 5/3 × exp − (2) v kT

Figure 1. A graphical overview of some factors ruling nucleation, growth, and spinodal decomposition processes. See text for details and explanations.

In this equation, k is the Boltzmann constant (k ∼ 138 × 10−23 J K−1 ) and GN is the energetic barrier to nucleation. The value of the pre-exponential factor J0 = J GN = 0
= D/v5/3 ∼ 10325 cm−3 s−1 has a clear physical interpretation. Accordingly, in the absence of any energetic barrier

Nanocrystals from Solutions and Gels

557

Table 1. Interfacial energies for water/solute interfaces derived from nucleation studies in aqueous solutions.  (mJ m−2 ) Halogens KCl KBr KI NH4 Cl NH4 Br NH4 I AgCl AgBr MgF2 CaF2 TlCl TlBr TlIO3

30 24 17 27 19 8 90 65 300 280 92 92 87

S, Se BaSO4 SrSO4 CaSO4 · 2H2 O PbSO4 Ag2 SO4 Na2 S2 O3 · 5H2 O BaSeO4 PbSeO4 TlSCN

135 85 117 100 96 16 88 71 65

Carbon BaCO3 PbCO3 CH3 COOAg SrC2 O4 · H2 O PbC2 O4

115 125 113 76 145

Oxides SiO2 TiO2 Mg(OH)2 Ca(OH)2

45 270 123 66

Cr, Mo, W PbCrO4 BaCrO4 Tl2 CrO4 Ag2 CrO4 SrMoO4 BaMoO4 CaMoO4 SrWO4 CaWO4 BaWO4

170 120 108 107 100 103 118 62 151 94

for incorporation into a crystalline embryo, the limiting step should be the diffusion of the solute species toward the embryo/solution interface. Consequently, the frequency of incorporation inc should scale like D/d 2 . This is because the solute species have just to diffuse over an area of size d × d to be immediately incorporated (recall that v ∼ d 3 and that GN ∼ 0). As the number of structural repeating units per unit volume is just 1/v ∼ 1/d 3 , the nucleation rate should scale like J0 ∼ inc /v ∼ D/d 5 ∼ D/v5/3 , in good agreement with Eq. (2). Obviously, J0 ∼ 10325 cm−3 s−1 is an incredibly high value and in most cases GN > 0, leading to a strongly reduced value. In fact, from a practical point of view,

nucleation rates can be experimentally measured only when 1 ≤ J ≤ 1015 cm−3 s−1 . Consequently, at room temperature, the nucleation barrier GN should be at least 80 kJ mol−1 and at most 170 kJ mol−1 .

2.4. Thermodynamic Barrier
G∗ Two terms are expected to contribute to the nucleation barrier. The first one G∗ has a thermodynamic origin. It corresponds to the energetic competition existing between reduction of the sursaturation ratio S = c/cs when S > 1 and the necessary increase in area A of the growing solid phase (energetic cost  × A). Identifying activities a to concentrations c ( = kT × Ln a ∼ kT × Ln c), one may write for the transfer of n moles of solute species from the solution to the crystal GnS → nC
= n ×  C − S
+  × A ≈  × A − nkT × ln S

(3)

The variation of G with n is visualized at the top right corner of Figure 1. If C ∼ S (i.e., c ∼ cs or S ∼ 1), the dominant contribution comes from the positive interfacial term (first monotonic curve in Figure 1). On the other hand, when S ≫ 1, the negative chemical contribution insures that a maximum should be reached for a given number n∗ of solute precursors (second curve in Figure 1). According to this analysis, only embryos such that n > n∗ are able to reach a macroscopic size as G < 0 for the n → n + 1
addition (growth). For other embryos such as n < n∗ the n → n + 1
association is characterized by G > 0, meaning that such species should dissolve back into the solution. Embryos such that n = n∗ are called critical nuclei, and the corresponding positive G∗ value defines the first thermodynamic contribution to the nucleation barrier GN . The magnitude of G∗ may be readily derived under the assumption that embryos undergo no change in shape (only the mean size is changing among similar bodies) during nucleation. Consequently, the ratio K = A/V 2/3 should remain invariant, and if the total volume of a given embryo is V ∼ n × v then the associated area A may be written A = K × n × v
2/3 . Inserting this value into Eq. (3) and looking for a maximum by setting dG
/dn = 0 leads to the following thermodynamic barrier G∗ =

=

n∗ × kT Ln S 2 A/3
3 V /2
2

n∗ = 2

 3 v2 kT Ln S
3 (4)

Table 2 gives the values of the dimensionless shape factor for some ideal geometries. As evidenced in Eq. (4) the only way to get G∗ = 0 is to have a vanishing interfacial energy. Such a condition ( ∼ 0) is easily met in polymer chemistry or in metallic alloys, and in this case phase segregation cannot lead to the sharp boundaries characterizing systems with  ≫ 0 (bottom left picture in Figure 1). Instead, for this spinodal decomposition scheme, deeply interconnected patterns

Nanocrystals from Solutions and Gels

558 Table 2. Dimensionless shape factors, for some common regular 3D geometries. Shape

Area A

Sphere

4a2

Volume V

Cube

6a2

4 3 a 3 5a3 2 6 √ a3 o¨ 4 5 2 √ 2 a3 3 a3

Parallelepiped 4a × 2a × a

28a2

8a3

Regular icosahedron Regular dodecahedron Regular octahedron

√ 5a2 3 √ 15a2 o¨ 3 √ 5 √ 2a2 3

√

2 12

Regular tetrahedron

√ a2 3

Rod 10a × a × a

42a2

10a3

Plate 10a × 10a × a

240a2

100a3

a3

= 4A3 /27V 2

16o¨ = 167552 3 √ 16 3 = 202162 4  √ 80 o¨ 5 = 222012 4 √ 16 3 = 277128 8 × 4 = 32
3 7 = 508148 4× 3 √ 32 3 = 554256 143 = 10976 25 5 4 = 2048 5

Note: Shapes are characterized by one size a and classified by increasing value. Here  stands for the golden √ ratio defined as the positive root of the 2 x = x + 1 equation, viz.  = 1 + 5
/2.

with rather diffuse boundaries are typically observed (bottom right picture in Figure 1).

2.5. Kinetic Barrier
G′ As indicated, the thermodynamic barrier G∗ is not the only contribution to the nucleation barrier GN . Another kinetic barrier G′ associated with chemical reactions occurring during jumps across the nucleus–solution interface should also be considered. Obviously, for a solid metallic alloy nucleating from a melt one may expect G′ ∼ 0 as the system undergoes only a disorder → order phase transition on going from the liquid to the solid state. During this transition local metallic chemical bonds between atoms are not fundamentally changed, explaining the low kinetic cost G′ ∼ 0. Such would not, however, be the case in other systems.

2.5.1. Quartz versus Glass Consider for instance the growth of quartz crystals from a silica melt. Besides the thermodynamic barrier associated with the quartz/fused silica interface and difference in chemical potentials, one has to take into account the fact that the local structure of both phases is very different. In the quartz crystal, SiO4 tetrahedra are strongly engaged into spiral chains, whereas in the melt small oligomeric species (which may be chains, cycles, or cages) are encountered. In order to be able to jump across the nucleus/liquid interface and fit inside the crystalline network, oligomeric species must first be reduced to monomeric SiO4 tetrahedra. This means that rather strong Si–O–Si bonds have to be broken, and that a corresponding activation energy G′ > 0 has to be provided before nucleation and growth of quartz crystals can occur.

Let us remark that this activation energy G′ has absolutely nothing to do with thermodynamics as almost the same Si–O–Si bonds are found before (oligomeric species) and after nucleation (spiral chains). This term is a true kinetic barrier inevitably associated with any kind of reconstructive transformation [11]. Accordingly, glass formation is intimately linked to the existence of a large kinetic barrier G′ which renders the cost of nucleation quite prohibitive just because GN = G∗ + G′ . If G∗ were the only contribution to nucleation, glasses would never be formed and should not exist.

2.5.2. Sodium Chloride: A Case Study The same kind of kinetic barrier G′ is expected when crystallization occurs from solutions or gels. First consider the simplest case: nucleation of a crystalline salt (NaCl for instance) from an aqueous solution. In solution we find ionic species such as [Na(OH2
x ]+ and [Cl(H2 O)x ]− . The precise value of the coordination number x does not matter here as it depends on the molar fraction of sodium chloride. Consequently, variation in x is well reflected by variation of the chemical potential S and should influence only the thermodynamic term G∗ . Our point of interest is rather how such solute species should be transformed in order to allow them to jump across the NaCl/water interface for further incorporation into the NaCl crystalline network. As this crystal is built from a regular stacking of naked Na+ and Cl− ions, it is clear that ion pairing should be the very first step: [Na(OH2
x ]+ + [Cl(H2 O)x ]− → [NaCl(OH2
2x ]. Having formed a neutral species, water removal may then occur: [NaCl(OH2
2x  → [NaCl] + 2x H2 O, and only after that incorporation into the network becomes possible. The origin of the kinetic barrier G′ lies here in the lability of water molecules around cations and anions.

2.5.3. Aquo Cations For the NaCl case, it is obvious that the lability around Na+ or Cl− ions is rather high (Fig. 2), meaning that G′ ∼ 0. But this clearly would not be the case for more charged cations, as it is well known that the kinetics of water splitting, [M(OH2
N ]z+ → [M(OH2
N −1 ]z+ + H2 O, is considerably slowed down if z increases or if the ionic radius of the metallic element decreases [12]. Also, for some transition metal elements such as Cr3+ , crystal field stabilization energies may be so high (leading to G′ ≫ 0) that nucleation of the corresponding hydroxide Cr(OH)3 cannot occur anymore. In such a case, gelation (the equivalent for solutions of glass formation from melts) instead of precipitation is usually observed. Figure 2 shows how the kinetic constant kw characterizing the removal of aquo ligand from the metal coordination sphere varies for usual ionic species. It was found that, at room temperature, it lies above 10−6 s−1 (V2+ or Cr3+ ions) and below 1010 s−1 (Ba2+ or Cs+ cations) [12, 13]. Now, if h stands for the Planck constant, the activated complex kinetic theory says that the rate constant may be written kw = kT /h
× exp−G0 /kT ) or for T = 300 K (G0 /kJ mol−1
= 73 − 571 × logkw /s−1
. Consequently, assuming that G′ ∼ G0 , the kinetic barrier for nucleation of a metallic salt or hydroxide should be in the range

Nanocrystals from Solutions and Gels

559 2.5.5. Metal Alkoxides The same kinetic barrier is encountered when the nanocrystal nucleates after decomposition of metallo-organic complexes in nonaqueous media. Figure 2 shows that, at least in the case of metallic alkoxides precursors, a considerable rearrangement of chemical bonds should also occur. As before, with hydrolysis of the metal alkoxide requiring a proton transfer, the kinetics depends strongly on the amount of acid or base added with the water. As shown in Table 3 for silicon alkoxides [17–22], the height of the kinetic barrier G′ is at least 90 kJ mol−1 explaining the difficulty to nucleate quartz directly from the solution at room temperature. Unfortunately, there is a paucity of kinetic data for other metallic cations. To the best of our knowledge, only a global kinetic constant k ∼ 30 s−1 is available concerning the nucleation of titanium or aluminum oxides from Ti(OEt)4 and Al(OBus
3 respectively [23]. This would mean that octahedral species may have significantly lower kinetic barriers (G′ ∼ 65 kJ mol−1
relative to tetrahedral silicate species.

2.6. Applications Figure 2. Chemical origin of the kinetic barrier G′ for oxide, hydroxide, or metallic salt nulceation. Top left: type of ligand expected around a metallic cation Mz+ in aqueous solutions as a function of pH. Top right: chemical bonds rearragements expected for oxy ions or metal alkoxides in the case of oxide nucleation. Bottom: techniques used for measuring lability of water molecules around metallic cations Mz+ (z ≤ 3, i.e., hydroxide or metallic salt nucleation).

15 < G′ < 110 kJ mol−1 , depending on the electronic configuration, ionic radius, or total charge of the metallic cation.

2.5.4. Oxo Anions Obviously this picture can be valid only for metallic cations having a low formal oxidation state (typically z ≤ 3) allowing them to keep the aquo ligand within their coordination sphere. As shown in Figure 2, cations displaying higher formal oxidation states z are on the contrary extensively hydrolyzed in aqueous solutions. These cations should then lead to oxide rather hydroxide precipitation [14]. Silicon (z = +4) is a good example of this behavior, as it is found under low pH conditions under the silanol form Si(OH)4 . As this form does not display any aquo ligand in its coordination sphere, a considerable rearrangement of chemical bonds is necessary (Fig. 2) in order to incorporate it in a network based on corner-sharing SiO4 tetrahedra (Si– O–Si bridges). However, as a proton transfer is needed in the transition state to form a good leaving group (water), the kinetics of condensation is expected to depend strongly on the solution pH. Accordingly, experiments [15] shows that the kinetic constants for silanol condensation vary from 1018 M−2 s−1 at pH ∼ 7 up to 1034 M−2 s−1 at pH ∼ 10. Proceeding as before, this corresponds to a kinetic barrier 50 kJ mol−1 < G′ < 70 kJ mol−1 . Another possibility of estimating this kinetic barrier G′ for nucleation from aqua and oxo ions is to use 18 O exchange studies [16] nH2 O∗ + MOH2
n z+ → nH2 O + M∗ OH2
n z+ nH2 O∗ + XOn z− → nH2 O + X∗ On z−

After this rapid outline of the classical nucleation theory, we may take time to see some typical examples, the relative orders of magnitude of the two kinetic barriers G∗ and G′ .

2.6.1. Ionic Crystals Consider first the case of a simple crystalline salt such as KCl for instance (M = 7455 g mol−1 ,  = 1984 g cm−3 , i.e., v = 10−282 m3 ). Referring to Figure 2, the characteristic residence time of water molecules around the potassium cation is  < 10−10 s. Consequently we may suppose that G′ ≤ 16 kJ mol−1 . A look at Table 1 shows that for the KCl/H2 O interface we have  ∼ 30 mJ m−2 . Consequently, at T = 300 K (kT = 41 × 10−21 J, or RT = 248 kJ mol−1
, we may write n∗ = 025

log S
3

and G∗ = 0714

kJ mol−1 log S
2 (5)

Owing to its cubic structure one may further assume a cube-shaped nucleus ( = 32); that is, n∗ = 8/log S
3 and Table 3. Measured rate constants and associated kinetic barrier G′ for the chemical reaction (RO)3 Si–OX + HOY → (RO)3 SiOY + HOX. XR


Y

Rate constant

pH

G′ kJ mol−1


Ref.

CH3 C2 H 5 C4 H 9 C6 H13 H(Me) H(Me) CH3 H(Me) CH3 C2 H 5 H(Et)

H H H H Si Si Si Si2 Si2 H Si

10 M s 10−13 M−1 s−1 H+ −1 10−17 M−1 s−1 H+ −1 10−21 M−1 s−1 H+ −1 10−4 M−1 s−1 10−21 M−1 s−1 10−48 M−1 s−1 10−47 M−1 s−1 10−49 M−1 s−1 10−42 M−1 s−1 10−53 M−1 s−1

5.5 7–2 7–2 7–2

104 120–92 123–94 125–96 96 85 100 100 101 97 103

[17] [18] [19] [19] [20] [17] [20] [21] [21] [22] [22]

−55

−1 −1

5.5 2.8 2.8 3.1 3.1

Note: Y = Si2 refers to the (RO)3 SiOSi(OH)(OR)2 dimer.

Nanocrystals from Solutions and Gels

560 G∗ = 228 kJ mol−1
/(log S
2 . Now for S = 2 (two times the solubility of KCl, cs ∼ 347 wt% at 20 C), it transpires that about 300 KCl pairs have to meet together, forming a sphere of radius r ∗ ∼ n∗ v
1/3 ∼ 26 Å, in order to nucleate the KCl network. The very low probability of such an event is easily understood by looking at the nucleation rate J . With G∗ = 252 kJ mol−1 and making the conservative choice G′ = 16 kJ mol−1 , it comes that GN = 268 kJ mol−1 leading to J = 1032 × exp−268/248
∼ 10−15 nuclei per cm3 and per second. Another statement of this matter of fact is to say that in order to see one such nucleus appear in 1 cm3 of solution, one has to wait about 32 Myr!!! Let us now see what would happen by just doubling the sursaturation ratio S. Now, with S = 4 the same equations tell us that n∗ = 37 (critical nucleus with radius ∼13 Å). If the thermodynamic barrier is still dominant, it has nevertheless been reduced down to G∗ = 63 kJ mol−1 . The new barrier to nucleation is thus GN = 79 kJ mol−1 leading to J = 1032 × exp−79/248
∼ 1018 cm−3 s−1 . Consequently, instead of waiting a virtual eternity to see just one nanocrystal, everything is finished (i.e., c ∼ cs corresponding to about 1022 nanocrystals with radius 13 Å) in about 8400 s or less than 3 hours.

2.6.2. Critical Sursaturation Ratio S∗ This example clearly demonstrates that nucleation is basically a critical phenomenon, as a mere doubling of one parameter S leads to a variation in J of about 33 orders of magnitude. These considerations are shown graphically at the bottom of Figure 1 (two middle diagrams). Below a critical value S ∗ , nothing can happen, while slightly just above this threshold, the nucleation rate displays a very fast divergence (middle left diagram). The main consequence of this burst of nuclei is that the sursaturation is strongly and rapidly reduced and may instantaneously become smaller than S ∗ (middle right diagram). This condition is usually met in moderately dilute solutions leading to just one burst of nuclei. Further removal of the remaining sursaturation would only be removed by growth processes (vide infra). This is the classical picture for the formation of monodisperse nanocrystals under homogeneous nucleation conditions [24, 25]. On the contrary, for strongly concentrated solutions, the first birth of nuclei is not able to reduce the sursaturation ratio below S ∗ , meaning that J > 0 during all the precipitation process. Under such circumstances, a polydispersed system is usually obtained and aggregation phenomena may become competitive for determining the final particle size or shape. From a practical viewpoint, the critical sursaturation ratio S ∗ may be defined as the value of S leading to J = 1 or log J = 0 [26] Ln S = ∗



kT
3

 3 v2 LnDv−5/3 gN
/1

(6)

2.6.3. Silica This example also illustrates the crucial role played by the kinetic barrier G′ . It is just because the thermodynamic term is dominant and the kinetic one negligible that it is so difficult to keep control over the nucleation

rate. To see that, let us consider the nucleation of silica SiO2 · 2H2 O (M = 7808 g mol−1 ,  = 22 g cm−3 , i.e., v = 10−282 m3 and  ∼ 45 mJ m−2 ) [8, 27]. For nuclei displaying spherical shapes, we may write n∗ = 14/log S
3 and G∗ = (40 kJ mol−1
/(log S
2 . Now, let us choose G′ ∼ 100 kJ mol−1 as a typical value characterizing the interfacial oxolation reaction (see Table 3): Si–OH + HO– Si → Si–O–Si + H2 O. This choice then leads to GN ∼ 540 kJ mol−1 for S = 2 and GN ∼ 210 kJ mol−1 for S = 4. Converting these values into nucleation rates according to J cm−3 s−1
= 1032 × exp−GN (kJ mol−1 )/2.48] leads to J ∼ 3 × 10−63 cm−3 s−1 (S = 2) and for S = 4, to only J ∼ 2 × 10−5 cm−3 s−1 . Consequently, doubling the sursaturation ratio as before has here no practical consequences, as the nucleation rate remains quite negligible (about 1 nucleus per cm3 after one day). Notice that even for S = 10 (10 times the solubility of amorphous silica, or c[Si(OH)4 ] ∼ 10−2 M) the nucleation rate is only J ∼ 3 × 107 cm−3 s−1 , a perfectly measurable value. The presence of a reasonable kinetic barrier then provides an efficient healing mechanism against the otherwise pathological behavior of the nucleation rate versus sursaturation variation. For oxide or hydroxide precipitation, the kinetic term is provided by the inertness of the metallic cation toward water substitution. It is thus possible to control the nucleation process under dilute conditions through thermal hydrolysis for instance [28].

2.6.4. Metallic Salts For metallic salts (such as carbonates, sulfates, or phosphates for instance) the kinetic barrier should be provided by the slow decomposition of a suitable preformed metallic complex [29]. Only in this case do we get a chance to keep some control over the nucleation process. Among the numerous chemical methods available to control the nucleation stage for sparingly soluble substances, we may cite [29–34]: (i) Variation of pH induced by temperature elevation (thermal solvolysis) is a method well suited for acidic media. In order to generate basic conditions a suitable organic reagent able to decompose into ammonia (urea, hexamethylene tetramine) upon heating should be added. (ii) Some anions such as [CO3 ]2− , [C2 O4 ]2− , [SO4 ]2− , [PO4 ]3− or S2− may also be generated in-situ after decomposition of various esters or amides. (iii) Metallic cations may be masked by formation of a suitable complex that may be carefully destroyed through pH changes or through exchange with another complex. (iv) The use of mixed solvents with one volatile component is another useful strategy. Upon aging or heating the volatile species will be evaporated inducing controlled precipitation. (v) For transition metals displaying several oxidation states, one may add a reducing reagent for highvalency forms or an oxidizing one for low-valency forms. (vi) Finally, if the temperature of the medium cannot be changed, the possibility remains to use a light sensitive molecule that should release one of the

Nanocrystals from Solutions and Gels

561

nucleating species upon irradiation with ultravioletvisible light or even -rays.

2.7. Heterogeneous Nucleation It is a well known experimental fact that nucleation from a vapor, melt, or solution is affected considerably by traces of impurities in the system. Indeed, some authors even say that true examples of homogeneous nucleation in the condensed state should be indeed very rare [26]. One may then wonder about the practical utility of this treatment. In fact, it has been shown [35] that if R is the radius of the foreign nucleating particle and if  stands for the contact angle with the foreign surface, then G∗ (hetero) = G∗ (homo) × cos  R/r ∗
with  ∼ 1 when R < 10 nm or when  = 180 . Above 100 nm  becomes independent of R (with  < 1) leading to [26]
 2−3cos +cos3  R → ⇒ G∗H = G∗ × 4 r∗ =

 2v sin kT lnS 2

(7)

The important point is that due to the existence of a kinetic barrier G′ , homogeneous nucleation is already governed by a sticking probability gN = exp−G′ /kT
J ∼ Dv−5/3 gN × exp−G∗ /kT


various mechanisms of these competing processes. Kinetically speaking, the dominant process just after nucleation has occurred should be crystal growth. Accordingly, Ostwald ripening is ruled by the Ostwald–Freundlich equation for two spheres with radii (a1 a2
characterized by their solubility (c1 and c2 respectively) [36] 
c 1 1 kT × ln 1 = 2v − (9) c2 a1 a2

It is then expected to become important only when a2 ≫ a1 in order to have c1 ≫ c2 (disappearance of small crystals and growth of larger ones). On the other hand, aggregation implies that nuclei have to move through diffusion in order to meet together. According to the Einstein equation D = kT /6a
, D should be, for a given solution viscosity , smaller for a nuclei displaying a size a much larger than that of active solute species for crystal growth. In the following, we consider a solution containing N nanocrystals per unit volume, all built from incorporation of nt
solute species. As n is changing with time, this means that some characteristic linear size rt
increases with time according to a given growth law drt
/dt = f c r
. Writing the volume V of each nanocrystal as V =  × r 3 , where  is a shape factor ( = 4/3 for spheres,  = 8 for cubes, etc.) means that nt
=  × rt
3 /v. According to literature [3, 5], four main mechanisms may be recognized for crystalline growth (see Fig. 1).

(8)

Now as we know that 0 ≤  ≤ 1 we may rewrite (7) as G∗ (hetero) = G∗ (homo) − Ecos  R/r ∗ ), with E the energetic correction that should be applied and such that E = 0 if R = 0 or  = 180 . As far as the nucleation rate is concerned, we then just have to define a modified sticking probability gH = expE − G′
/kT showing that only the pre-exponential term is affected. These two terms gN and gH could then help to understand the considerable spreading of the measured kinetic constant J0 = J G∗ = 0
observed in literature (10 ≤ J0 ≤ 1032 cm−3 s−1
[26]. Obviously, this spreading is no more surprising in view of the earlier analysis but just reflects the importance of the competition between the kinetic barrier G′ and the heterogeneous activation E. For this chapeter, we will still use (8) as the correct expression for the nucleation rate. However, the gN term should be interpreted as a nucleation probability whose value depends on at least three physicochemical parameters: G′ , cos , and R. With this new interpretation of the kinetic constant J0 , the exact nature of the nucleating event (either homogeneous or heterogeneous) becomes immaterial as soon as gN reflects the exact physicochemical state of the solution.

3. CRYSTALLINE GROWTH 3.1. Competitive Growth Ideally, it is desirable that the nucleation stage should be the whole story of nanocrystal formation from solutions and gels. From a practical point of view, one has then to ensure that other stages such as growth, Ostwald ripening, or aggregation should be avoided. In the following, we will try to understand how this should be possible by studying the

3.2. Diffusion Controlled Growth In this first mechanism, each partner is characterized by a linear size [rm for solute species and rt
for nanocrystals] and a diffusion coefficient (D for solute species and Dr for nanocrystals). If the solute species can be incorporated into the crystalline network immediately after sticking to a crystal, the limiting step should be diffusion of solute species toward the solid/liquid interface.

3.2.1. Growth Law Let d be the solid angle spanning the direction in which a sticking event has been observed. The rate for mass increase of the nanocrystals may be written d 2 n/dt d = k × c − cs
with k a kinetic constant which may be approximated by k ∼ Dr + D
r + rm
. As Dr ≪ D and r ≫ rm , it comes k ∼ D × r, or after integration over all the possible sticking directions: dn/dt = 4Dc − cs
r. But as nt
=  × r 3 /v
, we should also have dn/dt = 3 × r 2 /v
× dr/dt
. The final growth law for pure diffusion may thus be written

  4 4 Dvc − cs
dr 2 dr = ⇐⇒ = Dvc − cs
dt 3 r dt 3 (10) An interesting point is that irrespective of the original size r0 at nucleation time (t = 0) and provided that c ∼ c0 , then the square of the radius of all particles increases at the same rate [37]. Consequently, calling the absolute width of the distribution r for radius r and r0 for radius r0 , we have r × r ∼ r0 × r0 . This means that the absolute width of the distribution becomes narrower with time (r ∼ r0 /r × r0 ) while the relative width will decrease even faster (r/r ∼ r0 /r2 × r0 /r0 
.

Nanocrystals from Solutions and Gels

562 3.2.2. Chronomals From a practical point of view, one way to decide if such a rate law applies is to measure how the concentration in solute species ct
changes with time. Knowing ct
and the initial concentration c0 , one may write for spherical shapes  = 4/3
ct
= c0 − N × 4r 3 /3v
and compute the following quantity xD =

c0 − ct c0 − cs

⇐⇒

√ r = re 3 xD

(11)

Here re is the equilibrium size of the nanocrystals given by cs = c0 − N × 4re3 /3v
. Inserting (11) into (10) then leads to the characteristic t = KD ID law [5] with KD = re2 /3 Dvc0 − cs
 and ID =



xD 0

√ √ dx 1−xD 3 1 −1 3 tan − Ln = √ √ −1/3 1− 3 xD
3 1−x
3 x 2 2+xD (12)

A decisive test (named chronomal analysis) for validating this mechanism would be to observe a straight line for an ID = f t
plot leading to a diffusion coefficient D (computed from the observed slope KD ) close to 10−5 cm2 s−1 . Also notice that for crystals large enough the growth rate through diffusion may be increased by convection according to Dvc − cs
dr =F dt r

(13)

Here F is a complex hydrodynamic factor function of the solution viscosity [3].

3.2.3. Characteristic Time Besides knowing the growth law, one may also be interested in relating the final equilibrium size re to initial concentration in solution c0 . Let tD be the characteristic time below which the condition S > S ∗ remains satisfied. In this time domain J > 0 and if the initial concentration is high enough, we are allowed to write J c
∼ J c0
. When the condition S < S ∗ is reached due to solute species consumption during nucleation and growth (t > tD ), one should expect that J drops suddenly from J c0
down to zero. Further matter consumption should then be attributed to crystalline growth only. Now, from the growth law, we know that a nanocrystal nucleated at time  has reached, at time t, a size r t
=





t

f c r
× d

 ≤  ≤ t


(14)

Consequently, the concentration in solute species at time t should be given by t J c
r t
3 d v 0 t vc0 − ct
≈  J c0
r t
3 d

ct = c0 − ⇒

0

(15)

For diffusion growth, the magnitude of tD may be evaluated from (10) and (14) by setting c ∼ c0 , leading to  8Dvc0 r t
= t −
1/2 3  
 2J c0
8Dvc0 3/2 5/2 ⇒ vc0 −ct
≈ ×t (16) × 5 3 This shows that concentration of solute species decreases with time according ct = c0 1 − t/tD
5/2 ] with tD given by
2 1/5
  
1 5  1/5 3 3 tD ≈ × × 8 2 vc0 J c0
2/5 D3/5 
1 2  1/5 × 2/5 ≈ (17) 5 vc0 J0 D3/5

3.2.4. Equilibrium Size The next problem t is now to compute the total number of particles N ≈ 0 D J × dt formed during time tD . To do this, one may notice that from (8) and (4) that if n∗ changes slowly with time
n∗ c  3 v2  Ln J ∗ = n ⇒ J ≈ J c
= 2 0 3  Ln c kT Ln S
c0 (18) Thus, for the particular case of diffusion growth 
5/2  t c = c0 1 − ⇒ N ≈ J c0
tD f n∗
tD  1 n∗ 

k ∗ f n∗
= 1 − x5/2
n dx = (19) k + 2/5
0 k=1

Then, from the mass conservation law Nre 3 = c0 − cs
v ∼ c0 v, it becomes possible to get an analytical expression linking the average equilibrium size re  of the nanocrystals to the solid phase and solution properties  
1/5 1 94 vc0 3 = re  ≈ v11/5 c06/5 gN−3/5 N f n∗
 
3  3 v2 (20) × exp 5kT
3 Ln S0
2 Figure 3 shows the typical variation of re  with concentration in solute species c for several values of the dimensionless nucleation constant N =  3 v2 /kT Ln 10)3 . This fundamental curve shows that when N is low, re  remains roughly constant above S = 10, with typical sizes close to 1 nm. On the other hand, when N is high, nanocrystals with typical sizes close to 10 nm can be formed from concentrated solutions. In all cases, there is a critical concentration domain where average size suddenly drops from the millimetric range down to the nanoscopic one. One may also notice that for given physical conditions (sursaturation ratio S and nucleation constant N kept constant), chemistry can still play a determinant role on particle size through the sticking probability term (or kinetic barrier) gN . As seen in (20), decreasing the sticking probability (i.e., increasing the kinetic barrier) should lead to larger nanocrystals.

Nanocrystals from Solutions and Gels

563

concentration 10cs

100cs

1000cs

160

0

1.0

0,1 M ID

2

3

1M 4

5

6

1m

dr Dv(c - cs) r dt

0.6

-2

log re (m)

1

0.8

-1

xD

-3

0.4

-4

0.2

1 mm

0.0 -2

-5 -6

-1

log ID

0

1

re 1 µm

80

-7

60

-8 20

-9 -10

0

3.3.2. Thermodynamic
G ′ ∗ and Kinetic Barrier
G ′′

40 1 nm

10 23

1Å 24

25

26

27

log c (m-3) Figure 3. Crystalline growth controlled by diffusion. Boxed curve shows the shape of the chronomal function ID given by Eq. (9). Other decreasing curves were plotted according to Eq. (20) assuming a spherical shape ( = 4/3 = 4) with f n∗
 = 1/3 for six typical values of the nucleation constant N =  3 v2 /kT Ln 10
3 leading to log re = 1/3
log18v11/5 /4
+ 6/15
log c − 1/5
log gN + N /5log c/cs
2 . Other parameters were v = 10−285 m−3 , cS = 1022 m−3 , and g = exp−G′ /kT
= 10−5 . The key point is the final nanometer size of the crystal when precipitation occurs in concentrated solutions.

3.3. Interface-Limited Growth One has to remember that (20) applies only when diffusion in solution is the rate limiting step for crystalline growth. Physically, this means that if a sticking probability exists for nucleation (gN ≪ 1), the corresponding growth sticking probability gC should be close to unity. Obviously, systems satisfying simultaneously these two conditions are not very common, and in most experimental situations if gN ≪ 1, then gC ≪ 1 too. Consequently, we must look for a possible generalization of (20) in order to be able to cover the widest range of chemical systems.

3.3.1. Surface Nucleation Rate In this mechanism it is supposed that in order to form a new layer, solute species have to be adsorbed onto the crystal surface and make, through surface diffusion, a critical two-dimensional nucleus. As some chemical reactions must occur to lead to this new 2D nucleus, bulk diffusion in the solution cannot be the rate-determining step. As before, one may define a surface nucleation rate J ′ ∼ Dv−4/3 × exp−Gc /kT
with Gc = G′ ∗ + G′′ . Here, the preexponential factor has been written by taking into account that for Gc ∼ 0, the surface nucleation rate may be approximated by J ′ ∼ 1/d 2
× inc = Dv−4/3 ∼ 1025 cm−2 s−1 . Con′ cerning the thermodynamic barrier G ∗ , the prime symbol is a reminder for the fact that we are now treating a twodimensional (2D) problem instead of a 3D one. Similarly, the double prime symbol refers to the kinetic barrier associated with all chemical reactions needed to transfer a structural motif from the surface into the crystalline lattice.

At first sight, one may think that as the solute species responsible for crystalline growth are the same as those responsible for bulk nucleation, one should have G′′ ∼ G′ . In fact, this cannot be true. For bulk nucleation a crystal/solution interface first has to be created in order to initiate crystalline growth. For interface-limited growth, the solid interface is already there and by its mere presence may catalyze or inhibit the incorporation of the solute species inside the crystalline network. Consequently, in the most general case G′′ = G′ , the equality being realized only in the case of a completely neutral crystalline interface. As before, we should define gC = exp−G′′ /kT
as the growth sticking probability. Concerning the associated thermodynamic barrier to surface nucleation, it should depends on a frontier energy  ′ = G/L
p T (mJ m−1
similar to the surface energy concept  = G/A
p T (mJ m−2
. In order to simplify the problem we will assume that  ′ ∼ d = v1/3 . With this simplification the free energy variation for incorporation of m moles of surface species inside the crystal may be written GS mS → mC
= m ×  C − S
+  ′ × L ≈  ′ × L − mkT × Ln S

(21)

As for 3D nucleation, the magnitude of G′ ∗ may be readily derived under the assumption that surface embryos undergo no change in shape (only the mean size is changing among similar bodies) during surface nucleation. Consequently, the ratio K ′ = L/A1/2 should remain invariant, and if the total area of a given embryo is A ∼ m × a then its frontier L may be written L = K ′ × m
1/2 × v1/3 . Inserting this value into Eq. (21) and looking for a maximum by setting dGS
/dm = 0 leads to the following thermodynamic barrier ∗

G′ = m∗ ×kT LnS

m∗ = ′

 2 v4/3 kT LnS
2

′ =

L2 4A (22)

Table 4 gives the values of the dimensionless shape factor ′ for some ideal 2D geometries. Table 4. Dimensionless shape factors, for some common regular 2D geometries. Shape

Length L

Area A

′ = L2 /4A

Circle

2a

a√2 3a2 3 2 a2 2 2a √ a2 3 4 10a2

 = 31416 √ 2 3 = 34641

Regular hexagon

6a

Square Rectangle 2a × a

4a 6a

Equilateral triangle

3a

Rectangle 10a × a

22a

4.0000 9/2 = 45000 √ 3 3 = 51962

121/10 = 121000

Note: Shapes are characterized by one size a and classified by increasing ′ value.

Nanocrystals from Solutions and Gels

564 3.3.3. Timing Events Having a suitable expression for the rate of surface nucleation J ′ ∼ Dv−4/3 gC × exp−G′ ∗ /kT
, two characteristic times have to be considered: (i) The time N = J ′ Ai
−1 needed to nucleate a new layer on a perfectly planar interface displaying area Ai . (ii) The time C = ai /k′ needed to get a full coverage of the area Ai just after surface nucleation. Here ai stands for the largest linear size found on surface Ai and k′ = da/dt is the linear growth rate of surface nuclei. This rate should be proportional to the frequency of arrival of solute species at any edge of the surface times a sticking probability ( ′ ∼ D × gc /d) and should also depend on their average number at the interface [n′ ∼ c − cs
v]. Consequently, with d ∼ v1/3 , we should have k′ ∼ n′ × ′ ∼ Dgc v2/3 c − cs
. Now, depending on the relative values of N and C two limiting cases may be encountered.

3.3.4. Mononuclear Growth Law For mononuclear crystalline growth, the rate limiting step is the apparition of a new 2D nucleus on the surface (N ≫ C ). In this case, the linear growth rate dr/dt may be written d dr = ∼ J ′ ′ r 2 v1/3 dt N 
G′ ∗ 2 ′ −1 r = km × r 2 =  DgC v exp kT

(23)

Here ′ stands for the shape factor of the surface nucleus. It should be obvious that such a mechanism cannot be the whole story as if r with a size r0 , then at time t one tone starts would have km 0 dt = r0t dr/r 2 ; that is, rt = r0 /1 − km tr 0
]. It thus appears that rt → + for a finite time t → 1/km r0
. Such a divergence being unacceptable from a physical standpoint, this means that this mechanism should be valid only for very small sizes.

= vkp c p

Accordingly, when the exposed surface becomes large enough, the probability to observe several nucleation events on the same area Ai cannot be neglected and one has to switch to the polynuclear growth. In this alternative mechanism, the rate limiting step is the coverage of a surface by all its surface nuclei (N ≪ C ). If the whole area is covered at time t = C , then dr/dt = d/C for the linear growth rate. In order to find a reasonable value for this time one should consider that at any time 0 ≤  ≤ C , a certain amount of surface nuclei dn = N
−1 d = J ′ Ai
d have appeared. First assume an independent for each germ so that a growth if da/dt = k′ then a = 0 da = k′  C dt = k′  − 
. The total area covered by germs that have appeared at t =  may then be written

(24)

m∗ + 2 3

(25)

3.3.6. Chronomals and Equilibrium Size As for the diffusion law, one may introduce a chronomal analysis in order to find the order p for crystalline growth. Setting c = c0 1 − xp
and r = re xp1/3 then leads to t = Kp Ip , with  xp re Kp = and Ip = x−2/3 1 − x
−p dx (26) p 3vc0 kp 0 Figure 4 shows the variation of Ip with xp for increasing values of p. Concerning the integrated form of the polynuclear growth law, let tP be the characteristic time below which the condition S > S ∗ remains satisfied. From (14) and (15) it comes that ct = c0 1 − t/tP
4 ] with tP given by


12 ′ 

1/4

−1/4

J c0


D−3/4 gC−3/4 c0−1/4

G′ ∗ exp 4kT




(27)

As done, this induction time may be used to evaluate the number of nanocrystals N formed per unit volume 
4  t ⇒ N ≈ J c0
tP F n∗
c = c0 1 − tP  1 n∗ 

k (28) F n∗
= 1 − x4
n∗ dx = k + 1/4
0 k=1

This in turn leads to an analytical expression linking the average equilibrium size re  of the nanocrystals to the solid phase and solution properties re 3 =

2

p=

The expression in the right hand side of (25) comes from an alternative evaluation of J ′ based on the observation that from (22): Ln J ′
/Ln S
= m∗ ⇔ J ′ ∼ kC c m∗ . It is interesting to compare relations (10) and (25). In both cases, the growth rate depends on the diffusion coefficient D, and on the difference (c − cs ). The differences are that we have a power law for the concentration dependence with a growth rate independent of crystal size in the case of polynuclear growth. According to this last mechanism this means that the absolute width of the size distribution should be constant (r ∼ r0 ) at all times and that only the relative width may decrease with time (r/r ∼ r0 /r × r0 /r0 ).

tP =

3.3.5. Polynuclear Growth Law

dA = ′ a2 dn = ′ J ′ Ai k′  − 
2  C ′ 2 ⇒ Ai = dA = J ′ Ai k′ C3 3 0

Taking into account that in fact germs may overlap during growth we may write C = ′ J ′ k′ 2 /3
−1/3 , where  is the numerical geometric correction describing the geometric details of the various overlaps. For disk-shaped 2D germs it may be shown that  ∼ 06 [5]. This gives our final polynuclear growth law 
d G′ ∗ dr = = ′ v/3
1/3 DgC c − cs
2/3 exp − dt C 3kT


 
 1 ′  1/4 9/4 5/4 gC 3/4 v c 0 F n∗
12 gN 
′ 2 4/3 3 2

 v 3  v (29) − × exp 4kT
3 Ln S0
2 4kT
2 Ln S0



Nanocrystals from Solutions and Gels

565

Concentration 10cs

100cs

1000cs

log re (m)

0

0.8 0.6

-2

0.4

-3

0.2

-4

0.0

-5

p=1

p=2 p=3 p=4

dr = vk cp p dt 2

4

6

8

10

12

14

re

8 σc = 16

4

-7

1m

1 mm

Ip

0

-6

1 µm

2

-8 -9

1M

xp

1.0

-1

3.4.1. Growth Law 0,1M

1 nm

1

-10

1Å 23

24

25

26

27

log c (m-3) Figure 4. Crystalline growth controlled by the interface. Boxed curve shows the shape of the chronomal functions IP given by Eq. (24). Other decreasing curves were plotted according to Eq. (29) assuming a spherical shape ( = 4/3 = 4) for 3D nuclei and a disk shape (′ = ′ =   = 06) with F n∗
 = 3/8 for five typical values of the growth constant C = ′  2 v4/3 /kT Ln 10
2 . For these particular shapes we have N ∼ 3C
3/2 leading to log re = 1/3
log6v9/4 /4
+ 5/12
log c + √ 1/4
loggC /gN
+ C 9 C − logc/cs
/12log c/cs
2 . Other param−285 eters were v = 10 m−3 , cs = 1022 m−3 , and gC /gN = 10−6 . As for diffusion controlled growth, nanometer sized nanocrystals are formed when precipitation occurs in concentrated solutions.

Figure 4 shows the typical variation of re  with concentration in solute species c for several values of the dimensionless growth constant C = ′  2 v4/3 /kT Ln 10
2 . One 3 2 3 may notice  that as N =  v /kT P Ln 10
, the link N = ′ ′

C /
C /
exists between growth and nucleation constants. As in the case of diffusion, these curves show that when N is low, re  remains roughly constant above S = 10, with typical sizes close to 100 nm. On the other hand, when N is high, nanocrystals with typical sizes close to 1 m can be formed from concentrated solutions. Again, in all cases, there is a critical concentration domain where average size suddenly drops from the millimetric range down to the nanoscopic one. One may also see that nucleation and growth terms have antagonist effects on the equilibrium size. Concerning the role of chemistry at the interface larger crystals are expected when g ′ > g (i.e., when G′′ < G′ ).

3.4. Dislocation Controlled Growth When the sursaturation ratio S is smaller than about 2, the rate of surface nucleation may become so slow that the crystal growth rate becomes practically nil. In fact there are numerous examples in literature where crystalline growth is easily observed even when S = 101. In such cases, it has been shown [2] that growth may be initiated on crystalline defaults (dislocations or stacking faults). Among the possible defaults, it appears that the most efficient ones are screw dislocations leading to the formation of typical Archimedean spirals on the surface [2, 3].

In this mechanism, we have to estimate the normal velocity of advancement u for the leading step of the growing spiral. For a spiral of radius R, this rate should be proportional to the frequency of arrival of solute species at the leading edge times a sticking probability ( ′ ∼ D × gc /d) and should also depend on their average number at the interface [n′ ∼ c − cR
v]. Consequently, with d ∼ v1/3 , we should have u ∼ n′ × ′ ∼ Dcgc v2/3 1 − cR /c
. Now, from (22) with ′ = , it is possible to relate the concentration just outside the crystal surface c to a critical germ radius r ′ ∗ according to r ′ ∗
2 = m∗ × v2/3 ; that is, r ′ ∗ = v/kT Ln S
. Moreover, applying (9) with r2 → + (i.e., c2 → cs
and r1 = R or r ′ ∗ allows one to express the ratio cR /c as 

  cR v 1 1 1 v 1 = exp − − ≈1+ c kT R r ′ ∗ kT R r ′ ∗ ⇒

lim u =

R→+

Dv5/3 c g kT r ′ ∗ C

(30)

Assuming a spiral displaying a stationary shape and rotating with a constant angular velocity S rad s−1
, it may be shown that S = u /r ′ ∗
[38]. Consequently, each spiral turn should deposit a layer of thickness d ∼ v1/3 leading to the following linear rate of growth
 dr S DkTc ∼ Ln S
2 (31) × v1/3 = gC dt 2 2 2 

3.4.2. Two Limit Cases Two important limiting cases may be discussed. When c ∼ cs (i.e., S = 1 + ), then Ln S = Lnc/cs
∼ c − cs
/cs , and we get a nonvanishing growth rate dr/dt = gC DkT /2 2 cs
 × c − cs
2 . This explains the possibility of growing large single crystals from solutions even when S ∼ 1. On the other hand, when c ≫ cs the growth rate may become quite large, but very soon the diffusion of solute species toward the spiral leading edge will become rate determining. As both rate laws are known, it is possible to see for which value of S both processes should contribute equally to the crystalline growth gC

Dv1 − S −1
DkTc Ln S
2 ≈ 2 2  r
 2 2 2 v SLn S
≈ ⇒ S−1 gC kTr

(32)

Using  ∼ 01 J m−2 , v ∼ 10−285 m3 , and T = 300 K as typical values leads to the following condition: [gC rnm
 × SLn S
2 ≈ 152S − 1
. Setting gC ∼ 1 and r = 1 nm as reasonable limits leads to S = 47338. Under these conditions, diffusion should become the rate limiting step as soon as S > 50. However, if one considers a millimetersized single crystal (r ∼ 106 nm) with rather low kinetic barrier (gC ∼ 1 for purely ionic crystals such as NaCl or KCl) well developed spirals should be evidenced only if S ≈ 1 + 152/gc r
< 100001. However, if gC ∼ 10−7 (i.e., G′′ ∼ 40 kJ mol−1
, then for the same single crystal size diffusion toward  the interface will be rate determining only when S ≈ exp 152/gc r
 > 105 . A direct consequence of this fact is that nanocrystal formation (r ∼ 10 nm) should never be under diffusion control, unless gC > 01.

Nanocrystals from Solutions and Gels

566 3.4.3. Chronomals Interestingly enough, when spiral growth is the rate limiting step, the chronomal analysis shows that it would be rather difficult to distinguish between the spiral and the polynuclear growth. Proceeding as before by setting c = c0 1 − xS
p and r = re xS1/3 leads to t = KS Ip , where Ip is the polynuclear p chronomal limited to values of p = 1 or 2 and KS stands for two time constants given by re 2 2  × with 3gC DkTfp c0
 xS Ip = x−2/3 1 − x
−p dx p = 1 or 2

xs

1.0

p=1 p=z

0.8 0.6 0.4 0.2 0.0

Ip 0

2

4

6

8

10 12 14

p

KS =

0

(33)

The two p-values correspond to the two previously described limiting cases. When c ≫ cs , then c ∼ c − cs and Ln S ∼ Lnc0 /cs ) leading to t = KS1 I1 with f1 c0
= c0 Lnc0 /cs
2 . For the other limiting case c ∼ cs , we have t = KS2 I2 with f2 c0
= cs /c0 − cs
2 . On the other hand, if diffusion is the rate limiting step, then one should follow the ID chronomal given by (12). This analysis shows that observing a crystalline growth law following ID , I1 , or I2 chronomals does not rule out the spiral mechanism. In fact the only clear evidence for the occurrence of the polynuclear mechanism would be to follow chronomals IP characterized by p > 2. To the best of our knowledge such a situation has been scarcely reported in literature as most precipitation processes follow either ID or I1 chronomals. This result may just be a direct consequence of the very high efficiency of the spiral growth for nanocrystals which bypassing the bottleneck of surface nucleation run directly into the one of diffusion toward the interface. If this is true, then all nanocrystals grown using concentrated solutions should in fact be formed from numerous interpenetrating nanospirals and not from overlapping surface nuclei.

3.4.4. Equilibrium Sizes At last, we may consider the integrated form of the spiral growth law, characterized by a time tS below which the condition S > S ∗ remains satisfied. From (14) and (15) it comes that ct = c0 1 − t/tS
4 ] with tS given by 3/4 1/4 
1/4  v 2 2  4 (34) tS =  gC DkT Ln S0
2 c02 J c0
As the power dependence of concentration with time is similar than that observed for polynuclear growth, (28) holds with tP replaced by tS , leading to 
3/4  
1/4  1  kT Ln S0
2 3/4 2 3/2 gC re 3 = v c 0 F n∗
4 gN 2 2  
3  3 v2 (35) × exp 4kT
3 Ln S0
2

3.5. Discussion At this stage a comparison between Figures 3, 4, and 5 may be useful. All curves display the same characteristic L-shaped aspect, the decreasing left part coming from an exponential term and the increasing right part from a power

2

Figure 5. Spiral crystal growth controlled by screw dislocations. Boxed curve shows the shape of the two chronomal functions I1 and I2 given by Eq. (26). Other decreasing curves were plotted according to Eq. (35) assuming a spherical shape ( = 4/3 = 4) for 3D nuclei with F n∗
 = 1/2 for five typical values of the nucleation constant N =

 3 v2 /kT Ln 10
3 leading to log re = 1/3
log6v2 /4
+ 1/2
log c + 1/4
loggC /gN
+ 1/4
logkTLn S0
2 /2 2  + N /4log c/cs
2 . Other parameters were T = 300 K, v = 10−285 m−3 ,  = 01 J m−2 , cs = 1022 m−3 , and gC /gN = 105 . As for other growth mechanisms, nanometer sized nanocrystals are formed when precipitation occurs in concentrated solutions.

of the total concentration c0 . In these drawings the two sticking probabilities gN and gC (linked to the occurrence of kinetic barriers G′ and G′′ for nucleation and growth) have been adjusted in order to have re > 01 nm in all the investigated concentration range. In the case of diffusion controlled growth, one should have gC ∼ 1 and the size of the nanocrystals will be fixed by the relative values of N and gN . This is no more the case for the two other mechanisms, where the two competing terms are N and the (gC /gN
ratio. However, a clear difference exists between polynuclear and spiral growth as one must set (gC /gN
= 10−6 for the first mechanism and (gC /gN
= 10+5 for the second one in order to get reasonable sizes. The chemical interpretation of this phenomenon is straightforward. Polynuclear (or mononuclear at low S-values) growth should be suspected as soon as gC < gN , that is, when solute species are more easily incorporated when the nanocrystal has not yet appeared (embryo stage). Chemically speaking, we are in a situation where crystal growth is poisoned by the decomposition products of the solute precursors. In this case, it seems obvious that the highly reactive leading edge of a growing spiral will be much more affected by the poisonous material than a less reactive flat layer. Owing to the presence of the poison, the surface nucleation bottleneck is unavoidable and the polynuclear picture dominates crystallin growth. In the reverse situation, gC > gN , the decomposition products of solute precursors are expected to show no affinity for the crystal/solution interface. In this case, solute species absorption should occur at the most reactive sites, and one may expect a crossover between polynuclear and spiral growth. It then appears that Eqs. (20), (29), and (35) should cover both qualitatively and quantitatively all the pertinent physical and chemical aspects of nucleation and crystalline growth from solutions.

Nanocrystals from Solutions and Gels

4. LIMITED AND SECONDARY GROWTH It should be noticed that the previous equations have been reached without any explicit description of the structure of the solution/crystal interface. One should then expect that some other important aspects remain yet hidden, particularly in the interfacial energy . In the previous treatment, it was tacitly assumed that the interfacial energy  was a constant characterizing the solution/crystal interface. In this section, we will consider what should happen to the final equilibrium size of a nanocrystalline dispersion when  may change owing to variations in pH or ionic strength. To do this we must dispose of a faithful model of the crystal/solution interface. In the following, we will consider mainly oxide/water interfaces that are by far the most commonly encountered experimental cases. Generalizations for nonaqueous or nonoxide interfaces will not be discussed in this chapter.

4.1. Solute–Solvent Interface The best way to understand what the structure of an oxide water interface should be is to start from the molecular scale (Frank and Wen model [39]). Accordingly, X-ray or neutron diffraction studies of aqueous solutions [13, 40] have shown that the molecular environment around a given solute is not a random medium. At less than 0.3 nm, a first shell of frozen solvent molecules is very often found in close contact with the solute species. After this coordination shell, other more or less immobilized shells may be found. The spatial extension of this zone is typically between 0.3 and 0.5 nm. In this critical zone, it may be observed that the local order characterizing the solvent in a pure state may either be broken or strengthened. This comes from the fact that on one side solvent molecules are under the influence of the more or less rigid coordination shell around the solute. On the other side, they are under the influence of the random motions of their congeners that are far from any solute (this holds for sufficiently diluted solutions). When the solute coordination shell is highly structured and rigid, this orderliness tends to be propagated at longer distances. One then speaks of a structure-forming solute. Conversely, if the coordination shell is highly labile and disordered, the disorder will also be more or less transmitted to the neighboring shells breaking the original structure of the pure solvent. In this case, one may speak of structure-breaking species.

4.2. Oxide–Water Interface This classical picture of the solvent–solute interaction may serve as a faithful basis for a realistic model of the solid/liquid interface. As explained in previous sections, nucleation is an irreversible phenomenon by which several solute precursors meet together in a limited volume (thermodynamic barrier G∗ ). Within this confined state, they may further expel partly or fully their coordination shell (kinetic barrier G′ ) in order to build a new solid phase. Chemically speaking, the solid phase may be viewed as the result of a giant polymerization process of solute species. On this basis, one may expect that their respective coordination shells should also merge together defining the solid/liquid

567 interface. This merging process should, however, also apply to secondary coordination shells leading to a more or less compact layer, tightly bonded to the interface (called hereafter the “Stern” layer). Finally, just outside this more or less immobilized Stern layer a slipping plane for solvent molecules should exist, that may be identified as the “outer Helmholtz plane” (OHP). Adopting this “natural” model for the solid/liquid interface allows us to zoom more deeply into the crystalline growth process. For instance, the quite vague statement of “diffusion toward the solid/liquid interface” may then acquire an accurate formulation. Accordingly, diffusing solute species may be identified as any kind of fully solvated chemical entity not able to go beyond the OHP. One must realize that the last statement is not an assumption, but a logical consequence of the chosen model. Accordingly, assume that some kind of solute species has been able to jump over the OHP landing inside the Stern layer. The chemical consequence of this hopping should be that some spatial reorganization of the surrounding solvating shell was possible in order to fit the local structure of the Stern layer. If the energetic cost of this reorganization were too expensive, the jumping would have been a very unlikely process. Consequently, the solute species would show no tendency to cross the OHP, preferring to diffuse quietly far away from this unfriendly disturbing spatial zone. Conversely, it may happen that some otherwise quietly diffusing solute species may experience a very high structural affinity for the Stern layer. Such a species is then expected to cross the OHP ending up right inside the Stern layer. As a result of this jump, a quite new solvating shell more adapted to the local structure of the Stern layer should be adopted, leaving with no regret the older one more adapted to the structure of the bulk solvent. The automatic consequence of fact is the existence of a second kind of molecular plane, named the “inner Helmholtz plane” (IHP). By its very definition this plane lies somewhere outside the solid surface and well below the OHP. From a chemical standpoint, it is formed by the locus of all strongly adsorbed species that are not allowed to diffuse freely into the solution due to their particular affinity for the Stern layer. At last, chemical entities already localized within the Stern layer may exist and could display rather high chemical affinity for the crystalline surface. For such species, full reorganization of their coordination shell needed for a “perfect” sticking to the surface is not a problem. They are thus not expected to stay localized close to the IHP but may undergo a last final jump right inside the crystalline lattice. The immediate result of this ultimate sacrifice is an increase of the spatial extension of the interface by one step, another way of saying that crystalline growth has occurred.

4.3. Surface Charge The next step is now to see how the surface charges may develop. As shown in Figure 6 (top left) metallic cations exposed at the surface are found to be hydroxylated owing to water chimisorption. Consequently, like soluble aquo, hydroxo, or oxo complexes they may acquire or lose protons according to the polarizing power of the metallic cation ⊖ M–O⊖ + H3 O⊕ ⇌ M–O–H + H2 O ⇌ M–OH⊕ 2 + OH

Nanocrystals from Solutions and Gels

568

4.3.2. Point of Zero Charge

Figure 6. A natural model for the oxide/water interface. Top left shows the H-bonded structure of the Stern layer in close contact with the oxide surface. Top right illustrates the Frank and Wen model (scale t0 , one should have   r∗ Mt − t0
1/3 = 1 + r0∗ r0∗
3

⇒


∗ 4 r gt
= gt0
0∗ r

(55)

4.6.4. Size Distribution As the time variation of r ∗ and gt
is known, we may notice that 4Mgt
M dr ∗ dgt
=− = dt 3r ∗
3 dt 3r ∗
2
  M =− ⇒ t r 3r ∗
3

4.6.3. Time Dependence Let us approximate each crystal by a sphere with radius r lying in a spherical diffusion field of solute species. The solubility cr of the small crystals may be computed from (9) with r2 → + (i.e., c2 → cs ) and r1 = r. A similar relationship also applies to the concentration c in the bulk that is in metastable equilibrium with the nuclei of size r ∗ (now with r1 = r ∗ ). Expanding exponential terms up to second order then leads to the following linear growth rate:
 
2v 2v and c = c exp c = cs exp r s kTr ∗ kTr
 2 1 vDc − cr
2v Dcs 1 dr ⇒ − = ≈ (52) dt r kTr r∗ r

2v2 Dcs kT

M=

(56)

In order to get the size dependence h
of nr t
one may insert the growth rate given by (52) into (51) leading, after transformation of variable r =  × r ∗ and some tedious algebra involving relations (56), to the following differential equation:     dh
2 3 dh
6 h
+  + h
= 3
 − 1
d d ⇒

d ln h
3 − 62 /
= 3 d  / − 3 + 3

(57)

Nanocrystals from Solutions and Gels

572 At this stage, as from (53)  is a number depending on the integral of the yet unknown h function, the solution may seem not obvious. However, we may notice that the denominator in (57) has the simple form x3 = ax + b with a = −b = 3. A possible solution to this cubic equation is thus x = b/2
+ 1/2 1/3 + b/2
− 1/2 1/3 , with  = b/2
2 − a/3
3 (Cardan formula). Consequently, setting  = 9/4 would lead to  = 0 (i.e., to the most simple root x = −3), allowing one to factorize the denominator as ( + 3
 − 3/2
2 . Remembering that with h0
= 1, integration of (57) is straightforward: =

9 4

⇒

h
=

11/3 7/3
3 3/2 3+ 3/2 − 
  × exp − (58) 3/2 − 




Figure 7 illustrates the previous considerations for some typical values of the solid phase and solution parameters, assuming a monodispersed and nanosized dispersion of particles (r0∗ = 10 nm) at the beginning of the Ostwald ripening. It is worth noticing the considerable spreading of the size distribution at long time and its typical asymmetric shape. This characteristic shape emerges from the fact that small particles are disappearing from the solution, allowing the bigger crystals to become still bigger. Concerning the time scale of the ripening, it was mainly ruled by the selection of a diffusion-limited growth mechanism for describing the dissolution–reprecipitation events. Obviously, this should hold only for largely ionic crystals, and for other materials, one may expect an interface-limited ripening. In this case, a numerical solution to the ripening equations (51) and (54) may be the best way of handling the problem [41].

5. AGGREGATION It is now time to remove the assumption that nanocrystals may nucleate and grow independently of each other [i.e., that the N r t
function should be more or less constant during the whole precipitation process].

5.1. Position of the Problem As explained, this assumption was crucial for deriving relations (20), (29), (35), (47), or (54) relating sizes to solution conditions and solid phase properties. Accordingly, even if nanocrystals have their diffusion coefficient 2 or 3 orders of magnitude less than solute species they may nevertheless, from time to time, undergo some collisions between each other. If the sticking probability is high during a collision, aggregates may be formed decreasing by two units the value of N r t
. Obviously, if all possible processes influencing the mean size r of nanocrystalline dispersions (nucleation, growth, ripening, and aggregation) were occurring with the same rate, the situation would be hopeless. Fortunately, owing to the physics involved in each kind of phenomenon not all these processes are expected to occur simultaneously. For instance, for any kind of reaction to occur, reactants (either complexes or particles) first have to diffuse toward each other. According to the Stokes–Einstein equation

D = kT /6a
, the diffusion coefficient D in a solution displaying viscosity  at temperature T is expected to be the highest for species displaying the lowest size a. This means that aggregation, like Ostwald ripening, should be important only at long time. However, if ripening is concerned by the behavior of N r t
when r → 0 and t → , aggregation focuses on the behavior of this same function when both r and t become large.

5.2. Fuchs Integral Consequently, we now assume that after a rather long lapse of time during which the function N r t
has assumed an almost constant value hereafter noted N0 , this number may change owing to binary collisions between the crystals. Let VT R
be the potential function covering all possible physical or chemical interactions that may occur during the collision. If CR
is the concentration of crystals at distance R, the flux of particles (number of crystals per unit area and per unit time) jt
is given by the modified Fick law: dCR
CR
dVT − × dR kT dR 
VT dCR
expVT /kT
 = −Drel exp − kT dR

jt
= −Drel

(59)

In this relationship, the first term arises from the free Brownian motion of the particles. As some external forces FT = −dVT R
/dR may act on the moving particles, the second term takes care of the additional flux CR
× v, proportional to the speed v of the particles. As usual, speeds may be related to forces through a friction factor f v = FT /f
derived from the Einstein relationship D = kT /f . Notice that as we are considering a binary collision between two moving particles, the Fick law uses a relative diffusion coefficient Drel . The link between Drel and D1 or D2 , diffusion coefficients of one single particle, may be easily derived by reference to the fundamental law of Brownian motion: ri2  = 2Di t. Consequently for a collision between two particles at a relative distance (r1 − r2
we should have r12  + r22  − 2r1 · r2  = 2D1 t + 2D2 t = 2Drel t r1 − r2 2  =  ⇒

Drel = D1 + D2

(60)

Let us now apply (59) to compute the flux of identical particles across a sphere of radius R. From (60) Drel = 2D and considering the spherical symmetry of the problem, we may write J t
= 4R2 × jt
leading after integration to
  VT dR J  +

VT exp +A =− kT 8D R kT R2 
 +

VT dR = 8DC

(61) exp Jt kT R2 2a

CR
exp ⇒




Boundary conditions VT → 0 and CR
→ C ∼ N0 /V when R → + and C2a
= 0 have been applied to get the integration constant A.

Nanocrystals from Solutions and Gels

573

5.3. Hard Spheres Model Let us apply (61) to the most simple case of noninteracting particles characterized by VT ∼ 0 (i.e., by Jt = 16DaC). Now, for a given flux Jt , half particles are expected to move in a direction allowing a close encounter at R = 2a, while the other half are moving in the right opposite direction. As all encounters lead to an irreversible sticking of both partners, the rate of disappearance of the crystals through coagulation should be −

dC 1 = Jt C = 8DaC 2 dt 2 C0 ⇒ Ct
= 1 + 8DaC0 t


kf = 8Da =

4kT (62) 3

Here C0 = N0 /V is the total concentration at time t = 0 that governs the characteristic lifetime t1/2 [such that Ct1/2
= C0 /2] of the dispersion. As shown in (62), this time is given by t1/2 = kf C0
−1 = 3/4kT C0
. For water at T = 298 K, we have t1/2 s
= 2 × 1011 /C0 cm−3
, showing that the lifetime of dispersions characterized by VT ∼ 0 is quite short (a few milliseconds when C0 ∼ 1014 cm−3 ).

5.4. Attractive Forces In fact the solution given by (62) is a little bit unrealistic as the situation VT ∼ 0 is never encountered in reality and because the equation applies only to doublets formed by the association of two single particles. For instance, due to the universality of the London dispersion interaction, an attracting potential should always expected at any distance. Accordingly, quantum mechanical considerations have shown that the potential energy of attraction between two atoms A and B, separated by a distance r, is given by VA = −L/r 6 (London force) [45]. The constant L, characterizing the two interacting atoms, has been roughly estimated by London from the atomic polarizabilities  and first ionization energies I as L = 3/2
× A B /40
2  × IA−1 + IB−1
−1 [46].

who first did this analysis [44]. This a characteristic value for any condensed substance (whether solid or liquid) as shown in Table 5 [47]. As detailed previously for similar substances one would expect L ∝ 2 × I [i.e., A ∝ 2 × I
/v2 ]. Materials displaying high Hamaker constants are then those with large polarizability , high first ionization potential I, and high atomic density (1/v). This explains the extreme position of ice and diamond in Table 5. Obviously, Table 5 should be applicable only for vacuum interactions. When two particles of substance A (Hamaker constant A1
and B (Hamaker constant A2
are immersed in substance C (Hamaker constant A3
, one may show [48] that the resulting interaction A123 may be characterized by     A123 =  A1 − A3
 A2 − A3
  =  A1 − A3
2 if A1 = A2

The use of square roots in (63) comes from the fact that A ∝ /v
2 while polarizabilities  are additive quantities. According to this relationship, one may expect for water/oxide interfaces Hamaker constants that are ranging from 0.2 up to 0.5 eV. As far as nanocrystals are concerned the relationship VA = −A/12R2
cannot be very useful as it applies to two infinite planar interfaces. Performing the previous integration with spheres displaying radii a1 and a2 , with a center to center separation S, leads to [44]

A 2a1 a2 2a1 a2 VA S
= − + 2 6 S 2 − a1 + a2
2 S − a1 − a2
2  S 2 − a1 + a2
2 (64) + ln 2 S − a1 − a2
2 Table 5. Some Hamaker constants A for condensed matter characterizing the intensity of van der Waals interactions between microscopic objects. Matter

5.4.1. Hamaker Constant A This quantum mechanical expression for the dispersion interaction may readily be used for computing attractive forces between surfaces [42]. For instance, the interaction between an atom and a crown of matter (as always characterized by its molecular volume v
of radius a, width da, and thickness dx, is just d 2 V = −L/v
2adadx
r −6 . If R is the distance separating the atom from the surface, we may write using Pythagoras’s theorem r 2 = R + x
2 + a2 , leading after integration (from zero to infinity) against x and a to VA = −L/6v
/R3 . Considering that the previous atom belongs in fact to a plate of thickness dz situated at a distance (R + x) from the other surface leads to an energy of attraction per unit area dvA = −L/6v2
dz/R + z
3 . A last integration against z from zero to infinity leads to the well-known formula giving the van der Waals interaction between two infinitely large flat plates separated by a distance H VA = −A/12R2
. The parameter A =  2 L/v2
characterizing the macroscopic interaction is called the Hamaker constant after the man

(63)

Hexagonal ice Liquid water H2 O Et2 O, Me2 CO, MeCOOEt MeCOEt Benzene, nitrobenzene Chloroform, dioxan Toluene CCl4 , ethylene glycol CS2 Polyethylene oxide Glycerol Polystyrene CaF2 , polymethylmethacrylate Polyvinyl chloride SiO2 quartz -Fe2 O3 , haematite Silver chloride AgCl Silver bromide AgBr TiO2 , anatase TiO2 , rutile Diamond

A (eV) 0.19 0.23 0.26 0.29 0.36 0.37 0.38 0.39 0.41 0.42 0.46 0.49 0.51 0.62 0.92 1.02 1.07 1.14 1.28 1.41 2.81

574

Nanocrystals from Solutions and Gels

5.4.2. Application

5.5.1. Diffuse Layer Relaxation

In order to see the effect of such a potential on the rate constant kf , we may simplify (64) by writing S = R + a1 + a2 and assuming a very small distance R ≪ a1 + a2
between the two spheres. In such a case the first term of (64) is the leading one and VA R
∼ −A/6R
1/a1 + 1/a2 −1 = −Aa/12R for two spheres of equal sizes (a1 = a2 ). Replacement of VT by VA R
in (61) then leads after integration and expansion of the exponential term up to second order to Jt = 16DaC1 + A/48kT
. As A is at most a few tens of kT, the kinetic constant kf should be at most multiplied by a factor 2, relative to its value at VT ∼ 0. Even if this correction appears to be negligible relative to the 1011 cm−3 term, it nevertheless shows that aggregation is accelerated (particularly at low temperature and for materials displaying high Hamaker constant) relative to pure diffusion. Consequently, if the dispersion has to be stabilized against irreversible aggregation, one must play with experimental factors leading to a VT curve that must display some maximum positive value Vmax at a given distance R > 2a. Accordingly, the previous analysis has shown that the additional destabilization factor coming from the region where the potential displays strongly negative values was at most two. Such a modest contribution is then expected to be negligible even in the presence of a small maximum. This observation simplifies integration of Eq. (61) and allows one to write Jt = Jt /W , where W is the so-called stability ratio W ∼ expVmax /kT
. The higher Vmax , the higher W and the longer the lifetime of the dispersion as kf ∼ 8Da/W and t1/2 = kf C0
−1 ∝ W . For instance, the stabilization for about one year of a dispersion displaying about 1014 particles per cubic centimeter would require a stability ratio of about 10105 . This corresponds to a potential maximum of about 24 kT (∼60 kJ mol−1 or 0.6 eV at T = 298 K).

The first one is the Debye–Hückel relaxation by which counterions located in the diffuse layer move with a diffusion coefficient Di in order to adjust the structure of the double layer to the new conditions. The characteristic relaxation time DH associated with this process is equal to the average time needed for displacement of ions across the double layer: DH = −2 /2Di . For aqueous solutions at room temperature Di ∼ 10−9 m2 s−1 leading to DH ps
∼ 46/Imol l−1
. Consequently, one may safely assume that double layers are always at equilibrium during collision, as even for a = 1 nm we still have B ∼ 4DH .

5.5. Repulsive Electrical Forces The first method widely used to stabilize colloidal dispersion is to work with charged surfaces. For oxide materials, this is done by changing the pH or the ionic strength I. Owing to the rapid diffusion of protons is aqueous solutions, any pH variation instantaneously affects the surface potential 0 as evidenced by Eq. (38). By playing with the ionic strength, the spatial extensions of the double layers around colloidal particles are deeply affected. Consequently, let us evaluate the time duration B of a collision between two charged particles. During this collision, each particle must then diffuse over a length scale of the order of the Debye–Hückel length −1 given by Eq. (39). If Dp is the diffusion coefficient of the colliding partners (fixed by radius a and solution viscosity ), then B = −2 /2Dp = 3a/2 kT
. For water at room temperature (T = 298 K,  ∼ 10−3 Pa s), we should have B (ns) = 02 × anm
/Imol l−1
. For nanosized particles (1 ≤ a ≤ 100 nm) with typical ionic strength (10−3 ≤ I ≤ 1 M), a Brownian collision cannot be shorter than 100 ps (small particles or high ionic strength). On the other hand, for larger size or lower ionic strength it cannot be longer than 0.1 ms. Due to the geometric changes involved by the collision, several relaxation processes have to occur in order to restore complete equilibrium.

5.5.2. Stern Layer Relaxation The second mechanism relies on the adsorption or desorption of all the PDIs involved in the double layer equilibrium. This includes charged species located at the surface as well as those located at the IHP within the Stern layer. Here the rate of charge adjustment is ruled by the value of the exchange current density i0 between the surface and the Stern layer. For a given charge density (0 + 
the characteristic relaxation time may be evaluated as AD = 0 + 
/i0 . If the exact value of i0 is highly dependent on the detailed structure of the interface, 1 A cm−2 is surely a very high value and 10−10 A cm−2 is a very low one [49]. With 1 C cm−2 ≤ 0 + 
≤ 100 C cm−2 , it follows that PDI relaxation should take at least 1 s and may in the worst cases (high surface charge and low mobility of the charges) be as long as several days.

5.5.3. Interactions at Constant Charge or Potential It follows from the last analysis that two limiting cases will be encountered during a Brownian collision between two charged particles: (i) For large nanocrystals displaying a low surface charge with high mobility of the PDI and placed under low ionic strength conditions, one may safely assume that B > AD . In such a case, the interface, as a whole, is completely relaxed during the collision. This means that at all times the potential remains constant. (ii) In all other cases, one may expect that B ≪ AD , and in this case, only the diffuse part of the double layer is completely relaxed. As  = −0 d/dx, this means that at all times the slope at the surface of the  = f x
curve cannot change leading to an increase of the surface potential 0 . A consequence of this fact is that the distance of closest approach should not be S = 2a, but S = 2 × a + 
, where  is thickness of the Stern layer holding some of the PDI ions [49]. Obviously, this nonequilibrium situation holds only during a time B . If particles remain stuck together after the collision, a discharge current will appear allowing a reduction of the potential toward its equilibrium value. This relaxation allows a still closest approach between the two cores of the particles, until S = 2a at equilibrium. In this fully relaxed state particles are irreversibly bonded through the van der Waals interaction and cannot be separated anymore.

Nanocrystals from Solutions and Gels

575

This constant charge mechanism helps to explain the repeptization phenomenon by which a flocculated system may be dispersed again by changing the composition of the medium without stirring [49].

5.5.4. Osmotic Pressure and Repulsive Potential

5.5.5. DLVO Theory

From a physical point of view, the origin of the repulsion between two overlapping double layers comes from the existence of an osmotic pressure pR , which develops due to the accumulation of ions between the two surfaces. At sufficiently long distances, the two interacting particles may be viewed as almost planar interfaces. The actual force per unit area exerted on the plates is given by the difference in the osmotic pressure between the solution in the midplane m and that in the bulk (concentration c0 ). With c+ and c− the concentrations at the middle plane, we may write pR = RT c+ + c− − 2c0 ). Assuming that c± = c0 exp∓zF m /RT
and developing the exponential terms up to second order ( 21 ex + e−x  ∼ 1 + x2 /2 when x is small) allows one to 2 express this osmotic pressure as pR = 21 2 0 m . In the case of small overlap between the two plates, potentials are expected to be additive leading to m = 2x = R/2
∼ 2d exp−R/2
. The resulting repulsion potential is thus obtained after integration over R leading to VR =





R

pR dR =

42 0 2 

exp−R
dR d 2 R

= 20 d2 exp−R


reduce to an expression similar to (65): VR = VR = 20 ad2 exp−R
. Other expressions are available in the literature for situations where the approximations d  < 60 mV and a ≫ 1 are not valid [49, 52–55].

Figure 8 shows the resulting potential curves VT = VA + VR obtained when the Hamaker attraction given by (64) is added to the repulsive contribution VR coming from the overlap of double layers given by (66). As shown, four parameters have a deep influence on the occurrence of a maximum. The curve at the top left part of Figure 8 shows the effect of changing the ionic strength for a constant Hamaker constant, size, and surface potential. At high ionic strength or  there is no maximum and the dispersion should flocculate. This is not true at low ionic strength where the interaction is largely repulsive at all distances R > −1 . It is a well-known experimental fact (Schulze–Hardy rule) that the critical coagulation concentration (c.c.c.) displays a strong dependence on the valency z of the coagulating ion (c.c.c. ∝ 1/z6 ). This c.c.c. should correspond to the point at which the maximum in the total potential energy curve just touches the horizontal axis (VT = dVT /dR = 0). As coagulation implies a low potential and a rather small separation R, we may use VR = VR = 20 ad2 exp−R
and VA R
= −Aa/12R for the repulsive and attractive parts respectively. Applying

(65)

Equation (65) is useful as it shows that a high value of d is to be associated with a strong repulsive potential. It is, however, not very realistic, as it suggests that high ionic strength (i.e., high  values) also favors strong repulsion. Keeping the low potential approximation, but considering spheres of radii a1 and a2 instead of infinite plates, leads to a more satisfying result when d  < 60 mV and a ≫ 1 [50]: VR =

2 2 0 a1 a2 d1 + d2
a1 + a2

  1 + exp−R 2d1 d2 ln × 2 2 1 − exp−R d1 + d2
   + ln 1 + exp−2R

(66)

When a1 = a2 = a and d1 = d2 = d , (66) reduces to a much simpler expression, VR = 20 ad2 ln1 + exp−R
, that does not show the previously noted  dependence. The  superscript in the previous relationships refers to an overlap between double layers occurring at constant potential. For the other case (constant charge), an additional term has to be subtracted [51]: VR = VR −

2 20 a1 a2 d1

+ a1 + a2


2 d2


ln1 + exp−2R


(67)

Again, when a1 = a2 = a and d1 = d2 = d , (67) reduces to a much simpler expression, VR = −20 ad2 ln1− exp−R
. For large distances (R ≫ 1) both equations

Figure 8. DLVO (Deryaguin, Landau, Verwey, and Overbeek) theory for the stabilization of colloidal particles with radius a against van der Waals attraction (Hamaker constant A). The four curves were computed using a potential fixed by the PZC of the particles: 0 (mV) = 257 × (PZC − pH) = 257pH. These curves are the result of adding an attractive contribution, VA R
= −Aa/12R, to a repulsive one, VR (eV) = 0018 × a (nm) pH2 exp−R
coming from the osmotic pressure induced by the compression of the diffuse layers characterized by their thickness −1 .

576 the conditions VT = VA R
+ VR R
= dVT /dR = 0 to this particular form leads to  240 d2 2z2 F ccc R = 1 ⇒  = = Ae RT0  
4 
RT 0
3 d 288 2 (68) ⇒ ccc = e2 F 2 A2 z2

This interesting formula shows that the sensibility of a given dispersion to an increase in ionic strength is governed by two main factors. The first factor depends mainly on the physicochemical state of the solvent (temperature, dielectric constant ) and on the chemical nature of the dispersed solid phase (Hamaker constant A). Typically a high sensitivity (low c.c.c. value) is expected for materials displaying large polarizability and high density (high Hamaker constant) dispersed in apolar solvents (low dielectric constant) at low temperature. The second factor involves the valency z of the added ions and the electrokinetic potential ∼ d . Here, high sensitivity means low electrokinetic potential and large valency. Recalling that from the Poisson–Boltzmann equation we have d = −0 d with  ∝ z, we may expect that d ∝ z−1 , providing a nice explanation for the empirical Schulze–Hardy rule. The top right curve in Figure 8 shows the effect of changing the surface potential 0 = RT/F ln 10
PZC − pH
at constant Hamaker constant, size, and ionic strength. At the PZC, the attractive term dominates while for pH ≥ 1 a high maximum always exists. The bottom left curve illustrates the effect of changing the Hamaker constant at constant ionic strength, size, and surface potential. It shows that materials such that A > 1 eV are not good candidates for making stable dispersions. A very high surface charge or low ionic strength is mandatory in this case. At last, the bottom right curve shows the effect of the nanocrystal size at constant ionic strength, surface potential, and Hamaker constant. The interesting point is the absence of maximum for small particles. A direct consequence of this fact is that during a precipitation experiment involving aggregation, the smallest crystals should be selectively removed from the solution, while the largest ones would remain unaffected. This means that starting from a highly polydisperse system, a narrowing of the size distribution is to be expected. This explains the possibility of getting, for any kind of material, dispersions displaying very narrow size distributions as exemplified by the beautiful work of Matijevic [25]. Here, through a fine-tuning of experimental conditions it should be possible to heal by aggregation the inescapable polydispersity generated by nucleation and growth or Ostwald ripening processes.

5.6. Entropic Forces We have so far discussed only enthalpic stabilization. Another possible stabilization mechanism would be to play with entropy instead of enthalpy. In fact, steric stabilization through the adsorption of polymeric species is of widespread use. Accordingly, a polymer can adsorb on the surface of a colloidal particle because of various interactions (charge– charge, dipole–dipole, hydrogen bonding, or van der Waals) acting alone or in combination.

Nanocrystals from Solutions and Gels

5.6.1. Depletion Interaction When a surface exerts no net attraction on the polymer, a net attractive force may nevertheless exist between the two colloids coming from the unsymmetrical repartition of the osmotic pressure between the overlapping and nonoverlapping zones. Assuming a perfect gas behavior, this osmotic pressure  = r
kT should be proportional to temperature T and to the local density r
in solute species (here polymers) at distance r from the surface. Let us further assume that the local density r
is close to the bulk density of solute S , and that the distance d is smaller than the sum of the diameters of colloidal C and polymeric species S d < C + S
. Then it is clear that solute species considered as hard spheres should be excluded from the central volume Vc , leading to a nil osmotic pressure in this central zone. To this depletion phenomenon [56] is associated an attractive potential between the two surfaces given by Vdep = −Vc = −kTS Vc . It should be realized that this depletion force is not always attractive. If one looks the interaction at a larger distance (C + S
< d < C + 2S
, then one finds a repulsive potential. Accordingly, as there is now enough room between the two surfaces, a polymer approaching a surface is pushed against it (negative adsorption) by the osmotic pressure unbalance existing between the surface zone (no polymers) and the bulk solution (density S
. Consequently, the local density in polymers in the middle zone should be higher than in the two opposite directions, leading to a neat repulsive force. For a still larger distance (C + 2S
< d < C + 3S
, the potential is again attractive because we have now a system of two coated surfaces displaying again an excluded volume for other polymers in the middle zone. In fact, one may understand that the effective potential for a system of large hard spheres (colloids) in contact with a large amount of smaller hard spheres (polymers) is not a regular one. It always displays a strong attractive component at very small distance (naked surfaces) and displays regular oscillations (period ruled by S
at longer distances. The higher the volume fraction occupied by the small spheres, the more visible the oscillations and the longer the interaction range.

5.6.2. Positive Adsorption The depletion phenomenon (negative adsorption) will always exist when a polymer solution is near an inert surface. However, if the attractive forces are strong enough, the enthalpy gained overcomes the entropy lost, leading to the model at the bottom of Figure 9 [56]. In this model [57], the links in contact with the surface are called trains. These are adsorbed on the surface, by physical forces invoked above. The loops may be defined as parts of the polymer connecting two trains but displaying no contact with the surface. Finally, the tails are all the nonadsorbed chain ends pointing outside the particle. Using this model, one may define the bound fraction as that fraction of chains forming trains. At very low polymer concentrations, a “pancake” conformation having a relatively large bound fraction may be formed. In this conformation, as there is plenty of room at the surface, the chains in contact with the surface tend to spread out. At higher solution concentrations, severe competition for the available space may occur, leading to shorter

Nanocrystals from Solutions and Gels

577 Provided that a loop or a tail does not approach full extension, the probability that it extends randomly from an arbitrary point (x y z
to another point x + dx y + dy z + dz
should be given by a simple Gaussian distribution, leading to

10 9 8 7 6 5 4 3 2 1 0 0.0 0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

W =

n

Wi = N0

Figure 9. Colloidal interactions between solids and polymeric chains. For inert interfaces only depletion and negative adsorption may occur (existence of an excluded volume shown in black leading to osmotic pressure unbalance !). Positive adsorption and steric stabilization require some affinity between the surface and the chains. Bridging flocculation is very often observed with small nanocrystals.

trains. In this case, long tails and loops stick out into the solution leading to a “brush” conformation. At last another possibility is that when two particles are brought together it may occur for the loops and tails of one polymer to attach them to bare patches on the other approaching particles (top right drawing in Fig. 9). Obviously, this bridging process is favored if the adsorption density of the polymer is not too high and should be most successful for half-surface coverage and small colloid size.

5.6.3. Relaxation Time Let us now have a closer look at the dynamic aspect of the collision between particles coated with polymeric chains. Relaxation time involved in the adjustment of the conformation of an adsorbed chain or loop has been estimated to be 2 × 10−4 s for polystyrene (M = 50 000) in a liquid with viscosity 0.2 Pa s [49]. On the other hand, adjustment of the loop size distribution is a much slower process requiring several minutes. For these systems, the time involved in a Brownian encounter should be defined as the time needed to travel over a distance equal to the thickness of the adsorbed polymeric layer  B ∼ 2 /2Dp = 32 a/kT. For a viscosity  = 1 Pa s at T = 298 K, B  s
= 23 × anm
× nm
2 . With 1 ≤ a ≤ 100 nm and 1 ≤  ≤ 10 nm, we find that the conformation adjustment can follow the collision process. This is not the case for loop size redistribution.

5.6.4. Steric Stabilization With the help of this model, it is easy to track the entropic origin of the steric stabilization. Let W be the total number of conformations available to the system. Then the corresponding entropy S should be given by the Boltzmann definition S = k lnW
. Let n be the number of loops or tails, let N0 be the number of allowed conformations, and let r 2  be the mean square end-to-end distance for each loop or tail.

Wi

i=1




3 2r 2 

3/2

  3 xi2 + yi2 + z2i exp − (69) 2 r 2 


2 2 In this expression we may replace each xi2 y i , and zi n 2 by the mean-square average for the assembly: i=1 ui = u2  u = x y z
. The resulting conformation entropy SC may then readily be expressed as 3/2
3 SC = k ln W = nk ln N0 2r 2 

−

3nk x2  + y 2  + z2 
(70) 2r 2 

Let us now consider that the loops are deformed along the x direction at constant volume. Relation (70) still holds, but the mean-square averages are now xd = "x x yd = "x y zd = "x z, with "x "y "z = 1 "2y = "2z = 1/"x = 1/". For an isotropic medium all directions are equally probable, so that on the average x2  = y 2  = z2  = r 2 /3 leading to 
nk 2 2 S"
= SC − (71) " + −3 2 " Consequently, any deformation of the loops or tails occurring during a Brownian collision has an unfavorable entropic cost V = −2T S"
− S" = 1
 = nkT "2 + 2/" − 3
. The curve at the top of Figure 9 shows the steep increase in energy provided by the compression of the polymeric chains occurring when the distance between two coated particles is reduced. Obviously, this repulsive potential could play the same role as the repulsive V R
term of the DLVO theory. When added to the attractive VA R
term, it should also lead to a maximum at some distance R. Let us also notice that when the chains belonging to two particles start to overlap, this amounts to a local increase of the concentration. Consequently, besides the repulsive force coming from the volume restriction, another repulsive contribution coming from the apparition of an osmotic pressure is also present (depletion).

5.7. Kinetics of Aggregation We are now in a position to treat the full problem of irreversible aggregation by looking at the time evolution of the aggregate size distribution.

5.7.1. Von Smoluchowski Equations The kinetic equations that must be used had been derived as early as 1916 by Max Von Smoluchowski trough the use of aggregation kernels Kij associated with the following irreversible reaction: i + j → i + j. Here the notation [i] refers to the ensemble of aggregates containing i particles

Nanocrystals from Solutions and Gels

578 while the Kij carry all the physical and chemical aspects of the collision. The equations may be written by noticing that [k] species may appear in solution as a result of the reaction between [i] and [j] species such that i + j = k. But the same [k] species may also disappear after reaction with any [i] species. Consequently, if ck stands for the number of aggregates of size k per unit volume, the time evolution of this concentration should be given by  dck 1  Kik ci ck = K cc − dt 2 i+j=k ij i j i

allowing one to find an analytical solution. Inserting Kij = K into (72) allows one to perform the following factorizations: C=



k=1

ck

 dck K k−1 c c − Kck C = dt 2 i=1 i k−i

⇒

 dC KC 2 dck = = − KC 2 dt dt 2 k=1

⇒

(72)

=−

KC 2 2

(76)

The 21 factor is there to avoid double counting in the first sum describing the creation of species [k]. As all the physical and chemical aspects are encapsulated in the Kij kinetic constants, all the problems lie in getting a meaningful i j dependence of these kernels and in the resolution of an infinite system of coupled equations. As for the nucleation and growth steps, it may be useful to consider two limiting cases: diffusion-limited and interface-limited aggregation.

If we start from an initially monodispersed system (i.e., at t = 0 we have c1 = c0 and ck = 0 for k > 1), then the variations with time of the total concentration in clusters Ct
or in single particles c1 t
are obtained after integration of (76):

5.7.2. Pure Diffusion Kernel

c1 t
, c2 t
may be obtained from (76) by making k = 2 leading to dc2 /dt = c0 1 + t
−3 − 2c2 t
1 + t
−1 where we have set  = Kc0 /2
. A solution to this equation may be easily found by setting c2 t
= gt
c0 1 + t
−3 with gt
satisfying the differential equation dg/1 + g
=  dt/1 + t
[i.e., gt
= t]. These two particular solutions for k = 1 and k = 2 may be put under the following general form:

If there is no chemistry involved in the sticking of two aggregates, we may use diffusion equations to get a kernel Kij . Here, we are interested by the probability Pij that a cluster with radius Ri shall be hit in the time interval dt by another cluster of radius Rj having the concentration cj . This probability should be equal to the diffusion flux ij into a sphere of radius Rij = Ri + Rj
. From Fick’s first (j = −D × grad c
and second laws dc/dt = D# 2 c
the concentration c t
of species [j] at a distance  from the center of the sphere of radius Rij may be written
  Rij  − Rij c t
= cj 1 − 1 − erf √  2Dt
 c dc j ⇒ ij = lim = t→ d R ij =Rij

Ct
=

(73)

⇒

Kij = Di + Dj
Ri + Rj
(74)

Now, the Stokes–Einstein equation D = kT/6R and the fact that for compact objects Rk = R1 × k1/3 lead to Kij =

kT 1/3 i + j 1/3
i−1/3 + j −1/3
6

(75)

5.7.3. Getting an Analytical Solution Consequently, even for the simplest situation, the problem has no analytic solution. However, one may notice that there is no i j dependence when i = j as Kii = 4kT/6 = K. This suggests that one look at what the solution will be if Kij = K for all clusters (even those for which i = j). This is obviously a considerable simplification of the problem,

c1 t
=

c0 1 + Kc0 /2
t2 (77)

Kc0 t/2
k−1 1 + Kc0 t/2
k+1
2  k+1
k 1 4 k2 ck t
= 2 K c0 t 1 + 2/Kc0 t


(78)

It is easy to show that (78) is a general solution to (76) leading to dck t
=0 dt

As both clusters are undergoing Brownian motion, their relative diffusion coefficient should be Drel = Di + Dj
. With this diffusion flux of matter, the sticking probability may easily be evaluated as Pij dt = 4R2ij Drel ij dt

⇒

ck t
= c0 ⇒

c0 1 + Kc0 /2
t

k − 1
Kc0

⇒

t=

⇒

ckmax = 4c0

k − 1
k−1
k + 1
k+1


(79)

5.7.4. Discussion Consequently, for a given k > 1 ck goes through a maximum and tends to zero as t −2 for very large times. In this last case (i.e., when t ≫ 2/Kc0 ), the power term in (78) may be developed as 

1 1 + 2/Kc0 t


k+1


≈ 1 − 2/Kc0 t
k = expk ln1 − 2/Kc0 t


2k ≈ exp − Kc0 t

(80)

This shows that for a given time, the size distribution has the shape of a decreasing exponential whose decreasing rate becomes smaller as time increases. It also means that the

Nanocrystals from Solutions and Gels

579

size distribution function may be written under an interesting scaling form
 

kck c0 Kc0 k = k=1 = = 1 + t

Ct
2 k=1 ck
 k ⇒ ck t →
= k−2 exp − k

(81)

The important point in this relation is that the k and t dependence of ck t
is given through an universal function of a single variable k/kt
 ∝ k/t  that does not depend on the initial distribution. For the present case the scaling exponents were  = 2 and  = 1.

5.8. Fractal Geometry Obviously, the case of constant kernel Kij = K is interesting by its analytical solution but not very realistic and one may seek a more general solution. To do that we may notice that according to (75) Kij kernels are invariant under any scaling " as K"i "j
= Ki j
. In fact it may be shown that it is this very particular scaling relation that is responsible for the asymptotic scaling of the solutions ck ∼ k−2 exp−k/k
. Let us recall that (74) and (75) were derived under the assumptions of dense aggregates undergoing Brownian motion. A direct observation of these aggregates by transmission electron microscopy has in reality shown that their structures were highly ramified and tortuous exhibiting a scale invariance [59].

A direct measure of this scale invariance is provided by the associated fractal dimension  [60]. Figure 10 (top right) shows how a bidimensional (d = 2) fractal object may be generated by sticking together identical spherical particles of diameter . First, a seed sphere is centered at the origin. In a second step, four other spheres are added at the left, right, top, and bottom of the seed. In a third step, the ensemble of five spheres is considered a unique object, on which four similar objects may be added using the same rule as before. After a large number of iterations, it is easy to see that the resulting highly ramified object has a diameter L = 3p  and contains N = 5p spheres. Now a fractal object characterized by a fractal dimension  may be defined by the property that the minimal number of spheres N of diameter  needed to cover its matter scales like N ∼ − . In the present case we have L = 3p  ⇒ ⇒

N = 5p ln N lnL/
= p= ln 3 ln 5
 L ln 5 = ≈ 147 N =  ln 3

(82)

The fact that  < d = 2 is here the geometric expression that the object displays holes at any scale. The same model could be used to derive a fractal dimension when d = 3. At each step, we have to add two similar objects above and below the plane. For the same diameter, the number of spheres is just N = 7p−1 + 6 × 7p−1 = 7p , leading to  = ln 7/ ln 3 = 177 < d = 3. Table 6 gives the fractal dimensions observed for some physical processes leading to highly ramified structure. As can be seen the ideal value of  = 177 is found to be very close to that found for Brownian cluster–cluster aggregation. This should not be very surprising, as fractal dimensions are universal exponents that do not characterize uniquely the geometry of ramified objects. They just inform us about the kind of growth mechanism that has been used in order to generate the aggregates. Consequently, all objects grown using a hierarchical cluster– cluster mechanism should share the same fractal dimension close to 1.8, even if their local structure may appear quite different.

〈k〉

k/〈k〉

5.8.1. Fractal Dimension 

Table 6. Some fractal dimensions Df for physical processes leading to ramified structures (compiled from [59]). Structure

Figure 10. Aggregation and fractal aggregates. Top left shows the fractal ( < 1/2) and limited aggregation ( < 0) regimes. Bottom left illustrates the case ( ≥ 1/2) leading to a divergence of the mean cluster size (gelation regime). Top right diagram shows how a fractal object may be generated through iteration of simple rules. Bottom right is a 3D-percolation cluster characterized by the fractal exponents  = 253  = 184 $ = 130 = 187 = 1143 = 389  = 2548 and " = 1374. Black spheres correspond to the backbone ( = 187), large gray spheres to the red contacts ( = 1143), and small white spheres to dangling ends.

Solvated linear polymer Brownian cluster–cluster aggregation Ballistic cluster–cluster aggregation Ideal linear polymer Solvated branched polymer Chemical cluster–cluster aggregation Ideal branched polymer Diffusion limited aggregation Percolation cluster Ballistic limited aggregation Dense particle

Df 166 177 191 20 20 204 216 250 249 2.8–3.0 30

580 A direct consequence of this matter of fact is that fractal dimension gives only information on the total mass MR
∼ R found in a sphere of radius R. In particular, it does not give any details concerning the connectivity properties (the way masses are connected in space). In order to describe these connectivity properties, several other scaling exponents have to be used [59, 60].

5.8.2. Bond Dimension  The bond dimension " (or tortuosity) gives the minimal distance L = R" that must be covered by a walk entirely contained in the object in order to go from two points separated by a Pythagorean distance R. A high " value means that very tortuous circumvolutions are present between one point in space to another. However, since the minimum path in the cluster should not be shorter than the Pythagorean distance d nor longer than a complete random walk, we should have 1 ≤ " ≤ 2. Usually this exponent is evaluated through the use of numerical simulations. For instance, it has been found that diffusion-limited aggregates (" = 102) are much less tortuous than cluster–cluster aggregates (" = 125) [61]. The most tortuous structures are found with 3D-percolation clusters (" = 1374) [62] or branched polymers (" = 136) [63]. One may notice that both  and " are extrinsic dimensions as they link a fractal property of the object (mass M or minimal walk L
to a Pythagorean distance R measured in an Euclidian space.

5.8.3. Topological Dimension  The topological (also graph or chemical) dimension  gives the total mass ML
∼ L visited along a walk of distance L entirely contained in the object. It may be noticed that this last exponent is not independent of the two others  and . Accordingly, as L ∼ R" , M ∼ L ∼ R , it comes that  = ". For determinist fractals one usually have either  =  (e.g., the Sierpinski gasket displaying no tortuosity but only lacunarity) or  = 1 (e.g., the Koch curve that has no lacunarity but is tortuous at all length scales). In contrast, with the fractal or bond dimensions,  is an intrinsic dimension as it links two properties (mass M to the minimal length L
of a given fractal object. This fact is also expressed by the relation  = /" showing that the ratio of two extrinsic dimensions should be another intrinsic one.

5.8.4. Spectral Dimension  The spectral dimension $ affects all properties depending not only on space (static dimensions) but also on time (dynamical dimensions). This exponent is linked to the number S ∼ M ∼ t $/2 of distinct sites visited by a random walk (the de Gennes “ant” [64]) after t-steps, or to the probability P0 ∼ t −$/2 that the ant ends up at time t on its starting point. As $ is clearly an intrinsic property independent of the embedding space, it should also be expressed as the ratio of two other extrinsic dimensions. For example a random walk in ordinary homogeneous space is such that the mean square distance varies proportionally to the time R2  ∼ t, the proportionality factor R2 /t being the diffusion coefficient. Consequently, for a random walk occurring on a fractal structure one may expect that the mean

Nanocrystals from Solutions and Gels

square displacement will scale like R2  ∼ t 2/ , the extrinsic exponent > 2 characterizing the fractal walk. Writing that M ∼ R2 
/2 ∼ t / ∼ t $/2 shows that we have $ = 2/ . The intrinsic exponent $ ratio of two extrinsic dimensions affects all transport properties in a fractal medium. For example, the diffusion coefficient is no more a constant but must depend on the distance R as R2− . A similar relation holds for electrical conductivity that becomes size dependent. Concerning the low-energy behavior of the phonon density of states it becomes nE
= E $/2−1 , leading to a density of states per unit frequency 
= $−1 (fractons) [65]. A consequence is that the specific heat at low temperature T scales like T $ . One of the big advantages of the spectral dimension is that it can be either measured on real objects (mainly through light or neutron scattering) or computed from simulations by at least three independent ways. Strangely enough, very close $ values were found from two very different fractal models (percolation cluster with $ = 130 [66] and diffusion-limited aggregates with $ = 135 [61, 67]). This has led to the conjecture that some fundamental link should exist between static and dynamic exponents. For instance, following conjecture = 3/2 (or $ = 4/3 ∼ 133) [65] may help to explain these data. However, this conjecture does not hold for lattice animals that are characterized by $ = 119 [63] and for screened growth aggregates characterized by $ values ranging from 1.10 to 1.22 [68]. However, it may be demonstrated that for any fractal for which loops are irrelevant one should have $ = 2/ + 1
[68]. This shows that a relation between static and dynamical exponents may occur but only on structures displaying no loops. According to this last relationship, if cluster–cluster aggregation ( = 142) leads to fractals with no loops, their spectral dimension should be $ = 118.

5.8.5. Backbone Dimension  Another exponent has to be introduced for dealing with mechanical properties of fractal aggregates. This is backbone dimension that is always less than the fractal dimension . It allows one to differentiate between domains connected by at least two contacts and dangling ends that are connected to the backbone only once and hence have no effect on mechanical properties. The backbone dimension is thus the fractal dimension of the object generated by removing all dangling ends to a given fractal. Going one step further one may differentiate between red contacts and blobs. Red contacts are those parts of the backbone whose removal cuts the object into two independent pieces (particles with coordination number equal to 2). The isolated clusters remaining after this operation and made of all particles having a coordination number of at least 3 are the blobs. The new object made by all the blobs has its own blob dimension

that should again be less than the backbone dimension. The backbone and blob dimensions are currently known only from numerical simulation (for a 3D-percolation cluster  = 253 > = 187 > = 1143
[69].

5.8.6. Hull Dimension  Another important exponent for fractal interfaces is the hull dimension  [69]. This exponent relates the external perimeter or surface of the aggregates to the Euclidean distance

Nanocrystals from Solutions and Gels

between two points. The hull consists of all cluster sites that are connected to infinity through an uninterrupted chain or plane of empty sites. In other words, it is the negative imprint of the aggregates onto the embedding space. Like the backbone and blob dimensions it is currently known only from numerical simulation (for a 3D-percolation cluster  = 2548).

5.9. General Solution The previous section has shown that the properties of a fractal object cannot be obtained by rewriting formulae valid on 3D-Euclidean space and simply replacing 3 by . In fact, one may rigorously define an 3D-Euclidean object by the fact that  =  = $ = = = 3, =  = 2, and " = 1. This degeneracy no longer holds for a fractal such as a 3D-percolation cluster where  = 253  = 184 $ = 130 = 187 = 1143 = 389  = 2548, and " = 1374 [69]. Obviously, it would be hopeless to find an analytical model able to explain all these values from a given set of kernels Kij . For such situations, numerical simulations are obviously of considerable help. Also, it would have been a very naïve attitude to take the kernel (75) and simply write Kij = kT/6
i1/ + j 1/
i−1/ + j −1/
with the hope that everything will run fine. Even if such expression may be found in a large number of papers or textbooks, it is definitely not correct to use the consequence of a growth process (a fractal object characterized by various exponents   $  ") as the ultimate cause of its appearance. A more correct approach is to guess some general properties for the aggregation kernel Kij from physical principles. Then one may seek either an analytical or a numerical solution to see if the resulting fractal matches some experimental structure.

5.9.1. Realistic and Simple Kernel The really fundamental point in (75) lies in the scale invariance of the aggregation kernels K"i "j
= "0 Ki j
. In this last expression, we have emphasized the null power of the scale parameter " because one may expect that in the more general case we should have K"i "j
= "2 Ki j
. However, we have to restrict the values of  to the range  < 1, since the averaged active sites in an aggregate cannot grow faster than its size i or j [70]. Another physically acceptable assumption is that the largest cluster (say cluster [i]) that has the slowest diffusion rate would be explored for a possible reaction among its i accessible sites by the smallest one [j] that has the largest diffusion coefficient. As far as seeking the simplest model able to handle the largest number of experimental cases, there is no loss of generality to consider the homogeneous kernel: Kij = kT/6
i1 × j 2−1 . The advantage of this quite simple formulation is that all the complex interfacial physics and chemistry are contained in the single -exponent characterizing the most active aggregate. The restriction comes from the unit power affecting the i-cluster. This choice reflects the possibility for species [j] to have full time to explore all exposed sites on cluster [i]. Consequently, we are limiting ourselves to the case of interface-limited aggregation. The case of diffusionlimited aggregation and derived models will thus not be

581 covered. However, as it corresponds to an unphysical mechanism for nanocrystal aggregation in solution, we are safe. Another important physical aspect contained in our kernel is the existence of a critical -value ∗ = 21 for which the kernel becomes independent of j. At this critical point, any [j] species, whatever its size, is able to stick to any i site. This is a very clear signal for gelation to occur and one may understand that two qualitatively different regimes may be anticipated.

5.9.2. Gelation Regime Gelling kernels are characterized by  > 21 (Fig. 10). In this regime it may be shown [71] that there exists a finite time tg such that
 2 − 1 c k tg = ⇒ ck t < tg
= 0   2c0 k k with k ∝ tg − t
2/1−2


(83)

In this equation  ∝ kT/
is a kinetic constant function of temperature and solution viscosity and  is a universal scaling function such that x
∼ Bx− . The exponent  should be such that  = 2 + 3
/2 > 2 ensuring that the mass flux of finite particles (sol) to the infinite cluster (gel) remains bounded for any time t > tg . As  > 21 (83) shows that the mean cluster size k diverges when t → tg . In the postgel stage (t > tg ) the size distribution is qualitatively different and takes the form ck t
∼ Bt
k− with again  = 2 + 3
/2. For this regime,  > 21 , to occur the key point is that we should have a size selectivity mechanism favoring aggregation of large species over smaller ones. This is well evidenced by the power law ck t
∝ k− (with  > 2) or by the positive contribution in our Kij kernel of the power of [j] (2 − 1 > 0 ⇔  > 21 ). As soon as  < 21 , the power reactivity of [j] species becomes negative explaining the crossover toward a regime where the size selectivity mechanism favors small clusters over larger ones. Therefore, the probability of finding an infinite cluster occupying the whole solution volume becomes zero in which case only precipitates or sizelimited aggregates should be formed (Fig. 10).

5.9.3. Fractal Regime One of the amazing things in this  < 21 regime is that the scaling invariance K"i "j
= "2 Ki j
is sufficient to determine the asymptotic shape ck t
∼ k−2 k/k
of the solutions to the Smoluchowski kinetic equations [70, 71]. In this case, the average cluster size k ∼ t  carries all the time dependence, whereas the universal function  governs the shape of the size distribution. As this function is universal, it does not depend on the complex details of the aggregation kernels and can be determined by just looking at the most probable configuration coming out of a multinomial distribution [70]   Nc = knk nk N = k

k

⇒

nk =


 N − Nc 2 1 − 2
1−2 Nc2  1 − 2
N − Nc kNc
1−2
Nc k × 1− (84) N − Nc

Nanocrystals from Solutions and Gels

582 In this equation, nk = ck × V , where V is the total volume of the solution and  is the gamma function. This expression may be further simplified by introducing the reduced variable x = k/k = Nc k/N and taking the limit Nc → 0 leading to nk =

N x
k2

⇒

x
=

1 − 2
1−2 −2 exp−1 − 2
x x  1 − 2

1 < (85) 2

As explained, the mean cluster size k carries all the time dependence that may be expressed as [71] k =

N ∼ 1 + c0 1 − 2
t1/2−1
∼ t − Nc with  =

1 1 − 2

(86)

Figure 10 shows the behavior of the normalized size distribution x
for several values of the  exponent. It is seen that when 0 ≤  < 21 , the size distribution displays a monotonic exponential decreasing behavior. Interestingly enough, when  = 0, we recover exactly the analytical solution of the Von Smoluchowski equation at constant kernel Kij = K.

5.9.4. Limited Aggregation Regime When  < 0, another qualitative change of the size distribution occurs. Instead of a monotonic decrease, we have now a bell-shaped curve leading to a narrowing of size distribution as  decreases. This crossover between a highly polydisperse size distribution ( > 0) and a more monodispersed limited aggregation ( < 0) is again well modeled by the Kij ∝ i1 × j 2−1  kernel. Accordingly, when  = 0, we get the most symmetric kernel Kij = i/j meaning that no clear distinction exists between the explored cluster and the exploring one. This is no more the case when  > 0 modeling a situation where the most probable events are the sticking of the any [j] species onto the dangling ends of the slowly moving [i] clusters. One may then understand that the result of this site selectivity (dangling ends more reactive than the backbone) would be a polydisperse collection of objects displaying holes at any scale. The situation is also unsymmetrical when  < 0. In this regime, we may say that dangling ends are poisoned becoming inactive toward sticking. By consequence, the largest [j] species, being able to explore only the outer tips of the larger [i] clusters, cannot participate in the growth process anymore. In this case, the better the poisoning, the lower  and the better the selectivity both in size (only the smallest species are allowed) and in site (backbone free of poison becomes more reactive than dangling ends). This explains the crossover from a fractal regime toward a more compact regime characterized by a bell-shaped size distribution (larger clusters just “eat” the smallest ones without being eaten by their congeners). A high compactness may be expected for large negative values of , as only very small aggregates are allowed to explore the whole backbone in

order to fill any hole. The larger ones are still there but they are staying quietly outside waiting their turn to fill the holes. Because of this hierarchical mechanism based on a full osmosis between size and site selectivity, one may also expect a strong narrowing of the size distribution as evidenced in Figure 10.

6. CONCLUSION We are now at the very end of the description of all the physical and chemical processes that may cooperate to make nanocrystals appear from initially homogeneous solutions. We have taken care to cover the whole size range, from molecular solute species displaying sizes less 0.5 nm up to macroscopic crystals or gels having size higher then 1 cm. Besides size, one may notice that we have also covered the whole time range starting from nucleation (t = 0) and ending with ripened crystals or aggregates (t → ). At each step, analytical expressions have been derived, because this is the only way to be able to fully understand the underlying physics or chemistry involved in experimental or simulation results. At this stage, we want to stress that all the derived equations are in fact very new, even if they share the same analytical shape with already published ones. The novelty comes from the fact that they are all clearly interpreted on a molecular basis, leading to deep conceptual modifications in quite old theories. Moreover, as we have for the first time a full coverage of precipitation from solutions, it is worthwhile to emphasize where fundamental changes have occurred. This will also help us to point to the directions where considerable work remains to be done to get a fully quantitative picture. Concerning nucleation theories, the deep change was the explicit introduction of a kinetic barrier G′ linked to the complex underlying chemistry involved in embryo formation. This chemistry has been reviewed and orders of magnitude have been given. The presence of this additional contribution has led to the appearance of a sticking probability gN = exp−G′ /kT
in the pre-exponential factor J0 of the nucleation rate. It was the neglect of this fundamental factor in older theories that explains the differences often observed with experimental measurements [10]. It also allows one to close definitively the controversial discussion relative to homogeneous versus heterogeneous nucleation. Accordingly, for homogeneous nucleation, it was shown that gN carries purely chemical structural information. In all the cases where nucleation is really induced by impurities or defaults, gN carries additional information about the size of the heterogeneous host and about its chemical nature through a contact angle. In fact, with the introduction of this sticking probability, we have reached the ultimate correct formulation of equilibrium nucleation theory resolving most discrepancies with experiments. Going one step beyond would require one to use nonequilibrium thermodynamics based on entropic fluxes avoiding the use of thermodynamic potentials (Gibbs function) that just cannot be defined for irreversible processes such as nucleation. By symmetry, the same conceptual change has been introduced in interface-limited crystalline growth, through another kinetic barrier G′′ and through the associated growth sticking probability gC = exp−G′′ /kT
. Another

Nanocrystals from Solutions and Gels

quite novel aspect was the derivation of Eq. (35) for spiral growth that has never been published before. Here also a deep conceptual change has occurred. In older theories, the accepted paradigm was Solution → Embryo → Nucleus  Diffusion  

 Mononuclear → Interface → Polynuclear    Spiral


This widely accepted view has several conceptual difficulties. First, diffusion should be the rate limiting step only for gC ∼ 1, a very rare case. Second, the most efficient mechanism (spiral growth) plays a minor role in crystalline growth literature and is usually treated separately from diffusionlimited or interface-limited growth. A much more logical scheme emerges by taking into account the existence of a growth sticking probability: 

Diffusion  Spiral → Polynuclear Solution → Embryo → Nucleus →  Mononuclear ↑

Here, just after nucleation, the created interface may act either as a catalyst or as an inhibitor for solute species incorporation. In the first case (gC > gN ), we invoke the most efficient mechanism avoiding the bottleneck of surface nucleation. However, owing to its very high efficiency, it very soon ends up either in the bottleneck of diffusion (gC ≫ gN ) or switching to the polynuclear mechanism if several spirals compete (high sursaturation ratios) for the same limited crystal volume. The essential point is that as the final morphology is always governed by the slowest growth mechanism, the only chance to see well-developed spirals is to use the sursaturation ratio very close to unity. The second case (gC ≪ gN ) means that for some chemical reason, the most reactive growth sites are poisoned. The spiral growth relying on the presence of these active sites is then killed. The only way to proceed is to make somewhere in a flat zone a 2D nucleus that will first grow independently from other nuclei (mononuclear mechanism) and then start to overlap (polynuclear mechanism) at some stage. Future work along these lines would be to reinterpret already published data and design new experimental settings and experiences to validate definitively this simple and logical picture. Few novelties have been brought in the section devoted to the role of the interface on the growth process. This is because the most important conceptual change (existence of a point of zero interfacial energy or PZIE) has been treated elsewhere [41]. Nevertheless, some new features have been added in order to get a deeper coherence. First, we have pointed to a useful link between the Frank and Wen model of solute–water interface and the double layer model (Stern and diffuse layers) for oxide–water interfaces. Then explicit expressions for the PZIE, Eq. (46), and equilibrium sizes, Eq. (47), have been derived. Finally, a more logical derivation from the Cardan formula of the  = 9/4 value allowing one to solve ripening equations has been proposed. Future work should obviously be oriented toward the elaboration of simple models for nonoxide and nonaqueous solid–liquid

583 interfaces. New solutions for the ripening equations covering interface-limited growth also have to be derived. The last section dealing with aggregation processes is surely the most critical one. This is just because, due to their small sizes, individual nanocrystals cannot remain isolated from their congeners. For the same reasons that molecular species may be assembled through weak bonds into supramolecular species, nanocrystals should be used as building blocks for making supracrystalline aggregates. In fact, it may be realized that all major applications involving nanocrystals should handle a more or less ordered array of such objects and not a disordered set. Consequently, if the control of the final size distribution of an assembly of nanocrystals is of the utmost importance, the control of the corresponding mesoscopic or macroscopic texture is yet another scientific challenge. This is the reason we have tried to introduce the very confusing literature devoted to aggregation processes in the most logical and synthetic way. First, we have attempted to make a clear distinction between enthalpy and entropic effects. Furthermore, if we have presented the quite old and criticized DLVO theory, it is just because no other simple theory is yet available. Consequently, if quantum electrodynamics is known to heal some obvious defaults of DLVO theory [72], it is of little practical use, being useful only in some pathological cases. Another highly confusing area concerns the geometric description of fractal objects coming out from aggregation processes. Here most published papers are concerned with the determination or measure of the fractal dimension  of the aggregates. One must realize that reducing such complex objects to a single real number is a considerable loss of valuable information. Moreover, fractal dimension being an extrinsic parameter, it just points to the kind of mechanism that has been used to generate the aggregate in our three-dimensional Euclidian space. In particular, it can say nothing about the internal architecture or detailed topology of the aggregate. To get this kind of crucial information for all practical applications, one has to rely on an intrinsic dimension such as the topological dimension  or the spectral dimension $. Being the ratio of two extrinsic dimensions (one of them being always the fractal dimension ), these intrinsic parameters helps us to understand the chemical or physical properties of the associated fractal object. Unfortunately, accurate determination of these very interesting exponents is rather rare and currently limited to ideal mathematical objects coming from numerical simulations. Future work should then be oriented toward the collection of a wealth of experimental data concerning these intrinsic fractal dimensions. Concerning the kinetics of formation of aggregates, a major conceptual change has been introduced through the Kij = kT/6
i1 × j 2−1  kernel. The simplicity of this kernel relative to other awkward expressions found in the literature is noteworthy. First, it allows one to understand the existence of three qualitatively different aggregation regimes: gelation (apparition of an infinite cluster if  ≥ 21
, fractal growth (exponential distribution when 0 ≤  < 21
, and limited aggregation (bell-shaped distribution when  < 0). Second, it avoids the widespread practice of putting at the origin (kernel Kij
the result (fractal dimension ) of the growth process. Obviously, what is still lacking is an analytical link between the exponent  and the molecular

584 chemistry occurring at the solid–liquid interface. This should be a major challenge for the coming years.

GLOSSARY Aggregate Large particle formed after irreversible sticking of several small colloidal particles. Aggregation Irreversible sticking of particles undergoing Brownian motion. Aggregation kernel Kinetic constant describing the process by which two aggregates containing i and j particles respectively leads to an new aggregate formed of i + j
particles. Aging Change of structure, morphology, or porosity with time. Aquo cations Soluble cationic hydrated species [M(H2 O)N ]z+ displaying low electrical charge z < 3
and stable at low pH. Backbone dimension Fractal dimension of an object generated by removing all dangling ends to a given fractal object. Blob dimension Fractal dimension of an object generated by removing all red contacts to a given fractal object. Bond dimension See tortuosity. Chemical dimension See topological dimension. Chronomal Mathematical integral equation Ix
describing the time evolution t = K×Ix
of a phase separation process in terms of a dimensionless reaction coordinate x and a kinetic constant K, see Eqs. (12), (26), and (33). Classical nucleation theory (CNT) Theory for nucleation based on equilibrium thermodynamics and mean field approximations. Cluster See aggregate. Coagulation Precipitation of a colloidal solution after addition of an electrolyte. Colloidal solution or sol Dispersion of solid particles (diameter 1–100 nm) in a liquid. Critical coagulation concentration (c.c.c.) Minimal electrolyte concentration allowing spontaneous coagulation of a colloidal dispersion. Critical nucleus Unstable concentration fluctuations that either redissolve or undergo further growth. Crystal Solid matter characterized by a well ordered atomic structure allowing X-ray diffraction. Crystal field stabilization energy (CFSE) Stabilization energy associated to a non-spherical charge distribution of electrons around the atomic nucleus of a transition metal element. Debye–Hückel length Thickness of the double layer function of the solution ionic strength, see Eq. (39). Depletion interaction Net attractive or repulsive force between two colloidal particles induces by the presence of a polymer in the dispersion medium. Diffuse layer Layer of counter-ions around a colloidal particle compensating at the OHP the sum of the surface charges and of the charges located at the IHP. The thickness

Nanocrystals from Solutions and Gels

of this layer is mainly function of the ionic strength of the solution. Diffusion controlled growth Growth during which the rate-limiting step is the diffusion of solute species towards the solid/liquid interface. Diffusion-limited aggregation Aggregation process during which the rate-limiting step is the diffusion of colloidal particles in the dispersing medium. Dislocation controlled growth Growth initiated on screw dislocations and bypassing the formation of a surface nucleus. DLVO theory Mathematical law giving the interaction energy of two overlapping diffuse layer as a function of their separation derived by first by Derjaguin, Landau, Verwey and Overbeek, see Eqs. (65–67). Electrokinetic potential In a colloidal dispersion, the difference in potential between the immovable layer attached to the surface of the dispersed phase and the dispersion medium (usually approximated by the potential at the OHP). Embryos Unstable concentration fluctuations that are doomed to redissolve at equilibrium. Extrinsic dimension Scaling exponent linking a property of a given fractal object to a Pythagorean distance measured in an Euclidian space. Fick’s laws Two mathematical expressions relating through the diffusion coefficient (i) the flux of matter to a gradient concentration and (ii) the variation of concentration with time to its laplacian. Fractal Mathematical object displaying holes at any scale. Corresponding physical objects are porous solids displaying porosity over several order of magnitude of scale length. Fractal dimension Scaling exponent  giving for an aggregate the total mass found in a sphere of radius R  MR
∼ R . Frank and Wen model Assumption of the existence of at least three structured domains for a solvent around a given solute (A = solvent in direct contact, C = bulk solvent and B = A/C interface). Frontier energy Energy needed for increasing the length of a given surface perimeter by one length unit. Fuchs integral Total flux of particles towards a given colloidal particle corrected for their possible interaction energy. Gel Dispersion of liquid droplets (diameter 1–100 nm) in a solid. Gibbs equation Relationship linking the decrease in interfacial energy to the concentration of adsorbed species (see Eq. (1)). Glass Solid matter displaying no crystalline order showing only X-ray diffusion. Gouy–Chapman theory Mathematical law linking the diffuse layer charge density to its associated potential at the OHP, see Eq. (40). Graph dimension See topological dimension. Growth constant Dimensionless constant incorporating shape factor, frontier energy, molecular volume, temperature, and ruling the change of the growth thermodynamic barrier with the sursaturation.

Nanocrystals from Solutions and Gels

Growth rate Rate of matter addition to an already existing particle and characterized by the rate of change of a characteristic length with time. Hamaker constant Energy parameter measuring attractive forces between surfaces, see Table 5. Hull dimension Scaling exponent relating the external perimeter or surface of a fractal object to the Euclidian distance between two points. Inner Helmholtz plane (IHP) Mean plane defined by all ionic species adsorbed into the Stern layer and not in direct contact with the surface of the colloidal particle. Interface limited aggregation Aggregation process during which the rate-limiting step is the irreversible sticking of two colloidal particles. Interface limited growth Crystalline growth during which the rate-limiting step is the formation of a surface nucleus. Interfacial energy or surface energy Energy needed for increasing the area of a given interface by one surface unit. Intrinsic dimension Scaling exponent linking two properties of a given fractal object. Isoelectric point (IEP) Solution conditions for which there is an equal number of positive and negative adsorbed species. Kinetic barrier Energetic contribution to the nucleation activation energy associated to the crossing of the solid/liquid interface. Langmuir isotherm Mathematical law describing the adsorption at a given interace of a single layer of ions, see Eq. (42). Lattice animals Connected subsets of the square lattice tiling of the plane used for modeling the theta transition displayed by branched polymers in a solvent. Loops Parts of a polymer connecting two trains. Metal alkoxides Organometallic complexes of general formula M(OR)z R = Cn H2n+1 ) leading after hydrolysis and condensation to metallic oxides MOz/2  M(OR)z + z/2 H2 O → MOz/2  + z ROH. Mononuclear growth Growth process occurring at very low sursaturation and during which a single surface nucleus is able to cover a whole crystalline face. Nucleation Formation of thermodynamically stables particles (solid or liquid) from an initially homogeneous medium (solid, liquid, or gas). It may occur either in the bulk (homogeneous nucleation) or at an interface (heterogeneous nucleation). Nucleation constant Dimensionless constant incorporating shape factor, interfacial energy, molecular volume, temperature, and ruling the change of the nucleation thermodynamic barrier with the sursaturation. Nucleation rate Number of nuclei formed per unit volume and per unit time. Nuclei Stable concentration fluctuations allowed undergoing further growth leading to ultimately to phase separation. Ostwald–Freundlich equation Relationship relating solubility to particle size, see Eq. (9). Ostwald ripening Change upon aging of the size distribution of a collection of particles owing to the occurrence of redissolution/precipitation phenomena.

585 Outer Helmholtz plane (OHP) Slipping plane for solvent molecule defining the thickness of the Stern layer. Oxo anions Soluble anionic species [MON ]2N −z
− displaying high electrical charge z > 4
and stable at high pH. Percolation A simple model in which sites on a lattice are randomly chosen and occupied and where nearest-neighbor particles are then considered to be connected, resulting in clusters of varying sizes and geometries. Percolation cluster Fractal aggregate percolating through an entire lattice network and whose formation is accompanied by the divergence of the mean cluster size and the range of connectivity. Point of zero charge (PZC) Solution conditions for which the surface of a colloidal particle is completely free of ionic species. Point of zero interfacial energy (PZIE) Solution conditions for which full covering of a surface by ionic species occur. Poisson–Boltzmann equation Mathematical law relating the laplacian of the potential to the charge density in volume and to the dielectric constant of the medium. Polyanions Small oligomeric species formed after adding an acid to oxo-anions. Polycations Small oligomeric species formed after adding a base to aquo-cations. Polynuclear growth Growth process during which several surface nuclei compete to cover the same crystalline face. Potential determining ions (PDI) Ions adsorbed on the surface of a colloidal particle or adsorbed into the Stern layer. Red contact Parts of the backbone of a fractal object whose removal cuts this object into two independent pieces. Schulze–Hardy rule Empirical law stating that the c.c.c. of a colloidal dispersion should vary as in the sixth inverse power of the electrolyte valency. Screw dislocation Crystalline defect arising from the occurrence of atomic stacking faults and moving perpendicular to the line joining the two points of an open circuit surrounding the dislocation (Burger’s vector). Shape factor Dimensionless quantity linking the volume or the surface of a particle to the cube or the square of its characteristic length (see Tables 2 and 4). Sol See colloidal solution. Spectral dimension Scaling exponent $ giving for an aggregate the number of distinct vacant sites S visited by a random walk involving t-steps S ∼ t $/2 . Spiral growth Growth process following a screw dislocation. Stability ratio Dimensionless parameter ruling the life time of a colloidal dispersion. Steric stabilization Stabilization of a colloidal dispersion by entropic forces (usually through polymer adsorption). Stern layer Compact shell of solvent molecules moving with a colloidal particle. Sticking probability Probability of incorporation of solute species according to Maxwell–Boltzmann statistics.

Nanocrystals from Solutions and Gels

586 Stokes–Einstein equation Relationship linking diffusion coefficient of a particle to its radius and to medium viscosity. Surface charge Density of adsorbed ionic species in direct contact with the surface of the colloidal particle. Surface energy See interfacial energy. Sursaturation Ratio between the concentration in solution at time t and the concentration at equilibrium (solubility) when t → + . Tails Non-adsorbed chain ends of a polymer pointing towards the solution. Thermodynamic barrier Energetic contribution to the nucleation activation energy associated to concentration fluctuations. Topological dimension Scaling exponent  giving for an aggregate the total mass M visited along a walk of distance L entirely contained in the object: ML
∼ L . Tortuosity Scaling exponenet " giving for an aggregate the minimal distance L that must be covered in order to go from two points separated by a pythagorean distance R  LR
∼ R" . Trains Bonds of a polymer in direct contact with the surface of a colloidal particle. Von Smoluchowsky equations Infinite system of coupled differential equations describing the kinetic of aggregation of a colloidal dispersion.

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