NANO
LETTERS
Designing van der Waals Forces
between Nanocolloids
2005
Vol. 5, No. 1
169-173
Silvina M. Gatica,† Milton W. Cole,†,§ and Darrell Velegol*,‡,§
Department of Physics, Department of Chemical Engineering, and Materials Research
Institute, The PennsylVania State UniVersity, UniVersity Park, PennsylVania 16803
Received October 21, 2004; Revised Manuscript Received November 24, 2004
ABSTRACT
van der Waals (VDW) dispersion forces are often calculated between colloidal particles by combining the Dzyaloshinskii-Lifshitz-Pitaevskii
(DLP) theory with the Derjaguin approximation; however, several limitations prevent using this method for nanocolloids. Here we use the
Axilrod-Teller-Muto 3-body formulation to predict VDW forces between spherical, cubic, and core−shell nanoparticles in a vacuum. Results
suggest heuristics for “designing” nanocolloids to have improved stability.
Introduction. van der Waals (VDW) dispersion forces
between colloidal particles have been calculated using
Dzyaloshinskii-Lifshitz-Pitaevskii (DLP) theory1,2 for over
30 years. The usual scheme is to combine DLP3,4 with the
Derjaguin approximation5,6 to account for particle curvature
with spherical or rod-shaped particles. For nanoparticles, this
method of calculation has several critical shortcomings. (1)
Accurate limiting cases can be difficult to evaluate except
for particles either nearly-touching or far apart.7-9 For
intermediate separations, a common approximation is to use
an additive Hamaker approach.10 (2) The dielectric or
polarizability properties for nanocolloids are neither bulk nor
molecular,11-13 and even within a particle can be spatially
varying. (3) The discrete nature is usually ignored for the
constituent atoms in the nanocolloids or nanocluster. (4) DLP
provides little mechanistic insight into how to design more
stable nanocolloids.14
In this letter we use the Axilrod-Teller-Muto (ATM)
3-body formulation15-17 to predict VDW forces between
spherical, cubic, and core-shell nanoparticles18 in a
vacuum (Figure 1). We focus on points 1, 3, and 4 from
the Introduction. Our previous research has addressed
point 2,19 and we expect this to be an important avenue
of future research. The long-term goal of the work is
to develop heuristics for “designing” nanocolloids to
have the desired dispersion and assembly (e.g., quantum dots20 and fluorescent particles) by examining all four
points.
Method for Evaluating VDW Forces. A general formalism for calculating VDW forces is to consider atom-atom
* Corresponding author. E-mail [email protected].
† Department of Physics.
‡ Department of Chemical Engineering.
§ Materials Research Institute.
10.1021/nl048265p CCC: $30.25
Published on Web 12/17/2004
© 2005 American Chemical Society
interactions (two-body interactions), then three-body interactions, four-body interactions, etc. This is written3,4
V0 )
(3)
(3)
+ ∑∑Vijγ
) + ...
∑i ∑γ Viγ(2) + (∑i β>γ
∑ Viβγ
i<j γ
(1)
The two-body interactions are summed over all pairs in both
spheres, while the three-body interactions are summed with
one atom in the first sphere and two atoms in the second
sphere, then two atoms in the first sphere and one atom in
the second sphere. In this manuscript we will neglect all fourbody and higher interactions.
In 1943 Axilrod and Teller15 (and independently, Muto16)
extended the perturbation theory employed by London21 to
find the three-atom interaction. The London result for twobody interactions and the “ATM result” for three-body
interactions (Vijk) may be written
Vij ) -
C6
r6ij
1 + 3cos θi cos θj cos θk
Vijk ) C9
C6 )
C9 )
r3ij r3ik r3jk
I
∫0∞ R1(iω)R2(iω) dω ) 3p
π 6
3p
π
I
∫0∞ R1(iω)R2(iω)R3(iω) dω ) 3p
π 9
3p
π
(2)
(3)
where the angles θi are for the triangle formed by the three
atoms. The r terms are the center-to-center distances between
the atoms.
Table 1. Molecules Used in This Studya
substance chemical formula
hexane
silica
sapphire
water
C6H14
SiO2
Al2O3
H2O
MW
SG
86.18
60.08
101.96
18.01
0.660
2.20
3.99
1.00
n0 (#/A3) R0 (A3)
0.00461
0.0220
0.0236
0.0334
11.85
5.25
7.88
6.88
a The molecular weight (MW) and specific gravity (SG) of the materials
are given.
Figure 1. Core-shell nanoparticles interacting with a gap (δ) and
center-to-center separation (r). Both particle 1 and particle 2 are
composed of a core material A (core has radius R) and a shell
material B (of thickness w). The intervening material is vacuum.
A similar geometry exists for cubic core-shell particles.
The coefficients (C6 and C9) are calculable from the
polarizabilities (R) for the pertinent atomic species. In
principle, these R should be calculated as a function of size
for the nanocolloid (see point 2 in the Introduction).
However, for the purposes of this paper, in which we focus
on points 1, 2, and 4 from the Introduction, we estimate the
“atomic” polarizabilities from known dielectric spectra for
n-hexane6 (C6H14), fused silica6 (SiO2), sapphire6 (>99.9%
Al2O3), and water.8 The “atomic” polarizabilities are taken
for the entire unit (e.g., SiO2, Al2O3). The dielectric spectra
come from absorption measurements, giving the loss modulus
(′′) at real frequencies (ω); a Kramers-Kronig relation then
transforms this function to the real function (iω). The
complexity of the spectra (particularly, water) makes it
difficult to use a simple Drude model.
In this manuscript we have neglected changes and spatial
variations in polarizabilities (R) due to the nanosize of the
particle, and we make a further estimate, obtaining the
polarizability from (iω) using the Clausius-Mosotti relationship22
(iω) - 0
(iω) + 20
)
4π
n R(iω)
3 0
(4)
While this equation makes no particular assumption about
the form of the dielectric function (i.e., it does not depend
on a Drude model), nor does the model depend on the
substance density (n0), the equation does assume that the
material is a continuum. We recognize that the continuum
approximation is not correct for nanocolloids, since there
are so many surface atoms compared with interior atoms,
but combining this approximation with known spectra is the
best approximation available for the polarizabilities of these
nanoclusters. Table 1 lists values of the polarizabilities and
number densities for the atoms used in this letter. The
equation used to construct the function (iω) is23
(iω) ) 1 +
dj
∑j 1 + ωτ + ∑j
j
170
fj
ω
()
ω
1 + gj +
ωj
ωj
2
(5)
Figure 2. VDW forces between two silica spheres with n ) 619
atoms in a vacuum. The lattice constant a ) 3.569 Angstroms and
the distance of closest approach occurs for a center-to-center
distance of 10.192a. While the two-body forces capture the
qualitative trend of the VDW forces, the three-body forces are
essential for quantitative results, since they constitute roughly 20%
of the total VDW forces.
where the molecular dipoles (dj), the relaxation times (τj),
oscillator strengths (fj), resonance frequencies (ωj), and
bandwidths (gj) are known24 for the substances we examined.
Table 2 lists our calculated values of the I6 and I9 coefficients
defined in eq 3.
The adequacy of using only two-body and three-body
terms in eq 1 depends on having particles for which the
atomic density is sufficiently small. The authors have
previously shown that the two-body and three-body terms
are the first two terms in an expansion of DLP theory,17 and
this has been verified by others.14 The two-body and threebody interactions ignore quadrupole, octupole, and higher
order terms, and thus anisotropies that will arise at short
distances compared with the atomic radius. In addition, at
very small distances, four-body and higher atom interactions
can become important, along with higher order terms in the
separation (e.g., r-8 and r-10). It must be remembered,
however, that cluster separations smaller than the cluster size
can still be large on the atomic scale. As the figures in this
letter show below, the two-body and three-body VDW forces
capture the essential physics of many real material systems.
Results and Discussion. Figure 2 shows for silica spheres
the two-body, three-body, and two-body-plus-three-body
VDW interactions. An important point is that at intermediate
gaps, the three-body forces are as much as 21% of the twobody forces. Thus, it is important to account for the threebody forces for quantitative purposes. Furthermore, this ratio
(∼0.2) suggests that the four-body forces are likely to be
Nano Lett., Vol. 5, No. 1, 2005
Table 2. Evaluated Constants I6 and I9 for Equation 3a
a
system
I6 (A6/s)
system
I9 (A9/s)
silica-silica
silica-water
silica-hexane
silica-sapphire
sapphire-sapphire
sapphire-water
sapphire-hexane
water-water
water-hexane
hexane-hexane
1.635 × 1017
0.816
6.036
2.456
3.698
1.228
9.043
0.409
3.005
22.44
silica-silica-silica
silica-silica-water
silica-silica-hexane
silica-silica-sapphire
silica-water-water
silica-water-hexane
silica-water-sapphire
silica-hexane-hexane
silica-hexane-sapphire
silica-sapphire-sapphire
water-water-water
water-water-hexane
water-water-sapphire
water-hexane-hexane
water-hexane-sapphire
water-sapphire-sapphire
hexane-hexane-hexane
hexane-hexane-sapphire
hexane-sapphire-sapphire
sapphire-sapphire-sapphire
3.747 × 1017
1.863
13.87
5.564
0.931
6.870
2.769
51.85
20.54
8.275
0.469
3.415
1.383
25.62
10.18
4.12
195.3
76.63
30.46
12.32
The Clausius-Mosotti equation was used to estimate molecular polarizabilities from known dielectric data and expressions.24
Figure 3. VDW potentials between two cubic particles with the
given center-to-center separation (r/a). The cubes have 125 silica
atoms. The inset shows two of the many possible orientations (top
inset shows face-to-face; bottom inset shows corner-to-corner). In
the limit of large r/a, eqs 1-3 and I6 from Table 2 can be used to
find that the asymptotic ordinate value is -777 eV, since for large
r/a the three-body interactions should approach zero.
much smaller (i.e., ∼0.22). At very large gaps, we have
shown previously (numerically and analytically) that threebody forces for spherical particles become insignificant
compared with two-body forces, for reasons of symmetry.19
We emphasize that the current accuracy limitation in Figure
2 and in other calculations in this paper results primarily
from the accuracy of the available polarizability data, not
from neglecting four-body and higher forces. Our research
group continues to study changes in polarizability (including
spatial variations within the particles) for nanocolloids
compared with bulk or molecular values.
Figures 3 and 4 compare forces between silica spheres
and cubes. Figure 3 shows the VDW potential for two cubes
averaged over all orientations, while Figure 4 compares
potentials for cubes at various orientations, and also for
Nano Lett., Vol. 5, No. 1, 2005
Figure 4. Interaction energies between particles having various
relative orientations. All cases are normalized by the energy of two
cubes averaged over all orientations. The cubic particles have 125
silica atoms, while the spheres have N ) 123.
spheres. For purposes of the comparison, the cube has N )
125 atoms, and the sphere has N ) 123 atoms (i.e., nearly
the same). The small number of atoms gives small VDW
attractions (∼kT/15 at small separations), but enables calculation of VDW forces over many orientations in a
reasonable computation time. Nevertheless, it is important
in viewing Figure 3 to remember that the VDW interaction
scales roughly as N2. The eventual goal is to test heuristics
learned from these calculations experimentally, and thus
going from N ) 125 atoms to N ) 2500 atoms would give
attractive VDW interactions of roughly 25kT for the system
shown.
The interparticle distance (r) is center-to-center, since it
would otherwise not be possible to define the gap for the
various orientations of cubes. The cubes are examined for
several geometries: (1) when the faces are parallel to each
other, (2) when the corners of the cubes give the point of
closest approach, (3) when the edges of the cubes give the
171
Figure 5. VDW forces with various shells on a silica core, relative
to values for a particle of the same radius made from pure silica.
The core has N ) 515 atoms, while the shell has N ) 104 atoms.
Three-body forces are roughly 30% of the total energy. A layer of
hexane causes the spheres to attract more than a layer of water.
closest approach, and (4) averaged over all orientations of
particles 1 and 2, such that
〈V〉 )
∫ ∫∫ ∫
2π
0
π
0
2π
0
π
0
V(θ1, φ1, θ2, φ2)sin θ1 sin θ2 dθ1 dφ1 dθ2 dφ2
(4π)2
(6)
For a view of two of the orientations, see the inset to Figure
3. The orientations of the cubes were specified by identifying
the coordinates of every constituent of the cube, and then
rotating the coordinates about two independent axes using
tensor rotation matrices.25 The face-to-face orientation of the
cubes gives the smallest VDW attraction for a fixed centerto-center distance; the corner-to-corner orientation gives the
largest attraction (as expected, because this combination has
the closest approach). Thus, if the angular orientation of the
particle is fixed, a cubic shape gives either the largest or the
smallest attraction, depending on the orientation. On the other
hand, if Brownian motion is able to randomly orient the
cubes, then the cube has more attraction than the sphere.
Figure 5 compares interactions between spherical silica
particles with various shell layers. To simplify the calculation,
we made all atoms have the same size and scaled the real
atomic polarizability of the shell material to the polarizability
used in the calculation. The relation is Rcalc
shell )
/n
,
where
the
n
is
atomic
density.
As
expected,
nshellRreal
core
shell
the core particles with the lower polarizability material in
the shell layer have smaller VDW attractions. Adding the
three-body contribution is important for these systems, since
it is up to about 30% of the two-body value. Importantly,
the water shell gives smaller VDW attractions than the
hexane shell.
While calculations for Figure 5 were done in a vacuum,
the calculations also have ramifications for particles in a
liquid environment. Clearly, if the particles are in vacuum,
those with shells have greater mutual attraction than particles
without a shell. However, if the nanocolloids are immersed
in a solvent, then either that solvent surrounds the particle
or some other shell of material surrounds the particle. By
172
making the shell material with lower polarizability than the
bulk solvent that otherwise would surround the particle, we
can reduce VDW attractions between the particles, as shown
in Figure 5.
The results indicate that a cosolvent system, with a very
small amount of a second solvent dissolved in the primary
solvent, might greatly improve nanocolloid stability in three
ways. (1) The majority solvent can be chosen to minimize
VDW attractions. For example, putting silica particles in pure
octane instead of pure water can reduce VDW attractions
by a factor of 5.26 (2) If the minority cosolvent selectively
binds to the surface (e.g., water, for silica particles in an
octane-water mixture), the adsorbed layer will reduce the
VDW attractions further, as Figure 5 shows. (3) The adsorbed
minority cosolvent can add a repulsive solvation force.27,28
All three effects lead to improved particle stability. Yet a
fourth possibility is that the adsorbed cosolvent will reduce
surface reactions or dissolution, minimizing Ostwald ripening
of the particles.29-31 Currently, we are working on measurements of nanocolloid forces,32 in order to test our hypothesis
concerning cosolvents.
The ATM method is relatively simple to use for many
material systems. Computational constraints can, of course,
make the calculations time-consuming, since the number of
terms in the ATM model grows as N3 rather than as N2 (as
for two-body systems). In addition, the ATM method is quite
amenable to using calculations of more exact polarizability
data33,34 or calculations35,36 for nanoclusters. For denser
systems, four-body forces and others will need to be
considered; however, the three-body forces often give
reasonably quantitative results that can be used to design
particle systems. It is important to note that direct numerical
calculations using density functional theory37 or other
methods usually give energies with insufficient accuracy to
determine VDW energies; however, these methods can give
polarizabilities with sufficient accuracy to use with the ATM
method.13,19,36
Our calculations have not been compared with DLP theory,
since neither the nearly-touching limit (i.e., the Derjaguin
approximation) nor the far-field limit (i.e., r-6) apply. That
is, even at small gaps, where the Derjaguin approximation
would normally work, for nanocolloids the distance required
for the approximation is less than the distance between atoms,
invalidating the DLP model. However, previous investigators
have examined VDW interactions between two spheres or
clusters at arbitrary separations. Langbein developed a
general expression for the nonretarded VDW interactions
between two spheres at any separation,38 and Kiefer et al.39
simplified Langbein’s expression to a more easily calculated
form. However, these works rely on having geometrically
perfect spheres, rather than clusters or particles with discrete
atoms. These authors have also examined the case of particles
having internally varying polarizabilities (and thus, continuum core-shell particles).8,40
Marlow and co-workers have carried out extensive calculations of cluster VDW interactions, including for “discrete” systems and complex shapes.41,42 They have compared
results of alternative methods: (a) a continuum approach
Nano Lett., Vol. 5, No. 1, 2005
based on Lifshitz theory, (b) Hamaker theory for the spatial
variation, with a substitution of the asymptotic many-body
coefficient for the two-body interaction coefficient, and (c)
the VDW interaction derived from a set of coupled harmonic
oscillators. Each of these approaches has its merits and
disadvantages, and a definitive comparison is not feasible
here. Method (a) is suitable only at separations large
compared to the lattice spacing, where DLP theory applies.
Method (b), which is widely used, is known to exhibit
qualitative errors in related applications.43 Method (c)
includes all orders of particle interactions, but it is limited
to systems that follow the harmonic oscillator model (i.e.,
none of the systems studied in this letter) and is computationally more expensive than our approach. Marlow and coworkers have also evaluated a fourth model, a sum over
molecules of particle A of the individual many-body interactions with particle B. While appealing in some respects (e.g.,
accuracy at small separation), this approach does not yield
the known long-range interaction.
Conclusions. Two important, distinct, and yet equivalent
approaches in the literature for calculating VDW interactions
are (1) evaluating the effect of the perturbation of the free
electromagnetic field caused by the presence of the particles
(i.e., the Lifshitz approach) and (2) adding the two-body
(Hamaker), three-body (Axilrod-Teller-Muto), four-body, etc.
interactions. We have used the second approach, since it is
computationally tractable for nanocolloids and since it
provides mechanistic insight.
The ATM method provides a powerful method for using
polarizabilites to calculate the leading many-body contributions to VDW forces. Furthermore, the method overcomes
some of the limitations of current evaluations of the Lifshitz
theory. In sum, the calculations in this letter enable us to
hypothesize two heuristics that could lead to improved
particle stability: (a) use spherical particles instead of cubic
if the cubic particles can assume all orientations, and (b)
use a cosolvent system where one solvent selectively adsorbs
to the particle, creating a “shell” of low polarizability (small
attraction), and also a possible solvation layer (large repulsion).
Acknowledgment. The authors thank the National Science Foundation (NER grant CTS-0403646, NIRT grant
CCR-0303976) and the Ben Franklin Center of Excellence
for funding this work. We thank Jorge Sofo for helpful
discussions.
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