Ceramics I
Liquid Phase Processes
Precipitation
Size!
Shape!
Size distribution!
Agglomeration!
Ceramics I
Liquid Phase Processes
Precipitation
Four major questions:
1. Why do molecules/ions precipitate?
2. What determines the size?
3. What determines the size distribution?
4. What determines the shape?
We have to have a closer look at the precipitation process!
Ceramics I
Liquid Phase Processes
1. Why do molecules/ions precipitate?
Supersaturation as a First and Simple Thermodynamic Approach
The molar Gibbs Free Energy ΔG and the Supersaturation S are related as follows
(T, p are constant):
8
Δ G = - RT ln S
Actual concentration in solution
S=
Equilibrium concentration
=
(ci)v i
ci: Concentration of species i
Ksp
vi: Stoichiometric index (number of mols)
Δ G = 0 for S = 1
Δ G < 0 for S > 1
ΔG (kJ/mol)
6
Driving force for
precipitation
4
2
0
-2
0
1
2
3
4
5
-4
-6
Saturation, SS(a
/a equ)
(cact
act/cequ
The larger the supersaturation S, the larger
the driving force for precipitation!
The thermodynamic driving force for crystallization is the supersaturation of the solution!
Ceramics I
Liquid Phase Processes
Precipitation
Four major questions:
1. Why do molecules/ions precipitate?
2. What determines the size?
3. What determines the size distribution?
4. What determines the shape?
Ceramics I
Liquid Phase Processes
Precipitation
Nucleation and
growth
determine size,
size distribution,
shape and
agglomeration!
1. Nucleation
Let‘s have a closer
look at nucleation
and growth!
2. Growth
Ceramics I
Liquid Phase Processes
1. Nucleation
Homogeneous nucleation
seed
Heterogeneous nucleation
Ceramics I
Liquid Phase Processes
1.1 Homogeneous Nucleation: Thermodynamics
8
T = 25oC
P = 1 atm
ΔG (kJ/mol)
6
4
Δ G = - RT ln S
Equilibrium
2
0
-2
0
1
2
3
4
Driving force for precipitation
5
-4
is ΔG < 0 (S > 1).
-6
Saturation, S (aact/aequ)
10
V = 25 cm3/mol
However: Often S >> 1,
T = 25 oC
5
J)
S = 0.5
ΔG (10
-18
but still no precipitation!
0
0
10
20
30
S=2
-5
S = 100
S = 10
-10
Radius, r (Ao)
Why?
Ceramics I
Liquid Phase Processes
1.1 Homogeneous Nucleation: Thermodynamics
Why?
Because surface free energy is
not considered.
r
During nucleation, small clusters are formed with a large fraction of the
molecules/ions at the surface.
The free energy for the surface molecules/ions is larger than the free energy for
the interior molecules/ions.
This surface free energy has to be taken into account when calculating ΔG.
Ceramics I
Liquid Phase Processes
1.1 Homogeneous Nucleation: The Classical Crystallization Model
r
Δ G = - RT ln S
Nucleus
r > r*: Growing
Embryo
r < r*: Dissolving
8
Surface (r2)
T = 25 oC
V = Molar volume of the precipitate
ΔG (10
-18
J)
4
ΔGmax
2
0
-2
0
10
r*
γ = Surface free energy
20
-4
-6
V
RT ln S + γ A
v = Volume of the aggregate
V = 25 cm3/mol
6
ΔG=-
v
30
Total
Bulk (r3)
-8
Radius, r (Ao)
S < 1: ΔG always positive
S > 1: ΔG with maximum at r* (critical size)
A = Area of the aggregate
If the particle/aggregate is a sphere:
4πr3
ΔG=RT ln S + 4πr2γ
3V
Bulk („Interior“)
Surface
Ceramics I
Liquid Phase Processes
1.1 Homogeneous Nucleation: Thermodynamics
r
Nucleus
r > r*
Embryo
r < r*
Small supersaturation is not
enough to overcome the surface
free energy!
Nucleation only with large S!
4πr3
ΔG=RT ln S + 4πr2γ
3V
4
3
V = 25 cm3/mol
J)
-18
ΔG (10
S=2
T = 25 oC
2
γ = 0.15
1
J/m2
Bulk („Interior“)
Surface
0
-1
0
10
20
S = 10
-2
S = 20
-3
-4
30
S = 100
Radius, r (Ao)
One possibility to lower the
activation energy ΔGmax (or
to decrease r*) is to increase
the saturation S.
Ceramics I
Liquid Phase Processes
1.2 Heterogeneous Nucleation: Thermodynamics
r
High surface free energy:
Nucleus
r > r*
Embryo
r < r*
r* and ΔGmax are large.
Low surface free energy:
r* and ΔGmax are small
4πr3
ΔG=RT ln S + 4πr2γ
3V
ΔG (10
-18
J)
4
3
V = 25 cm3/mol
2
S = 10
T = 25 oC
γ = 0.20 J/m2
1
0
-1
-2
-3
-4
0
10
20
30
γ = 0.15
γ = 0.10
γ = 0.05
Radius, r (Ao)
Bulk („Interior“)
Surface
Another possibility to lower
the activation energy ΔGmax
(or to decrease r*) is to lower
the surface free energy γ.
Ceramics I
Liquid Phase Processes
1.1 Homogeneous Nucleation: Thermodynamics
4πr3
ΔG=RT ln S + 4πr2γ
3V
Determination of ΔGmax and r*:
3
-18
J)
2
ΔG (10
dr
=0
2γV
*
r =
RT ln S
4
r*: Small for high
saturation and/or
small surface energy
ΔGmax
1
0
-1
dΔG
0
10
r* 20
-2
30
16 π V2 γ3
ΔGmax =
3 (RT ln S)2
-3
-4
o
Radius, r (A )
Free energy necessary to
form a stable nucleus!
Ceramics I
Liquid Phase Processes
1.1 Homogeneous Nucleation: Exercise
r*
2γV
=
RT ln S
16 π V2 γ3
ΔGmax =
3 (RT ln S)2
Suppose we want to produce Mg(OH)2 by mixing MgCl2 and NH4OH. Assuming that we
start with a solution containing 0.25 M of MgCl2 and 0.55 M of NH4OH at 100°C.
Data for Mg(OH)2: Solubility in water at 100°C = 0.04 g/L, Mw = 58.33 g/mol, ρ = 2.36
g/cm3, γSL = 0.12 J/m2, R= 8.31 J/Kmol, NA = 6.02 · 1023 mol-1.
1) What is the critical nuclei size?
Assumption:
⎛ nlim,sp
S =⎜ e
⎜ nlim,sp
⎝
ν
⎞
⎟
⎟
⎠
Actual concentration
of limiting specie
Equilibrium concentration
of limiting specie
2) What is the number of molecules in a critical nuclei?
ν: stoichiometric index
(number of mols)
Ceramics I
Liquid Phase Processes
1.1 Homogeneous Nucleation: Thermodynamics is not enough!
4πr3
ΔG=RT ln S + 4πr2γ
3V
Determination of ΔGmax and r*:
dΔG
dr
No indication about the time!
2γV
*
r =
RT ln S
4
3
ΔG (10
-18
J)
2
r*: Small for high
saturation and/or
small surface energy
ΔGmax
1
0
-1
=0
0
10
r* 20
-2
30
16 π V2 γ3
ΔGmax =
3 (RT ln S)2
-3
-4
o
Radius, r (A )
Free energy necessary to
form a stable nucleus!
Ceramics I
Liquid Phase Processes
1.2 Homogeneous Nucleation: Kinetics
How fast does nucleation occur?
r
Ion
Nucleus
Embryo
Diffusion
Embryo
+
Step 1
Reaction
Embryo
Nucleus
Step 2
Rate of nucleation can be expressed in a simple
Boltzmann approach:
dN(r*)
J=
dt
= K exp -
ΔGmax
kT
K = Kinetic prefactor; difficult to determine, often empirical!
The nucleation rate J (number of nuclei formed per unit volume per unit time) is
dependent on the activation energy ΔGmax, i.e., on the „difficulty“ for embryos to
reach the critical radius of a stable nuclei!
Ceramics I
Liquid Phase Processes
1.2 Homogeneous Nucleation: Kinetics
How fast does nucleation occur?
J=
dN(r*)
dt
=
K exp -
ΔGmax
kT
Kinetic factor (~ 1010-1035 cm-3 sec-1)
Jmax = Maximum nucleation rate
J = Jmax exp -
ln(J/Jmax) = -
ΔGmax
kT
ΔGmax
kT
Ceramics I
Liquid Phase Processes
1.2 Homogeneous Nucleation: Kinetics
How fast does nucleation occur?
ΔGmax
ln(J/Jmax) = kT
16 π V2 γ3
ΔGmax =
3 (RT ln S)2
R = k · NA
ln(J/Jmax) = -
16 π V2 γ3
3 NA2 (kT)3
A
A
1
=2
(ln S)
(ln S)2
A ∝ γ3
Ceramics I
Liquid Phase Processes
1.2 Homogeneous Nucleation: Kinetics
How fast does nucleation occur?
16 π V2 γ3
3 NA2 (kT)3
0
1
-2
ln(J/Jmax)
Decreasing nucleation rate
ln(J/Jmax) = -
A
1
=2
(ln S)2
(ln S)
2
10
-4
If A is small (i.e., surface free
energy γ is small), then a small
saturation S and also a small
increase of S leads to fast
nucleation.
20
100
-6
A = 200
-8
400
-10
-12
1
10
100
Saturation ratio, S
1000
If A is large (i.e., the particles
exhibit a high surface free energy),
then even a large saturation S
does not result in fast nucleation.
Ceramics I
Liquid Phase Processes
1.3 Heterogeneous Nucleation: Kinetics
Decreasing A: From homogeneous to heterogeneous nucleation
hom
One possibility to get nucleation
at lower saturation S is to lower
the surface energy γ (and thus
A):
Het
Jmax ∞
Concentration
of seeds
J = Jmax
BaSO4
Homo
< Jmax
Heterogeneous
nucleation
hom
J << Jmax
1) Heterogenous nucleation takes only place as long as there are seeds present
2) Heterogeneous nucleation (A=2) takes place at lower saturation ratio (
Jmax is 105)
3) At higher saturation (A=200), homogeneous nucleation occurs with Jmax = 1030
4) Homogeneous nucleation: Very sensitive to slight changes in S, difficult to control
5) Heterogeneous nucleation: Better control over particle size distribution, because the
nucleation rate is almost independent of S
Ceramics I
Liquid Phase Processes
2. Growth
Additional
surface energy
Cylinder
r
New nucleus (nucleation):
d
ΔG = -
4π r3 RTln S
3V
Bulk
4
V = 25 cm3/mol
3
T = 25 oC
ΔG (10
-20
J)
2
π dr2 RTln S + 2πdr γ
ΔG = -
ΔGmax
V
0
-1
0
2
4
6
8
Bulk
10
S,
-2
-3
Surface
New layer on the surface (growth):
Surface
d = 10 Ao
1
+ 4πr2 γ
Total
Bulk
γ
Surface
ΔGmax
Similar to nucleation: S and γ are the
main parameters to be considered!
-4
o
Radius, r (A )
Main difference: Surface energy ∝ r!
Ceramics I
Liquid Phase Processes
2. Growth
Two „borderline“ cases: 1) rough and 2) smooth surfaces
1. Case:
Assumption that
•
surface energy γ is low, and
•
saturation ratio S is high
ΔGmax is small
Formation of rough surfaces
2. Case:
Assumption that
•
surface energy γ is high, and
•
saturation ratio S is low
ΔGmax is large
Formation of smooth surfaces
Ceramics I
Liquid Phase Processes
2. Growth
1. Case:
C∞
Assumption that
surface energy γ is low, and
•
saturation ratio S is high
Ceq
ΔGmax is small
Formation of rough surfaces
Growth
rate
T. Ring p. 197-198
=
Diffusion x Probability
rate
factor
For
spherical
coordinates
dr
dt
∝
Ceq
r
Diffusion controlled growth
Diffusion
=1
Fick‘s law:
Δ
•
δ
J = - D· C
Every ion/molecule that hits the
surface is attached to the nuclei
(because of the small ΔGmax).
Probability = 1
Growth rate only
dependent on diffusion rate
Diffusion rate dependent on
Ceq and r (concentration at
the crystal surface)
Diffusion controlled growth (when γ low, S high) dependent on
i) equilibrium concentration at the surface of the nucleus, and ii) radius of the nucleus
Ceramics I
Liquid Phase Processes
2. Growth
2. Case:
Assumption that
•
surface energy γ is high, and
•
saturation ratio S is low
ΔGmax is large (nucleation controlled!)
growth
Formation of smooth surfaces
growth
Mononuclear
growth
Polynuclear
growth
Ceramics I
Liquid Phase Processes
2. Growth
2. Case (large γ, low S):
growth
Mononuclear
growth
Growth
rate
=
dr
dt
∝
Diffusion x Probability
rate
factor
r2
ΔGmax
x exp RT
Nucleation
controlled
growth!
The high surface energy (
large ΔGmax) leads to the situation that only
very few molecules/ions are able to overcome the large activation energy
(nucleation controlled), and thus are able to attach to the growing nuclei. The
slow process does not result in a concentration gradient.
Diffusion
growth
Polynuclear
growth
dr
dt
∝
Ceq
ΔGmax
x exp RT
Nucleation
controlled
growth!
Due to the large number of nucleation sites on the surface, thermal
fluctuations and Fick‘s diffusion play a role: There is a concentration gradient.
The growth rate depends on thermal and Fick‘s diffusion [(Ceq) comes into
play] and on the activation energy ΔGmax, but it does NOT depend on the
radius r of the nuclei!
Ceramics I
Liquid Phase Processes
2. Growth: Summary
1. Case:
Diffusion
Controlled
C∞
δ
Diffusion
Ceq
(low γ, high S)
Mononuclear
(high γ, low S)
Nucleation controlled
2. Case:
dr
dt
∞ r -1
ΔGmax does not play
a big role!
Growth rate is inversely proportional to r!
Diffusion
growth
dr
dt
∞r2
Growth rate is directly proportional to r2!
Diffusion
Polynuclear
growth
dr
dt
∞r0
Growth rate does not depent on r!
Ceramics I
Liquid Phase Processes
3. What determines the size distribution?
Diffusion
Polynuclear
growth
growth
dr
dt
∞r0
What is the prediction regarding
size distribution?
Growth rate is not
dependent on r, i.e., all
nuclei grow with the
same speed.
Time
Initial absolute size
distribution remains
constant over time (0.1
μm) (but relative size
distribution decreases).
Ceramics I
Liquid Phase Processes
3. What determines the size distribution?
Mononuclear
growth
dr
dt
Diffusion
+
growth
∞r2
Polynuclear
growth
dr
dt
Diffusion
growth
∞r0
What is the prediction regarding size distribution?
Time
Absolute particle size
distribution broadens
with growth time, i.e,
larger particles grow
faster (growth rate
proportional to r2) than
the small ones.
Ceramics I
Liquid Phase Processes
3. What determines the size distribution?
C∞
Diffusion
controlled
δ
Diffusion
Ceq
dr
dt
∞ r -1
What do you expect regarding size distribution (small ones grow fast, large ones slow down)?
Particle size distribution
narrows with growth time,
i.e., smaller particles grow
faster than the larger
ones.
Time
Ceramics I
Liquid Phase Processes
Nucleation & Growth: Examples
Homogeneous Nucleation for Monodisperse Particles:
Hot Injection Method
The method involves the injection of a “cold”
(room temperature) solution of precursor
molecules into a hot liquid (300 °C).
Hot injection leads to instantaneous
nucleation (very high S), quenched by
fast cooling of the reaction mixture and
fast decrease of the supersaturation by
the nucleation burst. Further growth of
the nuclei into mature nanocrystals
occurs at a lower temperature, such that
new nucleation events do not occur:
Separation of nucleation and growth
Monodisperse nanoparticles
Small 2005, 1, 1152; Angew. Chem. In. Ed. 2007, 46, 4630
Ceramics I
Liquid Phase Processes
Nucleation & Growth: Examples
Homogeneous Nucleation for Monodisperse Particles:
Hot Injection Method
CdSe
Small 2005, 1, 1152
Ceramics I
Liquid Phase Processes
Nucleation & Growth: Examples
Heterogeneous Nucleation for Monodisperse Particles:
1)
Synthesis of monodisperse iron
nanoparticles (seeds) with sizes of 4, 8,
and 11 nm from reaction mixtures
containing 1:1, 1:2, and 1:3 molar ratios
of pentacarbonyliron and oleic acid.
2)
Solutions containing 1.5, 3.0, and 4.5
mmol of the iron oleate complex (low
concentration to suppress
homogeneous nucleation!) were
prepared by heating appropriate
amounts of pentacarbonyliron and oleic
acid in dioctyl ether at 403 K for 12 h.
After refluxing the mixtures generated
from the various combinations of the
iron nanoparticles and the iron oleate
solutions, it was possible to synthesize
monodisperse iron nanoparticles with
particle sizes of 6, 7, 9, 10, 12, 13, and
15 nm.
TEM images of a) 6-, b) 7-, c) 8-, d) 9-, e) 10-, f) 11-, g) 12-, and h) 13-nm-sized air-oxidized iron oxide
nanoparticles showing the one nanometer level increments in diameter.
Angew .Chem. Int. Ed. 2005, 44, 2872
Ceramics I
Liquid Phase Processes
4. What determines the shape?
What determines
the crystal
morphology/shape?
(a) Equilibrium shape:
thermodynamics-determined
- Low saturation ratio, S
(b) Kinetic shape:
kinetics-determined
- High saturation ratio, S
Ceramics I
Liquid Phase Processes
4. What determines the shape?
(a) Equilibrium shape:
thermodynamics-determined
What if the surface energy (γ) depends
on the crystallographic orientations ?
Low saturation ratio, S
Face with lowest surface energy
determines the crystal shape!
γ F1
γF2
F2
h1
F1
h2
γ F1 < γ F2
Wulff diagram
Aspect ratio =
b
c
=
γF1
γF2
The growth rate hi of each face is
proportional to its surface energy γi.
Thermodynamic equilibrium shape: Minimization of the surface free energy!
Ceramics I
Liquid Phase Processes
2. What determines the shape?
(b) Kinetic shape:
kinetics-determined
What if the growth rate depends
on the crystallographic orientations ?
F2
F1
b dr dt F 2 k 2 S m2
=
Aspect ratio = =
c dr dt F 1 k1S m1
High saturation ratio, S
Face with slowest growth rate
determines the crystal shape!
Ceramics I
Liquid Phase Processes
4. What determines the shape?
Surface energy can be tuned through the
adsorption of ions, surfactants, proteins !
Example:
Ceramics I
Liquid Phase Processes
4. What determines the shape?
Surface energy tuning through surfactants!
Surface modulation effects
induced by surface-selective
surfactants on either a)
anisotropic rod or b) disc
growth. When surfactant
molecules specifically bind to
the {100} and {110} surfaces of
a hexagonal structure,
preferential growth along the
<001> directions and therefore
rod growth is facilitated (a).
In contrast, when surfactant
molecules bind to the {001}
surfaces of a hexagonal
structure, it prevents growth
along the <001> direction and
therefore disc shapes are
obtained (b).
Angew. Chem. Int. Ed. 2006, 45, 3414
Ceramics I
Liquid Phase Processes
4. What determines the shape?
Surface energy tuning through polymers!
2μm
BaSO4 Morphogenesis at pH 5
0.5μm
PEG-b-PMAA-Asp
PEG-b-PEI-SO3H
2μm
0.5μm
No additive
2μm
PEG-b-PEI-COOH
Dr. Cölfen, MPI Potsdam
PEG-b-PMAA-PO3H2
Ceramics I
Liquid Phase Processes
Liquid Phase Processes: Summary
Synthesis of powders
(ex: Al2O3, ZrO2)
... involve the precipitation of
ions from a supersaturated
solution and can be used for:
... occur through nucleation
and growth mechanisms
which ultimately control:
Construction materials
(ex: concretes)
Fabrication of complex
microstructures
(ex: nacre)
Particle size distribution
Particle shape
Aggregation state
Ceramics I
Liquid Phase Processes
Liquid Phase Processes: Summary
Nucleation and growth are
thermally activated
processes (ΔGmax), which
can occur by a number of
different mechanisms:
The saturation ratio (S)
and the surface energy (γ)
are the main factors
determining the nucleation
and growth mechanisms
Nucleation:
• Homogenous
• Heterogeneous (seeds)
Growth:
• Diffusion-controlled
• Nucleation-controlled
Therefore, can be tailored
to control the particle
size distribution and crystal
morphology of precipitates
Ceramics I
Liquid Phase Processes
Liquid Phase Processes: Summary
Saturation
ratio, S
High
nucleation
rate
Diffusioncontrolled
growth
Size
distribution
narrows
during growth
Kinetic
shape
1 μm
Ex: Precipitated
submicron-sized SiO2
Low
nucleation
rate
1 --
Nucleationcontrolled
growth
Size
Equilibrium
distribution
shape
broadens
during growth
Ex: Big quartz crystals