van der Waals forces between nanocolloids
Silvina M. Gatica1, Milton W. Cole,1,3 and Darrell Velegol2,3,*.
1
Department of Physics, The Pennsylvania State University, University Park PA 16803.
2
Department of Chemical Engineering, The Pennsylvania State University, University Park PA 16803.
3
Materials Research Institute, The Pennsylvania State University.
*
to whom correspondence should be addressed. email velegol@psu.edu.
submitted to Nano Letters 19 Oct 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” nanoparticle stability.
keywords: nanocolloids, nanocolloids forces, Axilrod-Teller, Axilrod-Teller-Muto
Table of contents graphic
1
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 exist only for particles nearly-touching or widely-spaced.7,8,9 For intermediate separations a
common approximation is to use the additive Hamaker approach.10 2) The dielectric or polarizability
properties for nanocolloids are neither bulk nor molecular.11,12,13 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 formulation15,16,17 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 dispersion and assembly (e.g., quantum dots20 and fluorescent particles) by
examining all four points.
2R
B
2
1
A
Figure 1. Core-shell nanoparticles interacting
with a gap () 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 system exists for cubic core-shell
particles, with similar variables.

r
2R + 2w
2
Method for evaluating VDW forces
A general formalism for calculating VDW forces is to consider atom-atom interactions (2-body
interactions), then 3-body interactions, 4-body interactions, etc. This is written3,4


( 3)
V0  Vi( 2)   Vi
 Vij(3)   ...
i

i j 
 i  

(1)
The two-body interactions are summed over all pairs in both spheres, while the 3-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 4-body 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 2-body interactions and the “ATM
result” for 3-body interactions (Vijk) may be written
Vij  
1  3 cos  i cos  j cos  k
C6
, Vijk  C 9
6
rij
rij3 rik3 r jk3

C6 
(2)

3
3
3
3
 1 i 2 id 
I 6 , C9 
1 i 2 i 3 id 
I9


 0

 0

(3)
where the angles i are for the triangle formed by the three atoms. The r’s are the center-to-center
distances between the atoms.
The coefficient (C6 and C9) are calculable from the polarizabilities () for the pertinent atomic
species. In principle these  should be calculated as a function of size for the nanocolloid (see point 2 in
the Introduction), but since this letter focuses on point 1, we estimate the atomic polarizabilities from
known dielectric spectra for n-hexane6 (C6H14), fused silica6 (SiO2), sapphire6 (>99.9% Al2O3), and
water8.
These spectra come from absorption measurements, giving the loss modulus () at real
frequencies (); a Kramers-Kronig relation then transforms this function to the real function (i). The
complexity of the spectra (particularly, water) makes it difficult to use a simple Drude model.
3
Since we already have neglected changes in polarizabilities () due to the nanosize of the particle, we
make a further estimate, obtaining the polarizability from (i) using the Clausius-Mosotti relationship22
i   0
4

n 0 i
i  2 0
3
(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.
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. Table 1 lists values of the
polarizabilities and number densities for the atoms used in this letter. The equation used to construct the
function (i) is23
 i   1  
j
dj
1  

j
j
fj
  
1 g j

 j   j




2
(5)
where the molecular dipoles (dj), the relaxation times (j), oscillator strengths (fj), resonance frequencies
(j), and bandwidths (gj) are known24 for the substances we examined. Table 2 lists our calculated values
of the I6 and I9 coefficients defined in Equation 3.
Table 1. The molecules used in this study. The molecular weight (MW) and specific gravity (SG) of the
materials are given.
substance chemical formula
MW
SG
n0 (#/A3)
0 (A3)
hexane
C6H14
86.18
0.660
0.00461
11.85
silica
SiO2
60.08
2.20
0.0220
5.25
sapphire
Al2O3
101.96
3.99
0.0236
7.88
water
H2O
18.01
1.00
0.0334
6.88
The adequacy of using only two terms in Eq 1 depends upon having particles for which the atomic
density is sufficiently small. The authors have previously shown that the 2-body and 3-body terms are the
first two terms in an expansion of DLP theory,17, and this has been verified by others14. The 2-body and
4
3-body 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, 4-body and
higher atom interactions can become important, along with higher order terms in the separation (e.g.,
R-10). It must be remembered, however, that separations smaller than the cluster size can still be large on
the atomic scale. But as the figures in this letter show below, the 2-body and 3-body VDW forces capture
the essential physics of many real material systems.
Table 2: Evaluated constants I6 and I9 for Eq 3. The Clausius-Mosotti
molecular polarizabilities from known dielectric data and expressions.24
System
I6 (A6/s)
System
silica-silica
silica-silica-silica
1.635  1017
silica-water
silica-silica-water
0.816
silica-hexane
silica-silica-hexane
6.036
silica-sapphire
silica-silica-sapphire
2.456
sapphire-sapphire 3.698
silica-water-water
sapphire-water
silica-water-hexane
1.228
sapphire-hexane
silica-water-sapphire
9.043
water-water
silica-hexane-hexane
0.409
water-hexane
silica-hexane-sapphire
3.005
hexane-hexane
silica-sapphire-sapphire
22.44
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
equation was used to estimate
I9 (A9/s)
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
Results and discussion
Figure 2 shows for silica spheres the 2-body, 3-body, and 2-body-plus-3-body VDW interactions.
These calculations cannot be 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 nanoparticles the distance required for the
approximation is less than the distance between atoms, invalidating the DLP model. We are not aware of
5
another accurate method for calculating VDW forces between nanoparticles.
While the separation
between particles is small compared with the particle radius, the separation is comparable to or larger than
the lattice spacing, making the ATM method valid. An important point is that at intermediate gaps, the 3body forces are as much as 21% of the 2-body forces. Thus, it is important to account for the 3-body
forces for quantitative purposes. Furthermore, this ratio (~0.2) suggests that the 4-body forces are likely
to be much smaller (i.e., O(0.212)). At very large gaps, we have shown previously (numerically and
analytically) that 3-body forces for spherical particles become insignificant compared with 2-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 4-body and higher forces. Our research group continues to study changes in polarizability for
nanocolloids compared with bulk or molecular values.
0.1
0.05
3-body VDW
0
-0.05
total VDW
-0.1
V
(eV) -0.15
2-body VDW
-0.2
-0.25
-0.3
-0.35
-0.4
1
gap (/a)
10
Figure 2. VDW forces between 2 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 2-body forces capture the qualitative trend of the VDW forces, the 3-body forces are
essential for quantitative results, since they constitute roughly 20% of the total VDW forces.
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
6
orientations, and also for spheres. For purposes of the comparison, the cube has N = 125 atoms, and the
spheres has N = 123 atoms (i.e., nearly the same). 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 closest approach, and 4)
averaged over all orientations of particles 1 and 2, such that
2   2 
    V  ,  ,
1
V 
0 0 0 0
1
2
,  2  sin 1 sin  2 d1 d1 d 2 d 2
4 2
(6)
For a visual of these orientations, see the inset to Figure 3. The orientations of the cubes was obtained by
identifying the coordinates of every silica atom in 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 center-to-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 spheres.
Figure 5 compares spherical silica particles with various shell layers. In order to simplify the
calculation, we made all atoms have the same size, and scaled the real atomic polarizability of the shell
calc
real
material to the polarizability used in the calculation. The relation is  shell
 nshell shell
/ ncore , where the n’s
are atomic densities. As expected, the core particles with the lower polarizability material in the shell
layer have smaller VDW attractions. Adding the 3-body contribution is important for these systems,
since it is up to about 30% of the 2-body value. Importantly, note that the adsorbed water layer gives
smaller VDW attractions than adsorbed hexane.
7
Figure 3. VDW potentials between two
cubic particles with the given center-tocenter 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-tocorner). 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 -777 eV, since for
large r/a the 3-body interactions should
approach zero.
0
-200
(r/a)6 Vaverage cube
-400
r/a
-600
-800
-1000
-1200
-1400
-1600
-1800
-2000
10
15
20
25
r/a
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.
1.2
corners
1.15
cubes_avg
edges
Vi / Vavg cube
1.1
faces
1.05
spheres
1
0.95
0.9
0.85
0.8
10
15
20
25
r/a
While these calculations were done in a vacuum, the calculations also have ramifications for particles
in a liquid environment. The results indicate that a co-solvent system – with a very small amount of a
second solvent dissolved in the primary solvent – can greatly improve nanocolloid stability in three ways.
1) The majority solvent can be chosen to minimize VDW attractions. For example, putting silica particles
8
in pure octane instead of pure water can reduce VDW attractions by a factor of five.26 2) If the minority
co-solvent 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 4 shows. 3) The adsorbed minority cosolvent can add a repulsive solvation force.27,28 All three effects lead to improved particle stability.
Currently, we are developing methods for measuring nanocolloid forces, in order to verify this effect.
2
Vi / Vsilica
1.5
sapphire
silica
1
hexane
water
0.5
0
1
/a
10
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. 3-body
forces are roughly 30% of the total energy. A layer of hexane causes the spheres to attract more than a
layer of water.
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 2-body systems). In addition, the ATM method is quite amenable to
using calculations of more exact polarizability data29,30 or calculations31 for nanoclusters. For denser
systems 4-body forces and others will need to be considered; however, the 3-body forces often give
reasonably quantitative results that can be used to design particle systems. It is important to note that
direct calculations using density functional theory32 or other quantum methods usually give energies with
insufficient accuracy to determine VDW forces; however, these methods can give polarizabilities with
sufficient accuracy to use with the ATM method.13,19
9
Conclusions
The ATM method provides a powerful method for using polarizabilites to calculate VDW forces.
Furthermore, the method overcomes some of the limitations of current Lifshitz theory. In sum, the
calculations in this letter indicate two heuristics that can be used to improve particle stability: a) use
spherical particles, b) use a co-solvent 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).
Acknowledgments
The authors thank the National Science Foundation (NER grant CTS-0403646) and the Ben Franklin
Center of Excellence for funding this work.
References
1
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3
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White, L.R. J. Colloid Interface Sci., 95, 286 (1983).
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Mahanty, J.; Ninham, B.W. Dispersion Forces. New York: Academic Press (1976).
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10
Hamaker, H.C. Physica A, 4, 1058 (1937).
11
Knight, W.D.; Clemenger, K.; de Heer, W.A.; Saunders, W.A. Phys. Rev. B, 31, 2539 (1985).
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Arup Banerjee and Manoj K. Harbola, J. Chem. Phys. 117, 7845 (2002).
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Wennerstrom, Hakan. Colloids Surf. A, 228, 189-195 (2003).
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Axilrod, B.M.; Teller, E. J. Chem Phys., 11, 299-300 (1943).
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Muto, Y. Proc. Phys.-Math Soc. Japan, 17, 629-631 (1943).
17
Gatica, Silvina M.; Calbi, Mercedes M.; Cole, Milton W.; Velegol, Darrell. Physical Review B, 68, 205409-1 –
205409-8 (2003).
18
Mine, Eiichi; Yamada, Akira; Kobayashi, Yoshio; Konno, Mikio; Liz-Marzan, Luis M. J. Colloid Interface Sci.,
264, 385 (2003).
19
Calbi, M.M.; Gatica, S.M.; Velegol, D.; Cole, M.W. Phys. Rev. A, 67, 033201-1 – 033201-5 (2003).
20
Myung, Noseung; Bae, Yoonjung; Bard, Allen J. Nano Lett., 3, 1053 (2003).
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London, F. Z. Phys., 63, 245 (1930).
22
Jackson, J.D. Classical Electrodynamics, 2nd ed. Wiley: New York (1975).
23
Russel, W.B.; Saville, D.A.; Schowalter, W.R. Colloidal Dipsersions. Cambridge: New York (1989).
24
Hunter, Robert J. Foundations of Colloid Science, Vol. 1. Oxford: New York (1986). Chapter 4 discusses van der
Waals forces between colloidal particles.
25
Aris, Rutherford. Vectors, Tensors and the Basic Equations of Fluid Mechanics. Dover: New York (1990).
26
Israelachvili, Jacob N. Intermolecular & Surface Forces (2nd edition). New York: Academic Press (1992).
27
Wang, Jee-Ching; Fichthorn, Kristen A. Colloids Surf. A, 26, 267 (2002).
28
Wang, Jee-Ching; Fichthorn, Kristen A. J. Chem. Phys., 112, 8252 (2000).
29
M. Manninen, R.M. Nieminen, and M.J. Puska, Phys. Rev. B, 33, 4289 (1986).
30
W.D. Knight, K. Clemenger, W.A. de Heer, and W.A. Saunders, Phys. Rev. B 31, 2539 (1985).
31
Mahan, G.D. J. Chem. Phys., 76, 493 (1982).
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Parr, Robert G.; Yang, Weitao. Density-Functional Theory of Atoms and Molecules. Oxford: New York (1989).
11
Figure captions
Figure 1. Core-shell nanoparticles interacting with a gap (/a) 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 system exists for cubic core-shell particles,
with similar variables.
Figure 2. VDW forces between 2 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 2-body forces capture the qualitative trend of the VDW forces, the 3-body forces are essential
for quantitative results, since they constitute roughly 20% of the total VDW forces.
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 -777 eV, since for large r/a the 3-body interactions
should approach zero.
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.
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. 3-body
12
forces are roughly 30% of the total energy. A layer of hexane causes the spheres to attract more than a
layer of water.
13
Tables
Table 1. The molecules used in this study. The molecular weight (MW) and specific gravity (SG) of the
materials are given.
substance chemical formula
MW
SG
n0 (#/A3)
0 (A3)
hexane
C6H14
86.18
0.660
0.00461
11.85
silica
SiO2
60.08
2.20
0.0220
5.25
sapphire
Al2O3
101.96
3.99
0.0236
7.88
water
H2O
18.01
1.00
0.0334
6.88
Table 2: Evaluated constants I6 and I9 for Eq 3. The Clausius-Mosotti
molecular polarizabilities from known dielectric data and expressions.24
System
I6 (A6/s)
System
silica-silica
silica-silica-silica
1.635  1017
silica-water
silica-silica-water
0.816
silica-hexane
silica-silica-hexane
6.036
silica-sapphire
silica-silica-sapphire
2.456
sapphire-sapphire 3.698
silica-water-water
sapphire-water
silica-water-hexane
1.228
sapphire-hexane
silica-water-sapphire
9.043
water-water
silica-hexane-hexane
0.409
water-hexane
silica-hexane-sapphire
3.005
hexane-hexane
silica-sapphire-sapphire
22.44
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
equation was used to estimate
I9 (A9/s)
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
14
2R
B
2
1
A

r
2R + 2w
Figure 1, Gatica et al (2004)
15
0.1
0.05
3-body VDW
0
-0.05
total VDW
-0.1
V
(eV) -0.15
2-body VDW
-0.2
-0.25
-0.3
-0.35
-0.4
1
gap (/a)
10
Figure 2, Gatica et al (2004)
16
0
-200
(r/a)6 Vaverage cube
-400
r/a
-600
-800
-1000
-1200
-1400
-1600
-1800
-2000
10
15
20
25
r/a
Figure 3, Gatica et al (2004)
17
1.2
corners
1.15
cubes_avg
edges
Vi / Vavg cube
1.1
faces
1.05
spheres
1
0.95
0.9
0.85
0.8
10
15
20
25
r/a
Figure 4, Gatica et al (2004)
18
2
Vi / Vsilica
1.5
sapphire
silica
1
hexane
water
0.5
0
1
/a
10
Figure 5, Gatica et al (2004)
19