Download Role of dipole–dipole interaction on enhancing Brownian

THE JOURNAL OF CHEMICAL PHYSICS 134, 084501 (2011)
Role of dipole–dipole interaction on enhancing Brownian coagulation
of charge-neutral nanoparticles in the free molecular regime
Yiyang Zhang,1 Shuiqing Li,1,a) Wen Yan,1 Qiang Yao,1 and Stephen D. Tse2
1
Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of
Thermal Engineering, Tsinghua University, Beijing 100084, China
2
Department of Mechanical and Aerospace Engineering, Rutgers University, Piscataway, New Jersey 08854,
USA
(Received 2 November 2010; accepted 27 January 2011; published online 22 February 2011)
In contrast to van der Waals (vdW) forces, Coulombic dipolar forces may play a significant role in
the coagulation of nanoparticles (NPs) but has received little or no attention. In this work, the effect of dipole–dipole interaction on the enhancement of the coagulation of two spherically shaped
charge-neutral TiO2 NPs, in the free molecular regime, is studied using classical molecular dynamics (MD) simulation. The enhancement factor is evaluated by determining the critical capture radius
of two approaching NPs for different cases of initial dipole direction with respect to path (parallel/perpendicular) and orientation with respect to each other (co-orientated/counterorientated). As
particle diameter decreases, the enhancement of coagulation is augmented as the ratio of dipole–
dipole force to vdW force becomes larger. For 2-nm TiO2 NPs at 273 K, the MD simulation predicts
an average enhancement factor of about 8.59, which is much greater than the value of 3.78 when
only the vdW force is considered. Nevertheless, as temperature increases, the enhancement factor
due to dipole–dipole interaction drops quickly because the time-averaged dipole moment becomes
small due to increased thermal fluctuations (in both magnitude and direction) of the instantaneous
dipole moment. © 2011 American Institute of Physics. [doi:10.1063/1.3555633]
I. INTRODUCTION
Coagulation, a process where small particles collide with
one another to form larger coalesced or aggregated particles,
is of great fundamental and practical interest due to its inherent participation in various processes such as nanoparticle synthesis,1, 2 nanodusty plasma,3, 4 dust sedimentation,5, 6
colloidal dispersion,7 and even planet formation.8, 9 For example, in gas-phase synthesis of nanoparticles (NPs), such
as flame synthesis,10–12 the size of primary particles and the
morphology of aggregates are greatly influenced by the coagulation process. If collisions between particles occur faster
than their coalescence (sintering), the final products are usually aggregates with fractal-like structure; meanwhile, if coalescence occurs faster than collisions, then the products are
larger, spherical particles.13, 14 Another example is the sediment of aerosol particles, in which the rate of coagulation determines the residence time of dusts in the atmosphere along
with their aggregate structure prior to settling.15
There are several mechanisms for particle coagulation,
including Brownian motion, laminar shear, and turbulence.15
Brownian motion is dominating in most cases, especially, for
fine particles. The population balance model, introduced by
Smoluchowski,16 is well known for describing the evolution
of particles in coagulation, which can be given as,15
∞
1 dn k
=
β(v i , v j )n i n j −n k
β(v i , v k )n i ,
dt
2 i+ j=k
i=1
a) Electronic mail: [email protected].
0021-9606/2011/134(8)/084501/8/$30.00
where ni , nj , and nk are the number concentrations of particles
within volume vi , vj , and vk , respectively. β is the coagulation rate describing the frequency that particles collide with
each other. For particles in the continuum regime with small
Knudsen number (i.e., Kn < 0.1), it is given as,17
1
2kT
1
β(v i , v j ) =
+ 1/3 ,
(2)
3μ v i1/3
vj
where k is the Boltzmann constant, T is the temperature, and
μ is the viscosity of the fluid. For the free molecular regime,
the classical theory treats particles as noninteracting rigid
spheres. By applying kinetic theory for gas molecules, the coagulation rate of particles can be derived as
1/6 6kT 1/2 1 1 1/2 1/3 1/3 2
3
v i +v j
+
,
β(v i , v j ) =
4π
ρP
vi v j
(3)
where ρ p is the particle density.
Further investigation finds that attractive forces, such as
the van der Waals (vdW) force, enhance the coagulation rate
by increasing the capture radius of individual particles18–22
and aggregates,23 particularly, in the free molecular regime.
To quantitatively describe the enhancement, an enhancement
factor can be defined as,20
F = W ψ,
(1)
(4)
where F is the total flux, ψ is the flux without particle interactions, and W is the enhancement factor due to particle
interactions. Marlow18, 19 gave a rigorous derivation for the
enhancement factor by vdW force in the free molecular and
134, 084501-1
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084501-2
Zhang et al.
transition regimes. Okuyama et al.24 measured the coagulation rate of NaCl and ZnCl particles with a smallest diameter of about 8 nm, with enhancement factors found to
be between 1 and 4. The experiments are consistent with
Marlow’s derivation if the proper Hamaker constants are
used.
The effects of Coulomb interaction can be more complex than that simply from vdW. Coagulation of charged microparticles is important for understanding the nature of the
aggregation process.25 However, for particles with sizes under 20 nm, the charging efficiency by the diffusion mode
is known to be quite low, differing from that for micronsized particles.26 More importantly, permanent dipoles have
recently been found to exist even in charge-neutral crystalline NPs.27–33 Some, such as CdSe,27 arise from the polar
lattice in the bulk, while others, such as TiO2 (Ref. 28) and
ZnSe NPs,29 come from the ions distribution at the particle
surface. Experiments have indicated that such dipoles can significantly affect the behavior of NPs, e.g., resulting in orientated coagulation.29 In our previous work,28 the effect of
dipole moment on the interaction forces of two TiO2 NPs was
studied using molecular dynamics (MD) simulation, which
found that, at certain separations, the Coulombic force between dipoles can be much greater than the vdW force by
several orders of magnitude. We also showed that the associated Coulomb force between NPs during precontact approach can be attributed mainly to the dipole–dipole force,
except for in the close-contact zone of <0.2 nm.28 As such,
in this work, we will refer to the Coulomb forces associated
with such dipole interactions approximately as dipole–dipole
forces.
This dipole–dipole interaction is sensitive to temperature and size of NPs. In reality, for NPs containing permanent dipoles, both dipole–dipole and vdW forces coexist during their coagulation. To date, reports on the enhancement of
particle coagulation by the dipole–dipole force are not found
in the literature. Moreover, Marlow’s model derived for vdW
force is not suitable for those cases of dipole–dipole interactions, in which nanoparticles may experience more complex
motion such as mutual rotation (as described in our previous
work28 ) during their approach.
In this work, classical MD is applied to simulate the coagulation of two equisized anatase TiO2 NPs with diameters
less than 5 nm. The enhancement factor when both dipole–
dipole and vdW forces coexist in the free molecular regime is
quantitatively determined, and then compared with a scenario,
where only the vdW force is present. Finally, the effects of
both temperature and size of NPs on the enhancement factor
for NP capture are studied.
II. METHODOLOGY
A. Simulation method of MD
Classical MD based on the DL _ POLY code34, 35 (version
2.18) is employed to simulate the coagulation of two identical
TiO2 NPs. The Matsui–Akaogi (MA) potential,36 which has
been validated by x-ray absorption spectra data to be one of
the most suitable potentials for TiO2 structures,37, 38 is used to
J. Chem. Phys. 134, 084501 (2011)
TABLE I. The interaction parameters for Matsui–Akaogi potential.
Ti
O
q
A
B
C
+2.196
−1.098
1.1823
1.6339
0.077
0.117
22.5
54.0
describe the interaction between atoms. The MA potential is
given as
Ui j =
qi q j
Ci C j
− 6 + f (Bi + B j ) exp
ri j
ri j
Ai + A j − ri j
Bi + B j
,
(5)
where the three terms on the right-hand side of Eq. (5) represent electrostatic (Coulomb), dispersion (vdW), and repulsion
interactions, respectively. Note that the last two terms constitute a Buckingham potential. The retardation effect of the van
der Waals force is not considered in the potential. The values
of partial charges q and the parameters A, B, and C of atom i
(or j) in Eq. (5) are listed in Table I.
The MD simulation procedure for studying interaction
dynamics of TiO2 NPs is described in detail in our previous work.28 Here, we only briefly present it. First, a perfect
anatase TiO2 lattice is constructed; and then a spherically
shaped NP, termed as NP1, is “excised” from the lattice, while
removing excess titanium or oxygen ions to maintain charge
neutrality. Second, the neutral NP1 is simulated in the canonical ensemble for 3 ns (600 000 5-fs time steps) to achieve
an equilibrium state.28 Third, another NP, termed as NP2, is
replicated and translated to a new spatial position with specific initial conditions that will be discussed in Sec. II B. NP1
is set at rest, and NP2 is given an initial velocity based on
Brownian translation for the NP. The approach dynamics of
NP2 to NP1 is simulated using the microcanonical ensemble
with a time step of 5 fs.
B. Initial conditions of coagulation process
The objective of this work is to characterize the enhancement of the coagulation rate when dipole–dipole and
vdW forces are both present. In the free molecular regime,
the enhancement factor is associated with the capture radius
of two NPs. Figure 1(a) shows the initial conditions for the
coagulation of two TiO2 NPs, where red and black arrows
indicate dipole moment and initial velocity, respectively. Perpendicular to the initial velocity, different impact parameters,
i.e., offset distances of the two NPs (Lo ) as shown in Fig. 1,
are continuously reduced in the MD simulations to determine the critical value at which collision occurs. This critical value is considered to be the capture radius of coagulation and is obtained from the simulation with an accuracy of
0.1 nm.
For the initial conditions, besides Lo , other key parameters include initial velocity of NP2, initial Ld between NP1
and NP2, and initial directions of the dipole moments for
NP1 and NP2. In Brownian coagulation, Brownian motion
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084501-3
Dipolar interaction on enhancing coagulation
J. Chem. Phys. 134, 084501 (2011)
produces particles with relative velocities. The physical assumption, as proposed by Einstein,39 applies the principle of
equipartition of energy to the translational energy of the particles, where for each dimension
1 2
1
mu = kT.
2
2
(6)
In real systems, the measured velocities of Brownian particles
can vary several orders of magnitude.42 Thus, we utilize the
one-dimensional velocity corresponding to Eq. (6) as a characteristic translational velocity for Brownian
√ motion. The initial velocity of NP2 is determined to be 2 times the characteristic velocity, being the relative velocity between NP1 and
NP2.
For the initial distance (Ld ) between NP1 and NP2, the
particle path length (or persistence length) can be used as
an approximate estimate, which assumes that the motion of
particles depends on the integrated effect of collisions with
molecules,15
l pa =
(mkT )1/2
,
f
(7)
where m is the particle mass, and f is the friction coefficient.
The values for particle path length of anatase TiO2 NPs used
in this simulation vary from 75.4 to 462.1 nm, which are much
larger than the particle diameters themselves. Repeated simulation indicates that the change of initial distance Ld has little
effect on the coagulation capture radius when the distance is
larger than ten times that of the particle diameter because the
interaction forces involved are very small at large distances.
As such, an initial distance of 70 nm is used for all cases in
this work.
Initial dipole-moment orientations may influence the
trajectories of NPs and the capture radius. To quantitatively
evaluate this enhancement factor, four characteristic dipolemoment directions/orientations are examined, as illustrated in
Fig. 1. With respect to the particle path, the dipole moments
of NP1 and NP2 can be either parallel or perpendicular to the
initial approach velocity. With respect to each other, the dipole
moments of NP1 and NP2 can be either co-oriented or counteroriented. In the simulations, thermal rotation of NPs is not
included in the initial conditions. The effects of both particle
FIG. 1. Four initial dipole directions/orientations investigated in the
simulation.
TABLE II. The parameters for MD simulations on particle coagulation
process.
Case
number
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Diameter
(nm)
Temperature
(K)
2
2
2
2
2
2
2
2
2
2
1.5
1.5
2.5
2.5
3
3
3.5
3.5
273
273
273
273
573
573
973
973
1673
1673
273
273
273
273
273
273
273
273
Abbreviation of
dipole directions
Parallel-co
Parallel-counter
Perpendicular-co
Perpendicular-counter
Parallel-co
Parallel-counter
Parallel-co
Parallel-counter
Parallel-co
Parallel-counter
Parallel-co
Parallel-counter
Parallel-co
Parallel-counter
Parallel-co
Parallel-counter
Parallel-co
Parallel-counter
temperature and diameter are also considered. Temperatures
investigated are 273, 573, 973, and 1673 K. Particle diameters
examined are 2, 2.5, 3, and 3.5 nm, which all correspond to
Kn > 30, belonging to the free molecular regime. The main
parameters of all simulated cases are presented in Table II.
C. Comparison of dipole–dipole force versus
vdW force
In the dynamic interaction of NPs, all forces act together. To distinguish quantitatively the enhancement due to
the dipole–dipole force, we evaluate the enhancement factor when only the vdW force exists and then compare with
the full result from the MD simulation. Although Marlow’s
work18, 19 gives a good estimate for vdW force, the situation
treated in our MD simulation is a little different from Marlow’s derivation. Marlow considered particle velocity as a distribution, while we take the mean value in the simulation. For
a better comparison with our simulations, a two-body model
is utilized to evaluate the vdW force between particles. In this
model, two particles are considered to be rigid spheres; and
the Verlet algorithm, as employed in DL _ POLY, is used to calculate their trajectories. The vdW energy is described by the
classical theory from Hamaker, i.e.,40
4R 2
A
4R 2
vdW
W12
=−
+
2
2
12 D12
+ 4D12 R
D12
+ 4D12 R + 4R 2
2
D12
+ 4D12 R
+ 2 ln
,
(8)
2
D12
+ 4D12 R + 4R 2
where A is the Hamaker constant, R is the radius of a NP, and
D12 = (X 12 − 2R) is the separation distance of the closest
approach between NP1 and NP2, where X 12 is their centerto-center distance (mass basis).
The initial conditions utilized in the two-body model are
the same as those in the MD simulation. The capture radius
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084501-4
Zhang et al.
obtained by this two-body model is compared with the capture radius obtained in the MD simulation using the MA potential (which includes all relevant forces), to better isolate the
enhancement brought by the dipole–dipole force itself. The
results from Marlow’s work are also examined as a reference.
III. RESULTS AND DISCUSSIONS
A. Forces between two TiO2 NPs during the approach
process
The interaction force between particles plays an important role in the capture process of NPs. First, we investigate
the interaction forces between two 2-nm anatase TiO2 NPs at
273 K, with the impact parameter, as shown in Fig. 1(a), set
to zero (i.e., a head-on collision), for two situations of initial
dipole orientations, i.e., co-orientated and counterorientated,
where the initial dipole directions are parallel to the initial
particle velocity. Three forces are involved in the MA potential, i.e., Coulomb force, vdW force, and repulsive force.
Coulomb and vdW forces dominate precontact approach,
with repulsive force being negligible. The VdW force, which
originates from interactions of induced dipoles, is attractive
and increases smoothly as the two NPs approach each other.
As shown in Fig. 2, for both co-orientated and counterorientated cases, the vdW forces exhibit little difference. The
Coulombic forces here, described approximately as the
interactions of permanent dipoles, can be referred to as
dipole–dipole forces as previously mentioned, and are larger
than vdW forces at relatively large distances and exhibit
larger fluctuations. When the distance is ∼10 nm, the dipole–
dipole force is larger than the vdW force by several orders
of magnitude. Such large values of dipole–dipole force make
it reasonable to neglect possible retardation damping of the
vdW force, which is important at relatively long separation
distances as given in the literature9 versus the short distances
(i.e., 10 nm), where interaction effects are important as examined in this work. As the separation distance decreases, the
vdW force catches up to the dipole–dipole force, and these
two forces are of the same order of magnitude upon contact.
The counterorientated situation exhibits a smaller dipole–
FIG. 2. The interaction forces between two 2-nm anatase TiO2 NPs during
the head-on approach for co-orientated and counterorientated dipoles, with
both dipoles parallel to the initial velocity.
J. Chem. Phys. 134, 084501 (2011)
dipole force at the beginning. However, the difference is
eliminated as the approach process of the two NPs continues,
due to the rotation of particles. As we found in Ref. 28, for
two head-on approaching counteroriented nanoparticles, with
initially repulsive interaction, the nanoparticles experience
mutual rotations and the dipole–dipole force changes from
repulsive to attractive during translation.
B. Capture radius and enhancement factor
Since the dipole–dipole force is attractive most of the
time, it will enlarge the capture radius of NPs, just such as
the vdW force does. Different impact parameters [Lo , see
Fig. 1(a)] are tested to obtain the critical capture radius. Two
situations with slightly different impact parameters are shown
in Fig. 3 using a visual molecular dynamics program.41 The
NP diameter is about 2 nm, and the temperature is 273 K. The
two NPs have parallel-to-path, counterorientated dipole moments at the beginning. In the initial stage, NP1 is stationary;
and NP2 moves in a straight line with a nearly constant velocity, where both dipole–dipole and vdW forces are too small at
the large distance to affect particle velocity. When the distance
between particles falls below about 10 nm, both NP1 and NP2
start to rotate, satisfying the principle of minimum Coulomb
interaction energy. Upon closer approach, the trajectory of
NP2 bends toward NP1 due to the attractive force (combination of dipole–dipole and vdW forces) between them. The
final result is either capture or escape, depending on Lo . When
Lo is 5.45 nm, collision occurs, as shown in Fig. 3(a). When
Lo is 5.55 nm, miss occurs, as shown in Fig. 3(b). Therefore,
the capture radius is determined to be 5.5 nm with an accuracy of 0.1 nm. Since the capture radius is equal to the sum of
the radii of two particles if no interparticle forces are present,
for example, in classical particle dynamics, the enhancement
factor can be expressed as
2
rf
,
(9)
W =
ro
where ro is the capture radius when no interparticle interactions are present, and rf is the capture radius with interparticle
FIG. 3. The trajectories (see Ref. 41) of 2-nm TiO2 NPs in a case of parallelto-path, counterorientated dipoles at 273 K. (a) Capture (collision) occurs
when L 0 = 5.45 nm; (b) escape (miss) occurs when L 0 = 5.55 nm. The capture radius in this situation is determined to be 5.5 nm.
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084501-5
Dipolar interaction on enhancing coagulation
interactions. In this case (2-nm particle, 273 K, counterorientated dipoles), the enhancement factor is 7.56 when both
dipole–dipole force and vdW force are acting together.
To isolate the effect of the dipole–dipole interaction
force, the two-body model (Hamaker) described before is
used to obtain the enhancement factor when only the vdW
force is present. The capture radius is found to be 3.60 nm,
⎛
J. Chem. Phys. 134, 084501 (2011)
corresponding to the enhancement factor of 3.24. This value
is much less than the enhancement factor when both dipole–
dipole force and vdW force are in operation, indicating that
the dipole–dipole force greatly increases the coagulation rate
of TiO2 nanoparticles in the free molecular regime. Marlow’s
formula also gives a solution for the enhancement factor produced by vdW force alone,18
⎞
d E (σ )
1
σ d E (σ )
d 2 E (σ )
(σ
)
exp
−
+
σ
+
E
+
⎜
⎟
dσ
dσ 2
kT
2 dσ
1
⎜
⎟
r12
W = − 2
⎜
⎟,
r12 +δ
2
⎠
2r12 kT ⎝ (σ
)
(σ
)
(σ
)
1
d
σ
d
E
E
d
E
2
exp −
+σ
+ E (σ )
dσ
r12 + δ
2
dσ
dσ
kT
2 dσ
r12
r12 +δ
dσ σ 2
where E is the attractive potential, r12 is sum of the radii of
both particles, and δ is the jump distance, defined as
2
2
+ lpa2
,
(11)
δ = lpa1
where lpa1 and lpa2 are the particle free paths. Using this relation, the enhancement factor is determined to be 3.78, which
is close to the prediction given by the approximate Hamaker
approach above.
Expectedly, the initial dipole orientation affects the
dipole–dipole force between two particles. Figure 4 shows the
capture enhancement factors due to both dipole–dipole and
vdW forces under the four characteristic initial dipole configurations shown in Fig. 1. The case of parallel-to-path, cooriented dipoles has the largest enhancement factor of 9.61,
while the case of parallel-to-path, counterorientated dipoles
has the smallest factor of 7.56. The two perpendicular-topath cases, for both co-oriented and counterorientated, are
in-between these two extrema, in which the enhancement
(10)
factors are 9.00 and 7.84, respectively. For dipoles initially orientated at arbitrary angles, the enhancement factors are also expected to lie between the two extrema. This
phenomenon is consistent with the dipole–dipole interaction force predicted by the classical theory,7 i.e., the value
of dipole–dipole force is largest when the dipoles are coorientated and smallest when counterorientated. Moreover,
the enhancement factors do not vary much amongst the four
different initial dipole directions, ranging from 20% to 30%.
The enhancement for the cases where only vdW force exists
is evaluated by both the two-body model (Hamaker) and Marlow’s formula.18 , 19 . The predicted enhancement factor for the
cases is about 3.24 by two-body model (Hamaker) and 3.78 by
Marlow’s formula. These factors are much lower than those
in cases where both dipole–dipole and vdW forces coexist,
irrespective of the dipole configurations given in Fig. 1. The
rotation of particles, arising from the dipole–dipole interaction, quickly changes the dipole–dipole force to become attractive during the approach process, which causes the enhancement factor to lie in a relatively narrow range, i.e.,
7.56–9.61.
At the initial distance (i.e., 70 nm), the magnitude of the
dipole–dipole interaction energy of NPs is about 10−6 eV,
which is much less than the kinetic energy of NPs at about
10−2 eV, explaining why the enhancement-factor differences
are small for the four different initial dipole–dipole configurations. Therefore, we take the arithmetic mean of the factors
in both co-orientated and counterorientated cases (parallelto-path) to represent the overall enhancement factor where
dipole–dipole and vdW forces coexist, when we investigate
the effects of system temperature and particle diameter.
C. Influence of system temperature on capture
enhancement factor
FIG. 4. Enhancement factors for different initial dipole directions with respect to path (parallel/perpendicular) and orientation with respect to each
other (co-orientated/counterorientated).
Figure 5 plots enhancement factor as a function of system temperature, for 2-nm anatase TiO2 NPs. Subplots A
and B depict the interaction forces between the particles, in
which red open dots represent vdW force and black filled dots
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084501-6
Zhang et al.
FIG. 5. Influence of system temperature on enhancement factors of coagulation. Subplots A and B show the forces between anatase TiO2 NPs at 273 and
1273 K, respectively. Red open dots are for the vdW force, and black filled
dots for the total interaction force. The forces have been projected along the
line that connects the centers of the two nanoparticles.
represent the total force that is a sum of all three forces accounted for in the MA potential. The forces have been projected along the line connecting the centers of the two NPs.
Considering only the vdW force, both the two-body
model (Hamaker) and Marlow’s formula indicate that the capture enhancement factor decreases slightly as the temperature
increases from 273 to 1273 K, with the vdW force between
particles varying little with temperature. Meanwhile, the average Brownian velocity of particles increases, and the capture radius becomes smaller. When the dipole–dipole force
acts together with the vdW force, the enhancement factor at
273 K is as large as 8.56, which corresponds to an averagedvalue of both co-orientated and counterorientated cases for
the parallel-to-path dipole configuration. As system temperature increases, the enhancement factor decreases dramatically.
At 1273 K, the enhancement factor in the presence of both
dipole–dipole and vdW forces is nearly the same as that of
the case with only vdW force. We, therefore, conclude that the
influence of dipole–dipole interaction on the coagulation rate
is significant at low temperatures, particularly, below 573 K,
but minor at high temperatures, becoming almost negligible
above 973 K. Such strong temperature dependence originates
from the nature of dipoles in crystalline TiO2 NPs, which are
sensitive to the positions of Ti and O ions at the particle surfaces. When temperature is low, lattice vibrations are small;
and fluctuations of dipole (in magnitude and direction) are
also small, leading to a steady attraction between particles, as
shown in subplot A of Fig. 5. As temperature increases, lattice vibrations become larger, resulting in larger fluctuations
of dipole moment. The dipole–dipole force fluctuates rapidly
between positive and negative and is spread widely for small
distances, resulting in vacillating attraction and repulsion, as
shown in subplot B of Fig. 5. With the time-averaged dipole
moment diminishing with increasing temperature, so does the
integrated dipole–dipole force; and the coagulation rate is reduced.
D. Influence of particle diameter on capture
enhancement factor
Figure 6 plots the enhancement factor as a function of
particle diameter, for anatase TiO2 NPs. When only vdW
J. Chem. Phys. 134, 084501 (2011)
FIG. 6. Influence of particle diameter on the enhancement factors of coagulation. The black filled dots are obtained by MD simulation considering both
dipole–dipole force and vdW force. The bule star dots are obtained by the
two-body model. The red open dots are obtained by Marlow’s formula with
only vdW force.
force is present, the enhancement factor is constant for the
particle sizes examined in the free molecular regime. Nevertheless, for larger particle sizes corresponding to the transition regime, the literature reports a drop in the enhancement
factor.21
When both dipole–dipole and vdW forces coexist, the enhancement factor for coagulation increases from 8.59 to 9.28
for particle diameters from 2 to 2.5 nm, and then decreases
for larger particle diameters, e.g., down to 5.68 at 3.5 nm,
implying weakening dipole–dipole interactions for larger diameters.
The changes in the dipole–dipole and vdW forces in relation to particle diameter can explain the nonmonotonic trend
shown in Fig. 6. Assuming that two dipoles are aligned in line
and in the same direction, the attractive force between them
can be expressed as
Fdipole =
3u 1 u 2
,
4
2π ε0 ε X 12
(12)
where u1 and u2 are the moments of the dipoles, and X 12 is the
center-to-center distance of the dipoles. While bulk anatase
TiO2 has no dipole despite its noncentrosymmetric lattice,
a dipole moment can exist for nanoparticles, as previously
mentioned, due to the asymmetric distribution of Ti and O
ions at the surfaces. As a result, the mean dipole moment of
NPs is approximately proportional to the surface area of the
particles.28 For fixed directions, the dipole–dipole force is approximately proportional to the fourth power of particle diameter.
The vdW force between two equal-sized spherical particles can be expressed as40
FvdW =
1
A
.
2R x 2 (x + 1)3 (x + 2)2
(13)
As aforementioned, A is the Hamaker constant, R is particle
radius, D12 is the separation distance between the surfaces
of the particles, and x ≡ D12 /2R. Figure 7 plots the ratio
of dipole–dipole force to vdW force, which is calculated using Eqs. (12) and (13), respectively. To meet the operational
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084501-7
Dipolar interaction on enhancing coagulation
FIG. 7. The ratio of dipole–dipole force to vdW force for TiO2 NPs with different diameters. The forces are calculated using Eqs. (12) and (13). The
dipole moment is assumed to be proportional to the surface area of the
nanoparticle (see Ref. 28). The distances between the surfaces of the nanoparticles are examined at 0.5, 1, 2, and 4 nm, respectively.
range of vdW interaction, the separation distances between
the surfaces of the particles are fixed at 0.5, 1, 2, and 4 nm,
respectively. For all separation distances, the ratio of dipole–
dipole force to vdW force decreases as the particle diameter
increases from 1 to 5 nm, indicating that the dipole–dipole
interaction, in contrast to vdW, becomes more important for
smaller particles and less important for larger particles, as
shown in Fig. 7. In addition, during the approach, the rotation
of smaller particles driven by the existing dipoles becomes
easier, since the moment of inertia is proportional to the fifth
power of particle diameter, resulting in increased particle coagulation.
IV. CONCLUSION
The effect of dipole–dipole interaction on the enhancement of coagulation of crystalline TiO2 NPs in the free
molecular regime is quantitatively characterized by using MD
simulation. The results indicate that the Coulombic dipolar
(approximately dipole–dipole) force, originating from permanent dipoles in the nanoparticles, can be the dominating interaction force between two approaching TiO2 NPs. The enhancement factor is determined directly from the critical capture radius of two coagulated particles. For 2-nm TiO2 NPs at
273 K, the classical Marlow’s estimation gives the enhancement factor of 3.78 by only considering vdW forces; while the
MD simulation gives a value of 8.59 due to the additional role
of dipole–dipole forces. This implies that, in the free molecular regime, the coagulation rate of crystalline NPs may be
greatly underestimated without considering the dipole–dipole
interaction between NPs.
Initial dipole directions of twin NPs in the field have significant effects on the enhancement factors for particle coagulation. The parallel-to-path, co-orientated dipoles exhibit the
most enhancement, while parallel-to-path, counterorientated
dipoles display the least. Comparatively, the perpendicularto-path NPs with dipoles, either co-orientated or counterorientated, straddle the two extrema, respectively.
J. Chem. Phys. 134, 084501 (2011)
In contrast to previous work on coagulation enhancement
by the vdW force alone, the enhancement originating from
the dipole–dipole force is strongly dependent on the temperature. As the temperature increases from 273 to 1273 K, the enhancement factor drops rapidly from 8.59 to 2.64. It can be explained that the strong fluctuations of dipole moment of NPs
at high temperature greatly reduce the time-averaged dipole
moment and thus the interparticle dipole–dipole forces. The
effect of particle diameter of NPs on the enhancement factor
is also apparent. Smaller particles (e.g., 2 to 2.5 nm) possess a
much greater enhancement factor compared to the larger particles (e.g., 3.5 nm), which can be attributed to the increased
ratio of dipole–dipole force to vdW force. As such, this effect
can be especially important in Brownian coagulation of primary particles during gas-phase synthesis at moderate to low
temperatures (e.g., <600 K for anatase titania).
ACKNOWLEDGMENTS
This work was supported mainly by the National Natural Science Fund of China (Grant No. 50776054) and the
National Program for New Century Excellent Talents in University (2009), and partially by the US Office of Naval Research (Grant No. N00014-08-1-1029) for coauthor, S.D.T.
We acknowledge Dr. Koparde at Vanderbilt for kindly providing his Ph.D. thesis. Special thanks are due to Professor
Pratim Biswas at WUSTL, Professor Gang Chen at MIT, and
Professor Jeff Marshall at UVM, for stimulating and helpful
discussions.
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