Lecturas, Jul 2011

8/3/2019 Lecturas, Jul 2011

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Designing van der Waals Forcesbetween Nanocolloids

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 coreshell 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 isto develop heuristics for designing nanocolloids to

have the desired dispersion and assembly (e.g., quan-

tum dots20 and fluorescent particles) by examining all four

points.

Method for Evaluating VDW Forces. A general formal-

ism for calculating VDW forces is to consider atom-atom

interactions (two-body interactions), then three-body interac-

tions, four-body interactions, etc. This is written3,4

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 four-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 two-

body interactions and the ATM result for three-body

interactions (Vijk) may be written

where the angles i are for the triangle formed by the threeatoms. The rterms are the center-to-center distances between

the atoms.

* Corresponding author. E-mail [email protected]. Department of Physics. Department of Chemical Engineering. Materials Research Institute.

V0 )i

Vi(2)+ (

i

>

Vi(3)+

i


2/5

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 (); a Kramers-Kronig relation thentransforms this function to the real function (i). Thecomplexity 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 (i) using the Clausius-

Mosotti rela-tionship22

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 (i) is23

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 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 three-

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, 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 sizecan 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 two-

body forces. Thus, it is important to account for the three-

body forces for quantitative purposes. Furthermore, this ratio

(0.2) suggests that the four-body forces are likely to be

Figure 1. Core-shell nanoparticles interacting with a gap () andcenter-to-center separation (r). Both particle 1 and particle 2 arecomposed of a core material A (core has radius R) and a shellmaterial B (of thickness w). The intervening material is vacuum.A similar geometry exists for cubic core-shell particles.

Table 1. Molecules Used in This Studya

substance chemical formula MW SG n0 (#/A3) R0 (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

a The molecular weight (MW) and specific gravity (SG) of the materialsare given.

Figure 2. VDW forces between two silica spheres with n ) 619atoms in a vacuum. The lattice constant a ) 3.569 Angstroms andthe distance of closest approach occurs for a center-to-centerdistance of 10.192a. While the two-body forces capture thequalitative trend of the VDW forces, the three-body forces areessential for quantitative results, since they constitute roughly 20%of the total VDW forces.

(i) - 0

(i) + 20)

4

3n0R(i) (4)

(i) ) 1 +j

dj

1 + j+

j

fj

1 + gj

j+ (

j)

2(5)

170 Nano Lett., Vol. 5, No. 1, 2005


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much smaller (i.e., 0.22). At very large gaps, we have

shown previously (numerically and analytically) that three-

body 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

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 cal-

culation 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 25kTfor 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

Table 2. Evaluated Constants I6 and I9 for Equation 3a

system I6 (A6 /s) system I9 (A9/s)

silica-silica 1.635 1017 silica-silica-silica 3.747 1017

silica-water 0.816 silica-silica-water 1.863

silica-hexane 6.036 silica-silica-hexane 13.87

silica-sapphire 2.456 silica-silica-sapphire 5.564

sapphire-sapphire 3.698 silica-water-water 0.931

sapphire-water 1.228 silica-water-hexane 6.870

sapphire-hexane 9.043 silica-water-sapphire 2.769

water-water 0.409 silica-hexane-hexane 51.85

water-hexane 3.005 silica-hexane-sapphire 20.54

hexane-hexane 22.44 silica-sapphire-sapphire 8.275

water-water-water 0.469

water-water-hexane 3.415

water-water-sapphire 1.383

water-hexane-hexane 25.62

water-hexane-sapphire 10.18

water-sapphire-sapphire 4.12

hexane-hexane-hexane 195.3

hexane-hexane-sapphire 76.63

hexane-sapphire-sapphire 30.46

sapphire-sapphire-sapphire 12.32

a 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 thegiven center-to-center separation (r/a). The cubes have 125 silicaatoms. The inset shows two of the many possible orientations (topinset shows face-to-face; bottom inset shows corner-to-corner). Inthe limit of large r/a, eqs 1-3 and I6 from Table 2 can be used tofind that the asymptotic ordinate value is -777 eV, since for larger/a the three-body interactions should approach zero.

Figure 4. Interaction energies between particles having variousrelative orientations. All cases are normalized by the energy of twocubes averaged over all orientations. The cubic particles have 125silica atoms, while the spheres have N) 123.

Nano Lett., Vol. 5, No. 1, 2005 171


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closest approach, and (4) averaged over all orientations of

particles 1 and 2, such that

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 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 theparticle 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 Rshellcalc

)

nshellRshellreal

/ncore, where the n is atomic density. As expected,

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

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 attractionsby 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 measure-

ments 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 givepolarizabilitieswith 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 Langbeins 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, con-

tinuum core-shell particles).8,40

Marlow and co-workers have carried out extensive cal-

culations of cluster VDW interactions, including for dis-

crete systems and complex shapes.41,42 They have compared

results of alternative methods: (a) a continuum approach

Figure 5. VDW forces with various shells on a silica core, relativeto 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 ofhexane causes the spheres to attract more than a layer of water.

V )

02

0

02

0

V(1, 1, 2, 2)sin 1 sin 2 d1 d1 d2 d2

(4)2

(6)

172 Nano Lett., Vol. 5, No. 1, 2005


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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 computa-

tionally more expensive than our approach. Marlow and co-

workers have also evaluated a fourth model, a sum over

molecules of particle A of the individual many-body interac-

tions 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 contribu-

tions 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 repul-

sion).

Acknowledgment. The authors thank the National Sci-

ence 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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