Role of steric stabilization on the arrested growth of silver nanocrystals in supercritical carbon dioxide

Article abstract of DOI:10.1021/jp026180b

Perfluorodecanethiol-stabilized silver (Ag) nanocrystals were synthesized in supercritical (sc)-CO₂ through arrested precipitation, by reducing silver acetylacetonate (Ag(acac)) with hydrogen in the presence of fluorinated thiol. The CO₂

Full text of DOI:10.1021/jp026180b

12178
J. Phys. Chem. B 2002, 106, 12178-12185
Role of Steric Stabilization on the Arrested Growth of Silver Nanocrystals in Supercritical
Carbon Dioxide
Parag S. Shah, Shabbir Husain, Keith P. Johnston, and Brian A. Korgel*
Department of Chemical Engineering and Texas Materials Institute, Center for Nano- and Molecular Science
and Technology, The UniVersity of Texas, Austin, Texas 78712-1062
ReceiVed: May 24, 2002; In Final Form: September 7, 2002
Perfluorodecanethiol-stabilized silver (Ag) nanocrystals were synthesized in supercritical (sc)-CO2 through
arrested precipitation, by reducing silver acetylacetonate (Ag(acac)) with hydrogen in the presence of fluorinated
thiol. The CO2 density used during synthesis controls the particle size and polydispersity. At high solvent
densities, (P > 250 bar, T ) 80 °C), the ligands provide a strong steric barrier that maintains small particles
with a 20 Å diameter. At lower solvent densities (P < 250 bar, T ) 80 °C), the osmotic repulsions between
capping ligands are weak, resulting in 40 Å diameter nanocrystals with higher polydispersity. At early stages
in the growth process, metal core coagulation competes with ligand adsorption. Conditions effective for steric
stabilization, such as long ligands and high solvent density, quench nanocrystal growth at relatively low
ligand binding densities, which leads to smaller nanocrystals. Under poor solvent conditions, particles grow
to larger sizes before the coverage of capping ligand is sufficient to prevent coagulation of metal particles.
Perfluorodecanethiol-coated silver nanocrystals, synthesized in either good or poor solvent conditions, readily
redisperse in acetone, fluorinated solvents, and sc-CO2 (at high density). The precursor concentration, thiol:
precursor ratio, and reaction time do not affect the nanocrystal size appreciably, although these parameters
do affect the polydispersity.
Introduction
ever, the low polarizability per volume of CO2 gives rise to
weak van der Waals forces and sterically stabilized systems in
sc-CO2 require “CO2-philic” groups with low cohesive energy
densities to provide favorable energetic interactions with the
solvent.16-19 Although examples of colloids stabilized by
hydrocarbon surfactants in sc-CO2 exist,20,21 generally, fluori-
nated groups are needed.22,23 To minimize solvation challenges,
several groups have now utilized water-in-CO2 microemulsions
as nanoreactors for synthesizing various types of nanocrystals.8,9
However, these approaches yield particles with weakly bound
surfactants and the resulting nanocrystals cannot be collected
without irreversible flocculation; the CdS nanocrystals developed
by Holmes et al. are an exception to this case.8 Recently, we
showed that silver nanocrystals capped with partially fluorinated
ligands (1H,1H,2H,2H-perfluorodecanethiol) are redispersible
in sc-CO2.24 Subsequently, we demonstrated arrested precipita-
tion in pure CO2 using hydrogen to reduce an organometallic
precursor25 in the presence of fluorinated ligands (1H,1H,2H,2H-
perfluorooctanethiol);10 however, the particles formed in these
studies were relatively large (∼55 Å) and very polydisperse
(∼47%). Furthermore, although these nanocrystals were well-
capped with the ligand and dispersed in acetone, the nanocrystals
did not redisperse in sc-CO2, as their chain lengths were too
short to overcome the relatively strong van der Waals attraction
between metal particles of this size in sc-CO2.
Nanocrystals ranging from 20 to 100 Å in diameter represent
a unique class of colloids, capable of exhibiting a variety of
size-dependent optical and electronic properties that could be
used in a variety of technologies, including coatings, environ-
mental, chemical processing, medical, electronic, and sensing
applications.1-3 To effectively utilize their size-dependent
properties, the nanocrystal size and polydispersity must be
controlled. For this purpose, organic stabilizing ligands with
proper binding strengths for the substrate4-6 are often used to
passivate the nanocrystal surfaces and quench particle growth.2
The ligands determine the particle dispersibility and enable
postsynthesis separation techniques, such as antisolvent-based
size selective precipitation. After the solvent is evaporated, the
stabilizing ligands control the edge-to-edge separation between
the metal cores and permit redispersion. In some cases, the
capping ligands can affect the nanocrystal shape.7
Recently, nanocrystals have been synthesized in supercritical
fluids (SCFs).8-11 SCFs provide a unique reaction medium with
density tunable solvation properties, which in some cases can
be exploited to manipulate nanocrystal and nanowire morphol-
ogy through temperature and pressure changes in the reaction
medium.12,13 Nanocrystal dispersibility in SCFs can be tuned
as well, since the ligand-solvent interaction strength, which is
responsible for colloid stabilization, relates directly to the solvent
density.14,15 Supercritical carbon dioxide (sc-CO2) provides a
potentially useful environment for nanocrystal synthesis, as it
is nontoxic, nonflammable, and environmentally benign. How-
Herein, we reveal that longer chain lengths (1H,1H,2H,2H-
perfluorodecanethiol; C10 vs C8) produce small nanocrystals,
20 Å in diameter with 37% polydispersity under good solvent
conditions (i.e., high CO2 density). The nanocrystal size and
polydispersity depend sensitively on the supercritical solvent
conditions. Low solvent density yields large nanocrystals with
* To whom correspondence should be addressed. Tel: (512)471-5633.
Fax: (512)471-7060. E-mail: korgel@mail.che.utexas.edu.
10.1021/jp026180b CCC: $22.00 © 2002 American Chemical Society
Published on Web 11/05/2002

Arrested Growth of Silver Nanocrystals
J. Phys. Chem. B, Vol. 106, No. 47, 2002 12179
very broad size distributions. The size and size distribution of
the nanocrystals synthesized under “poor” solvent conditions
match those obtained earlier10 using the shorter C8 stabilizing
ligands at high CO2 density. These observations are consistent
with the model for nanocrystal steric stabilization presented in
ref 14. High ligand capping density provides a thick steric barrier
that prevents metal core coagulation under both good and poor
solvent conditions. However, strong osmotic repulsions between
ligands in good solvent conditions slow nanocrystal growth at
a much lower capping density. Long ligands and high solvent
density produce small nanocrystals; ineffective stabilization at
low CO2 density leads to larger more polydisperse nanocrystals.
Our model for particle coagulation as a function of ligand
binding density reveals that relatively low ligand surface
coverages dramatically slow coagulation under conditions
providing effective steric stabilization.
the arithmetic mean radius, r1 ) ∑ ri/N∞, cube mean radius, r3
) 3/ ∑r 3/N , and harmonic mean radius, rh ) N∞/∑ (1/ri),
x
i
∞
where N∞ is the total number of particles. The moments of the
size distribution, µ1 ) r3/rh and µ3 ) r1/r3, were determined
from these measurements.
UV-Visible Absorbance Spectroscopy. UV-visible ab-
sorbance spectra were measured in a high-pressure optical cell
equipped with sapphire windows on opposing sides with an
optical path length of 0.9 cm with a 1.75 cm aperture. The cell
was loaded with nanocrystals by evaporating a drop of an
acetone dispersion into the cell. CO2 was slowly added to the
cell using a computer-controlled syringe pump (ISCO Corp.)
until reaching the desired pressure. An inline ball check valve
(High Pressure Equipment Corp.) was used to prevent nano-
crystal diffusion back into the pump. The cell was wrapped with
heating tape connected to an Omega CN76000 temperature
controller. The absorbance measurements were acquired on a
Beckman DU-40 UV-vis spectrophotometer at 1 nm intervals
from 200 to 800 nm. It was not feasible to measure spectra
during reaction due to the absorbance of the organic moieties
in the precursor.
Perfluorodecanethiol Cloud Point Measurements. The
solubility of perfluorodecanethiol in sc-CO2 was determined
using a high-pressure variable volume view cell equipped with
a front-facing sapphire window. Initially, 1 mL of perfluoro-
decanethiol was added to the cell. The thiol concentration was
controlled by the addition of CO2 to the cell using a computer-
controlled syringe pump (ISCO Corp.). After the desired
concentration was reached, the cell was pressurized to 172 bar
by filling the back of the cell with CO2. The temperature was
controlled by immersing the cell into an 80 °C water bath. After
the cell equilibrated, it was depressurized at ∼7 bar/min until
reaching the cloud point. The cloud point was defined as the
pressure at which the piston was no longer visible. Cloud point
determinations were repeated three times for accuracy. Carbon
dioxide was then added to the front of the cell to dilute the
thiol concentration, and the process was repeated.
Experimental Section
Materials. The silver precursor, silver acetylacetonate (Ag-
(acac), Aldrich Chemical Co.), 1H,1H,2H,2H-perfluorode-
canethiol (Oakwood Products Inc.), heptane (Aldrich Chemical
Co.), and Omnisolve spectrophotometry grade acetone (EM
Science) were used as received. High-grade carbon dioxide
(purity >99.99%, Matheson Gas Products) and hydrogen (purity
>99.999%, Praxair Inc.) were used.
Nanocrystal Synthesis. In a typical experiment, 12-15 mg
of Ag(acac) was loaded into the front of a variable volume view
cell. The front of the cell was filled with pressurized liquid CO2
(138 bar and 21 °C) using a manual pressure generator (High
Pressure Equipment Corp.) until reaching the desired precursor
concentration. CO2 was added into the back of the cell using a
computer-controlled syringe pump (Dionex model 501) to
increase the system pressure. The system was heated by
wrapping the cell with heating tape attached to an Omega
CN76000 temperature controller. Once the cell reached the
desired reaction conditions, 1H,1H,2H,2H-perfluorodecanethiol
and hydrogen were added to the cell through two injection loops
attached in series to the front of the cell. Injection loops of
various lengths were used to control the volume of added thiol
from 7.6 to 150 µL. The hydrogen pressure was varied from
21 to 69 bar in an 800 µL injection loop to maintain a fixed
hydrogen:precursor concentration of 30:1. Typical reaction times
were 3 h. Then, the cell was cooled and depressurized by venting
the CO2 from the back of the cell. CO2 was slowly vented from
the front of the cell, leaving only the nanocrystals, excess thiol,
and reaction byproducts. The nanocrystals were collected from
the cell using acetone. To remove reaction byproducts and
excess thiol, heptane was added as an antisolvent to precipitate
the nanocrystals, which were subsequently isolated by centrifu-
gation. The nanocrystals could then be redispersed in acetone,
sc-CO2, or fluorinated solvents (Fluorinert or Freon).
Transmission Electron Microscopy (TEM). High-resolution
TEM images were obtained using a JEOL 2010F microscope
with 1.4 Å point-to-point resolution equipped with a GATAN
digital photography system for imaging and operating with a
200 kV accelerating voltage. Low-resolution TEM images were
acquired on a Phillips EM280 microscope with a 4.5 Å point-
to-point resolution operating with an 80 kV accelerating voltage.
For imaging, the nanocrystals were deposited from acetone onto
200 mesh carbon-coated copper TEM grids. Average particle
size and polydispersity were determined using Scion Image for
Windows software. At least 300 nanocrystals were measured
to determine the average nanocrystal size, standard deviation,
Results
Perfluorodecanethiol-stabilized silver nanocrystals were syn-
thesized at 80 °C in sc-CO2 by reducing Ag(acac) with hydrogen
at a variety of precursor and thiol concentrations and pressures.
The reaction conditions explored in this study are summarized
in Table 1. The nanocrystals were generally well-passivated by
the ligands. They redisperse in acetone and fluorinated solvents
such as freon and fluorinert, which are good solvents for the
ligands. The TEM images show distinct particles separated by
the adsorbed ligands. For example, the perfluorodecanethiol-
coated silver nanocrystals in Figure 1 are separated by an
average edge-to-edge distance of 21 Å. This interparticle
separation is larger than what is found for hydrocarbon-stabilized
nanocrystals (decanethiol gives ∼12.5 Å26), due to the increased
stiffness of the fluorinated ligand.27 The lattice spacing in the
high-resolution TEM images in Figure 1 of 2.3 Å corresponds
to the d111 spacing in silver.
The significant finding in this paper is the distinct dependence
of the nanocrystal size and polydispersity on the solvent density
during particle formation. Well-passivated silver nanocrystals
could be synthesized at a wide range of solvent pressures, from
207 to 345 bar. However, they do not redisperse at pressures
less than 276 bar (at 80 °C)sas observed in the UV-visible
absorbance spectra in Figure 2. Figure 3 shows the size and
polydispersity of nanocrystals synthesized at 80 °C, 3.5 mM

12180 J. Phys. Chem. B, Vol. 106, No. 47, 2002
Shah et al.
Figure 2. UV-visible absorbance measurements of silver nanocrystals
dispersed in sc-CO2 at 80 °C and pressures of 207, 241, 276, 310, and
345 bar. The decrease in peak intensity with decreased pressure indicates
reduced solubility of the particles.
Figure 3. Data points correspond to experiments A-F. (a) Average
nanocrystal diameter (b) and size distribution moments, µ1 (2) and µ3
(1), of perfluorodecanethiol-coated silver nanocrystals synthesized at
80 °C, with a precursor concentration of 3.5 mM and thiol:precursor
ratio of 2.5 as a function of solvent pressure. The error bars represent
the standard deviation of the samples. µ1 for experiment D (3.1) has
not been shown for clarity.
Figure 1. Representative TEM images of silver nanocrystals capped
with perfluorodecanethiol synthesized in sc-CO2. Panels a and b show
images of nanocrystals grown at 80 °C and 276 bar (experiment B),
which have formed rafts of close packed particles. Panels c-e show
high-resolution TEM images exhibiting crystalline cores.
size and the polydispersity increase with decreasing pressure
(Figure 4). Nanocrystals synthesized below the dispersibility
limit of 276 bar are still well-passivated by the ligands and
redisperse in acetone, fluorinated solvents, and sc-CO2 at 80
°C and pressures greater than 276 bar.
Ag(acac), and thiol:precursor ratio of 2.5. Above 276 bar, the
particles exhibit an average diameter of 20 Å and the synthesis
pressure does not noticeably affect the nanocrystal size or
polydispersity. However, below 276 bar, both the nanocrystal
TABLE 1: Summary of the Conditions Used for Nanocrystal Growth
exp
no.
prec concn
(mM)
thiol/prec ratio
(mol/mol)
temp
(°C)
pressure
(bar)
density
(g/mL)
reaction
time (h)
avg diameter
(Å)
standard
deviation (Å)
% standard
deviation
µ1
µ3
A
B
C
D
E
F
G
H
I
3.5
3.5
3.4
3.5
3.5
3.5
0.8
1.8
0.9
1.7
3.5
3.5
3.5
3.4
3.5
3.5
2.4
2.5
2.4
2.3
2.6
2.5
2.6
2.5
2.2
2.5
2.4
2.5
0.3
0.5
4.9
5.0
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
80
345
276
259
241
223
207
276
276
207
207
276
276
276
276
276
276
0.786
0.720
0.698
0.674
0.643
0.610
0.720
0.720
0.610
0.610
0.720
0.720
0.720
0.720
0.720
0.720
3
3
3
3
3
3
3
3
3
3
6
12
3
3
3
3
17
18
17
24
34
40
19
12
27
27
17
18
18
12
14
20
6
7
11
22
21
21
4
33
38
66
91
62
52
21
34
51
65
92
61
37
37
39
52
1.2
1.3
1.9
3.1
1.8
1.8
1.1
1.3
1.5
1.8
2.6
1.8
1.3
1.3
1.3
1.5
0.9
0.9
0.7
0.6
0.7
0.8
1.0
0.9
0.8
0.7
0.6
0.7
0.9
0.9
0.9
0.8
4
14
18
15
11
7
4
6
J
K
L
M
N
O
P
10

Arrested Growth of Silver Nanocrystals
J. Phys. Chem. B, Vol. 106, No. 47, 2002 12181
Figure 4. Data points correspond to experiments A-F. Particle size
distributions as a function of solvent density. The size distribution
histograms have been normalized (frequency/max frequency) and offset
for clarity.
Figure 5. Cloud point measurements of perfluorodecanethiol in sc-
CO2 at 80 °C.
The size distribution moments, µ1 and µ3, reflect the particle
growth mechanism. Nanocrystals formed through condensation
of free atoms, or small oligomers, onto growing metal cores
are relatively monodisperse with µ1 ) µ3 ) 1, whereas
coagulative growth results in broad size distributions with µ1
> 1.25 and µ3 < 0.905.28 The size distribution moments indicate
that particles grow by coagulation at pressures below 276 bar.29
Above 276 bar, µ1 and µ3 approach 1, revealing that growth
occurs through a combination of both coagulation and conden-
sation. These results indicate that high solvent density slows
particle aggregation, which largely prevents coagulation and
stabilizes particles in the small size range. At lower solvent
density, where ligand solvation is poor, particles grow primarily
by coagulation.
Because colloid stability is provided by the adsorbed ligands,
understanding ligand solvation is very important. As particles
collide, well-solvated ligands repel each other through osmotic
forces. To stabilize the nanocrystal dispersion, these forces must
overcome the long range attraction between particles due to van
der Waals forces. For colloids in conventional solvents, which
exhibit upper critical solution temperatures (UCST), good and
bad solvent regions are defined by the θ temperature, which
relates to the critical flocculation temperature. Analogously,
colloids in SCFs exhibit lower critical solution temperatures
(LCST), and the strength of the steric repulsion is most closely
related to the density of the solvent.30,31 The cohesive energy
density of the solvent is higher at increased solvent density,
resulting in better ligand-solvent interactions.16 In SCFs, the
critical flocculation density (CFD) defines the solvent conditions
where the dispersion is no longer stable. Monte Carlo simula-
tions have shown that the CFD in SCFs corresponds to the
density where the stabilizer ligand is soluble at all concentra-
tions, defined as the upper critical solution density (UCSD).32-34
The UCSD for perfluorodecanethiol in sc-CO2 was determined
to be ∼135 bar at 80 °C, as shown in Figure 5. Figure 2,
however, reveals that pressures greater than ∼276 bar are
required for nanocrystal stabilization, indicating that the CFD
does not correspond to the UCSD. It is important to note that
the simulation results in ref 34 do not account for colloidal
core-core attractions, which are relatively strong for metal
nanocrystals. For polydisperse systems, the CFD can be blurred,
due to the fact that larger particles have stronger core-core
attractions than smaller particles.14 As shown in Figure 2, the
absorbance peak is greatest at the highest pressure, indicating
that the most nanocrystals have dispersed in CO2.35 The peak
Figure 6. Average diameter (b) and size distribution moments, µ1
(2) and µ3 (1), of perfluorodecanethiol-stabilized silver nanocrystals
grown at 80 °C with a thiol:precursor ratio of 2.5:1, as a function of
precursor concentration. The closed symbols correspond to 276 bar
and experiments B, G, and H, while the open symbols correspond to
207 bar and experiments F, I, and J. The error bars represent the
standard deviation of the samples.
height decreases as the pressure is lowered, indicating that the
largest nanocrystals have flocculated at that condition. As the
pressure decreases below 276 bar, the absorbance disappears
altogether, as the steric repulsion has become so weak that none
of the particles are stabilized at this condition.
At good solvent conditions, 276 bar, and 80 °C, the reaction
time and thiol:precursor ratio do not affect the average nano-
crystal size. Increased reaction time simply leads to a slight
broadening of the particle size distribution, perhaps as the result
of Ostwald ripening (see Supporting Information, Figures 1S
and 2S). Therefore, the reaction times studied here were limited
to 3 h. The thiol:precursor ratio could be varied widely, from
1:3 to 5:1 (thiol:precursor mole ratio), with little effect on the
nanocrystal size and size distribution (Supporting Information,
Figure 3S). The thiol:precursor ratio needed for total surface
coverage of 20 Å diameter silver nanocrystals, assuming an
average area per thiol of 16 Å2, is approximately 1:3.2,26 Thiol
diffusion to the nanocrystal surface does not appear to be size-
limiting.
The precursor concentration, however, can affect the nano-
crystal size and size distribution. Under poor solvent conditions,
both the size and the polydispersity are affected. Under good
solvent conditions, only the size distribution appears affected
by the precursor concentrationshigher precursor concentration
leads to broader size distributions. Figure 6 plots the particle

12182 J. Phys. Chem. B, Vol. 106, No. 47, 2002
Shah et al.
size and polydispersity for nanocrystals synthesized under poor
solvation conditions. As the precursor concentration increases,
µ1 and µ3 deviate increasingly from 1, signifying that these
conditions favor coagulation. A higher precursor concentration
increases the initial concentration of silver nuclei, which
increases collision rates at early times in the growth process.
At lower solvent density, an increased precursor concentration
results in larger nanocrystals as the number of initial collisions
increases. The size distribution moments indicate that coagu-
lative growth is greater at lower pressure for all precursor
concentrations.
A121 relates directly to the Hamaker constants for CO2 and silver
across vacuum, A11 and A22, respectively; A121 ≈ ( A
-
x
11
A )2. A22 has been determined to be 2.2 eV.36 A11 can be
x
22
determined using equations of state for the density-dependent
38
dielectric constant, ꢀ1, and refractive index, n1, for CO2 and
Lifshitz theory:36,39
ꢀ1 - 1
ꢀ1 + 1
3hνe (n12 - 1)2
2
3
4
A11
)
kT
+
(3)
(
)
x
16 2
(n 2 + 1)3/2
1
In eq 3, h is Planck’s constant and νe is the maximum electronic
ultraviolet adsorption frequency, typically assumed to be 3 ×
1015 s-1. The Hamaker constant for silver across sc-CO2 at 276
Discussion
bar and 80 °C is 1.5 eV.36 Taking A121 ) 1.5 eV, W ) 0.71 to
On the basis of the fact that µ1 and µ3 deviate significantly
from 1 under all of the reaction conditions studied, nanocrystal
coagulation is clearly an important growth mechanism to
understand. Coagulative nanocrystal growth requires interparticle
collisions that bring the metal cores close enough to promote
metal fusion.36 Particles with complete ligand surface coverage
do not fuse because the metal cores are sterically prevented from
touchingseven in a poor solvent (or no solvent for that matter).
Therefore, almost certainly at very early times in the particle
formation process, nanocrystals are only partially coated with
ligands in order for coagulation to occur. To gain an appreciation
for the combined influences of ligand surface coverage and
solvent strength on the growth process, the coagulation rate
between particles can be calculated. On the basis of the obvious
influence of coagulation, it is assumed that at very early times
in the particle formation process, nanocrystals will be only
partially coated with ligands and particle coagulation will
compete with ligand adsorption. Although the ligand adsorption
rate or ligand binding constants between thiols and silver in
sc-CO2 are unknown, the coagulation rate between particles can
be calculated with respect to particle size, solvent strength, and
ligand surface coverage in order to better understand the growth
process.
give K12 ) KBrownian/W ) 3.1 × 10
cm3/sec.
- 10
12
Of course, sterically stabilized nanocrystals undergo many
collisions without coagulation and particle growth. Therefore,
steric stabilization must be properly accounted for to determine
the coagulation rate. As a general guideline, the metal cores
fuse if they approach to within 5 Å (δc) of each other.36 Because
the interparticle separation for perfluorodecanethiol-coated
nanocrystals in the absence of solvent is 21 Å, well-passivated
nanocrystals are protected from fusion and coagulative growth
under all solvent conditions. Partially passivated nanocrystals,
however, have the possibility of fusing into larger clusters if
the cores approach to within δc upon collision. Early in the
growth process, particle coagulation competes with ligand
adsorption. Therefore, to prevent coagulative nanocrystal growths
which leads to large particle sizes and polydispersitysthe
solvent must enable the ligands to fully extend and prevent
cluster coagulation between partially capped cores until capping
becomes sufficiently high. The repulsive force between particles
provided by the ligands can be included in W using a model
for steric stabilization.
Vincent et al. proposed that steric interactions between
particles result from osmotic and elastic repulsions.40,41 The
osmotic repulsive energy (Φosm) results from the unfavorable
exclusion of solvent between colliding nanocrystals, and the
elastic term evolves from the mechanical compression of the
layer that prevents the metal cores from touching. Φosm depends
on the particle radius R and the center-to-center separation
distance x:
The nanocrystal coagulation rate (J12) of particles of size “1”
with particles of size “2” depends on their initial concentrations,
N1 and N2: J12 ) K12N1N2.37 K12 is the coagulation coefficient.
Considering the simplest case when particle collisions occur
through Brownian motion, K12 ) KBrownian ) 4π(R1 + R2)(D1
12
+ D2). The cluster diffusivities, Di, can be estimated using the
Stokes-Einstein equation, Di ) kT/6πµRi, where k is Boltz-
mann’s constant, T is the temperature, and µ is the CO2 viscosity.
4πRkbT
2 1
2
x - 2R 2
Φosm
)
φ
- ø l -
l < d - 2R < 2l
(4)
(
)(
)
Vsolv
2
For equally sized silver particles in sc-CO2 at 80 °C and 276
Brownian
12
bar, K
) 2.2 × 10-10 cm3/sec. Note that K1B2rownian is
4πRkbT
independent of the core radius when equally sized particles are
considered. Attractive or repulsive interactions with energy Φ(x),
where x is the center-to-center separation distance, affect the
collision rate and must be accounted for using a correction factor
W:37
2 1
2
x - 2R
1
4
x - 2R
Φosm
)
φ
- ø l2
- - ln
(
)
[
(
(
))
]
Vsolv
2l
l
d - 2R < l (5)
In eqs 4 and 5, Vsolv is the molecular volume of the solvent and
φ is the volume fraction profile of the stabilizer extending from
the particle surface. Important parameters in eqs 4 and 5 include
the ligand length, l, and the Flory-Huggins interaction param-
eter, ø, between the ligand and the solvent. In Flory-Huggins
theory, ø ) 1/2 typically represents the boundary between a
good solvent (ø < 1/2) and a poor solvent (ø > 1/2). Φosm can
be significant even at low surface coverage of ligand in a good
solvent. When ø > 1/2, Φosm becomes attractive due to the poor
solubility of the ligands. To determine solvent conditions in
SCFs, the compressibility must be accounted for and effective
(density-dependent) ø parameters can be used.42,43 For sc-CO2,
however, the density dependence of ø is not well-known.
Φ(x)
kT
∞
W ) (R + R )
exp
x-2 dx
(1)
∫
(
)
1
2
R1+R2
Subsequently, K12 ) KBrownian/W.
12
The van der Waals interaction between two equally sized
particles of radius R depends on the Hamaker constant, A121
,
for silver acting across CO2:36
2R2
2R2
x2
x2 - 4R2
A121
ΦvdW ) -
+
+ ln
(2)
2
2
(
)
]
x2
[
6
x - 4R

Arrested Growth of Silver Nanocrystals
J. Phys. Chem. B, Vol. 106, No. 47, 2002 12183
Figure 7. Ligand volume fraction profiles as a function of radial
distance from particle surface. Particle sizes shown are 20 Å diameter
(s), 40 Å diameter (- - -), and 60 Å in diameter (‚ ‚ ‚) at full surface
coverage θthiol ) 16 Å2. Also shown are volume fraction profiles for
Figure 8. K12 for 20 Å diameter perfluorodecanethiol-stabilized silver
nanocrystals as a function of thiol surface coverage (φs) at varying
solvent conditions (ø). Solvent conditions shown are ø ) 0, 0.125, 0.25,
0.375, 0.5 (θ), 0.75, and 1.0. Brownian limit corresponds to the
coagulation coefficient determined without any interparticle interactions
(2.2 × 10-10 cm3/sec). The van der Waals limit corresponds to the
coagulation coefficient with the incorporation of interparticle van der
Waals attractions (3.1 × 10-10 cm3/sec), without any steric effect from
the ligand. The hard sphere limit is the surface coverage necessary to
prevent core fusion (separation distance > 5 Å).
20 Å nanocrystals at varying surface coverages, θthiol ) 16 Å2, θthiol
)
24 Å2, and θthiol ) 36 Å2
.
Therefore, in the analysis presented here, ø parameters simply
provide guidelines for understanding how good, θ, and poor
solvent conditions affect particle growth, rather than providing
an explicit relationship to the CO2 density. Although more
sophisticated models have been developed for solvent-tail
interactions in SCFs to include effects of compressibility,32-34
the simpler model presented below captures many of the key
features needed to explain the trends in coagulation rates in this
paper.
The volume fraction profile (φ) of the stabilizing ligands
decreases radially from the metal surface and depends on the
particle size and “footprint” area per headgroup. The radial
decay of the volume fraction profile increases with decreasing
particle size due to the increased interfacial curvature, as shown
in Figure 7. The φ curves in Figure 7 were calculated using the
geometric equation for cylinders with ligand cross-section area
(SAthiol) of 14.5 Å,226 extending radially from a curved surface
with radius Rp + z:
SAthiolRp
Figure 9. Coagulation coefficient (K12) as a function of thiol surface
coverage (φs) at varying particle sizes, 20 Å diameter (s), 40 Å diameter
(- - -), and 60 Å diameter (‚ ‚ ‚). Solvent conditions shown are ø ) 0,
0.25, and 0.5 (θ). The hard sphere limits correspond to the surface
coverage needed to prevent core fusion (separation distance > 5 Å).
φ(z) )
(6)
θ
thiol(Rp + z)
where θthiol is the surface area per thiol headgroup, which
represents the binding density, and z is the radial distance from
the metal surface. Coagulation rates are plotted as a function
of the ligand volume fraction at the metal surface, φs ) φz)0 in
Figures 8 and 9, taking the close-packed layer to have a density
of 16 Å2/thiol.26
Low molecular weight alkane and fluoroalkane stabilizers
undergo only a very slight amount of compression, as the only
ligand mobility is rotation around a few carbon-carbon bonds.
For this reason, the elastic term given by Vincent is not
applicable; instead, a modified hard sphere model is used to
represent the repulsion due to high packing density of thiol on
the nanocrystal surface. The metal surfaces of colliding particles
Figure 7 shows that xmsd increases with higher binding density
and increasing diameter. If the metal cores approach to within
the critical fusion distance δc, they will fuse. If xmsd g δc,
nanocrystal aggregation will not lead to fusion and the coagu-
lative growth will stop. The ligand surface coverage (φs) when
this occurs in a θ solvent (ø ) 1/2) is called the hard sphere
limit, denoted as φs,HS. Note that even at poor solvent conditions,
the hard sphere repulsion provided by the adsorbed ligand can
protect the cores from fusion. Notice in Figure 9 that φs,HS
decreases with increasing particle diameter. In other words, θthiol
at the hard sphere limit for coagulation is considerably less for
larger particles than for smaller particles.
Figure 8 shows K12 calculated for 20 Å diameter perfluoro-
decanethiol-stabilized silver nanocrystals as a function of φs
by inserting Φtotal ) Φvdw + Φosm into eq 1. In a good solvent
(ø < 1/2), K12 decreases as φs increases due to the improved
steric barrier between nanocrystals. When ø ) 1/2, Φosm ) 0
and K12 does not depend on φs until reaching the hard sphere
cannot approach closer than a minimum separation distance xmsd
,
at which point the ligand volume fraction in the interfacial
region, φtotal, reaches 1. The volume fraction profile in the
interfacial region is determined by using the volume fraction
profiles of opposing particles at a given separation distance,
determined using eq 6, and adding the volume fractions in the
region of ligand overlap. xmsd can be thought of as an effective
hard sphere separation distance between colliding particles.

12184 J. Phys. Chem. B, Vol. 106, No. 47, 2002
Shah et al.
limit at φs,HS ) 0.69. Note that at complete coverage φs ) 0.91.
When ø > 1/2, the poorly solvated ligands increase the
interparticle attractions and K12 increases with higher φs, until
reaching the hard sphere limit when coagulative growth stops.
At all solvent conditions, nanocrystal coagulation terminates at
the hard sphere limit. However, nanocrystal growth in a good
solvent terminates at ligand surface coverages significantly lower
than φs,HS. The φs amount required to prevent particle coagula-
tion increases significantly as the solvent quality decreases due
to the loss of osmotic repulsion between the nanocrystals. Under
ideal solvent conditions (ø ) 0), coagulative nanocrystal growth
is quenched at φs ) 0.41.
Figure 9 shows K12 calculated as a function of nanocrystal
size and solvent condition. The value of φs necessary to stop
nanocrystal growth is lower for larger nanocrystals for all ø.
For example, in a ø ) 0 solvent, 20 Å particles require a surface
coverage of φs ) 0.41, while 60 Å particles require only φs )
0.34. Larger particles exhibit less surface curvature than smaller
particles, which enhances the steric repulsion between particles.
Consistent with the nanocrystal absorbance spectra in Figure
2, ø decreases as the solvent density increases. In fact, the system
is below the CFD at the lowest pressures. Therefore, provided
that ligand binding rates do not vary significantly with pressure,
nanocrystal coagulation terminates earlier in the growth process
when using higher pressures, thus leading to smaller particles
with lower polydispersity.
In ref 10, silver nanocrystals were synthesized in CO2 using
perfluorooctanethiol (C8) as a capping ligand. The particles were
relatively large and polydisperse and did not redisperse in CO2,
even at high density. At a precursor concentration of 3.5 mM,
the average particle diameter was 57 ( 28 Å (polydispersity of
∼47%) for perfluorooctanethiol, while it was only 16 ( 6 Å
(polydispersity of 39%) for perfluorodecanethiol. µ1 ) 1.5 and
µ3 ) 0.8 for the particles coated with perfluorooctanethiol, as
compared to values of µ1 ) 1.3 and µ3 ) 0.9 for the particles
capped with perfluorodecanethiol. The C8 chains are much less
effective as a steric stabilizer as compared with the C10 chains,
and particle coagulation dominates growth when using the
shorter chains. The transition from coagulation-dominated
particle growth to controlled growth and stabilization of particles
in the small size range occurs when going from low CO2 density
to high CO2 density and from increasing the ligand chain length
from C8 to C10. For example, when osmotic steric stabilization
is lost at a reaction pressure of 207 bar, the average particle
diameter increases to 40 ( 21 Å, which is much closer to the
sizes synthesized in ref 10. The longer ligand, which also has
a greater ratio of CF2 to CH2 groups, is more effective at
preventing coagulative collisions between particle cores. How-
ever, fully passivated cores are protected from irreversible
flocculation even at conditions where the ligand cannot provide
dispersibilitysconsider, for example, the deposited nanocrystals
in the TEM images and the fact that well-capped nanocrystals
of all sizes can be flocculated and precipitated using antisolvents
without irreversible coagulation or change in nanocrystal size.3,26
These results show the importance of steric stabilization during
the growth stage on maintaining control over particle size and
polydispersity.
solvents, and sc-CO2. The unique density tunable solvation
characteristics of SCFs provide a means to examine the effect
of steric stabilization on the arrested precipitation of nanocrys-
tals.
At high solvent densities, the ligands are adequately solvated
and capable of preventing coagulative collisions between
growing clusters with relatively low ligand surface coverage.
The resulting nanoparticles are in the range of 20 Å. Decreasing
the solvent pressure to conditions where the ligands are no
longer capable of stabilizing the nanocrystals causes the average
particle size to increase significantly. The growth mechanism
becomes coagulation-dominated, and a much higher ligand
surface coverage is needed to provide the thick steric layer
necessary to prevent fusion between metal cores. Coagulation
during the time it takes to achieve this higher surface coverage
results in larger particles with higher polydispersity. The large
differences for silver nanocrystal size and redispersibility for
the perfluorooctanethiol vs perfluorodecanethiol ligand result
from the improved steric stabilization provided by the latter. In
addition to solvent quality, other parameters typically found to
affect particle size were examined. In contrast with our previous
study with perfluorooctanethiol,10 the average particle size is
independent of precursor concentration under good solvent
conditions. Lowering the thiol:precursor ratio does not increase
the particle size, indicating that thiol diffusion to the nanocrystal
surface is rapid. The influence of effective ligand stabilization
as described by the experimental data and model in this study
should be universal for all single-phase nanocrystal synthesis,
even in conventional solvents.
Acknowledgment. This work is supported in part by the
STC Program of the National Science Foundation under
Agreement No. CHE-9876674 and the Welch Foundation.
B.A.K. also thanks NSF (Agreement No. CTS-9984396) for
support through a CAREER Award.
Supporting Information Available: Full description of the
materials. This material is available free of charge via the
Internet at http://pubs.acs.org.
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