Dipole–Dipole Interaction Model for Oriented Attachment of BaTiO3

ARTICLE pubs.acs.org/JPCC

Dipole
Dipole Interaction Model for Oriented Attachment of BaTiO3 Nanocrystals: A Route to Mesocrystal Formation Kyuichi Yasui* and Kazumi Kato National Institute of Advanced Industrial Science and Technology (AIST), 2266-98 Anagahora, Shimoshidami, Moriyama-ku, Nagoya 463-8560, Japan ABSTRACT: Although mechanisms of mesocrystal formation have been studied by several research groups, they are still under debate; Why do nanocrystals aggregate with their crystal axes aligned? In the present Article, the electric dipole
dipole interaction model has been studied for collisions between two particles by numerical simulations to study the mechanism of oriented attachment of BaTiO3 nanocrystals under the experimental condition of their ultrasound-assisted synthesis. If 5 nm BaTiO3 nanocrystals have some spontaneous polarization, then only small particles aggregate with other particles and their crystal axes are aligned by the dipole
dipole interaction, which is consistent with the experimental results. It suggests that 5 nm BaTiO3 nanocrystals may have some spontaneous polarization, although it has been reported that below 10 nm the crystal structure of BaTiO3 is cubic.

1. INTRODUCTION Mesocrystals are a new class of crystalline materials that have been intensively studied since 2005.1
5 The electron diffraction pattern of a mesocrystal is similar to that of a single crystal, whereas a mesocrystal is an aggregate of hundreds or thousands of nanocrystals. In a mesocrystal, many nanocrystals attach each other between specific crystal faces with the crystal axes aligned. Several mechanisms of mesocrystal formation have been proposed: biomineralization, roles of organic additives, alignment by capillary forces, hydrophobic forces, a mechanical stress field, magnetic fields, dipole and polarization forces, external electric fields, minimization of the interfacial energy, and so on.1
4,6
8 The mechanisms are, however, still under debate. Recently, Dang et al.9
11 reported the ultrasound-assisted synthesis of BaTiO3 nanoparticles in aqueous solutions. The reaction formula in their experiment is as follows

nanoparticles were mesocrystals, which was confirmed by the electron diffraction pattern. With mechanical stirring without ultrasound, it needed 8 h to obtain BaTiO3 particles that were much larger (1 to
2 μm) than those synthesized under ultrasound (0.2 to 0.4 μm) and were single crystals rather than aggregates of nanocrystals. The size of a primary particle (nanocrystal) synthesized under ultrasound was ∼5 nm. In the previous paper,12 we have modeled the ultrasoundassisted production of BaTiO3 nanoparticles and performed numerical simulations based on the model. The model consists of three steps; chemical reactions, nucleation of BaTiO3 nanocrystals, and their aggregation. With regard to aggregation of nanocrystals, two models were tested.12 One is a normal model that any particles aggregate with other particles when they collide. The other is a new model that only primary particles (nanocrystals) aggregate with other particles. The new model reproduced the experimental data of the particle (aggregate) size distribution, whereas the normal model did not. It was concluded that only small particles aggregate with other particles in the experiment of Dang et al.9
11 The reason for the aggregation only for small particles was, however, unknown as well as roles of ultrasound in the synthesis of BaTiO3 nanoparticles.12 In the present study, electric dipole
dipole interaction model for a collision between two particles has been numerically studied to see the mechanism of oriented attachment of BaTiO3 nanocrystals under the experimental condition of Dang et al.9
11 In the model, it has been assumed that a BaTiO3 nanocrystal has

BaCl2 þ TiCl4 þ 6H2 O f BaðOHÞ2 þ TiðOHÞ4 þ 6HCl

ð1aÞ 6HCl þ 6NaOH f 6NaCl þ 6H2 O

ð1bÞ

BaðOHÞ2 þ TiðOHÞ4 f BaTiO3 þ 3H2 O

ð1cÞ

Distilled water was sparged with argon gas for 30 min. BaCl2 was dissolved in it. Then, TiCl4 was added to it, resulting in a Tibased sol suspension. The atomic ratio of Ti to Ba was 1. A 5 M NaOH aqueous solution was added to it at room temperature to make the pH 14. The suspension was irradiated with an ultrasonic horn at 20 kHz in open air for 20 min at 80 °C. Then, BaTiO3 nanoparticles were formed. Surprisingly, the BaTiO3 r 2011 American Chemical Society

Received: September 13, 2011 Revised: November 23, 2011 Published: November 30, 2011 319

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particles (aggregates or primary particles) may be mostly due to that between closest primary particles, which are parts of aggregates. Now we consider a collision between two particles (Figure 1). We observe the collision in the center of mass system of the two particles. The z axis is taken on the line between two particles with its origin at the center of the mass. The xy plane is taken perpendicular to the z axis. For simplicity, both dipole moments of the two particles are assumed on the planes parallel to the xy plane. The angle between the dipole moment and the x axis is designated by θ1 and θ2 for particles 1 and 2, respectively. The potential energy of the dipole
dipole interaction is given by eq 2.35 Vdipole ¼ Figure 1. Configuration of a collision between two particles.

p2 cosðθ1
θ2 Þe
kl 4πεl3

ð2Þ

where k is the Debye
H€uckel parameter and is given by eq 336 sffiffiffiffiffiffiffiffiffiffi 2el 2 ni k¼ ð3Þ εkB T

an electric dipole moment (spontaneous polarization). It has been experimentally reported, however, that the crystal structure of BaTiO3 is cubic below certain critical size of BaTiO3 nanocrystals.13
19 It means that BaTiO3 nanocrystals below the critical size may not have spontaneous polarization.20 The critical size strongly depends on amounts of defects and impurities.21 The reported critical size for BaTiO3 ranges from 10 to 120 nm.13
19 There has been no experimental report on the presence of spontaneous polarization at the crystal size of ∼5 nm partially because it is difficult to measure spontaneous polarization for such a small size.22,23 The dipole
dipole interaction has been suggested as the mechanism of mesocrystal formation.24
27 There has been no report, however, on the collision between two particles with the dipole
dipole interaction except the study of Alimohammadi and Fichthorn.28 They performed molecular dynamics simulations of aggregation of various titanium dioxide nanocrystals in vacuum and concluded that the dipole
dipole interactions do not direct aggregation. It should be noted, however, that their simulations were performed for aggregation in vacuum and that the situation may be quite different in aqueous solutions of pH 14 in the experiment of Dang et al.9
11 It should also be noted that there are some other theoretical studies on the mechanisms of oriented attachment of nanocrystals and mesocrystal formation.29
34

where el is the electron charge (el = 1.6  10
19(C)), ni is the number concentration of ion in the aqueous solution (ni = 103NA (m
3) for 1 (mol/L) NaOH aqueous solution, where NA is Avogadro’s number (NA = 6.0  1023(mol
1))), ε is the permittivity of the aqueous solution (ε = 61.03ε0 for pure water, where ε0 is the permittivity of vacuum (ε0 = 8.85  10
12(F/m)), kB is the Boltzmann constant (kB = 1.38  10
23(J/K)), and T is the temperature of the aqueous solution. The Debye
H€uckel parameter means the strength of the electronic screening by the ions in the aqueous solution. The distance of the inverse of the Debye
H€uckel parameter (1/k) is often called the Debye length, which indicates the maximum distance for the electric interaction to occur between charged surfaces in aqueous electrolyte solution. Under the condition of the experiment of Dang et al.,9
11 the Debye length is only (1/k) = 2.9  10
10 (m) = 0.29 (nm). Therefore, the electric interaction occurs only when the two particles are close each other. In eq 2, l is the distance between the surfaces of the two particles and is estimated by eq 4 l ¼ r þ lmin
a1
a2

2. MODEL In the present study, collisions between two particles have been investigated. The particles are either primary particles (nanocrystals) or aggregates. In the present model, it is assumed that each primary particle of the same size has the same electric dipole moment. The electric dipole moment for a primary particle is a parameter in the present study and is designated by p. Aggregates are assumed to be mesocrystals as in the experiment of Dang et al.9
11 The electric dipole moment for an aggregate is assumed to be the same as that for a primary particle. The reason for this assumption is as follows. Any pair of neighboring dipoles of primary particles in an aggregate (mesocrystal) is in antiparallel, and the dipole moments cancel each other. In addition, the Debye screening in an aqueous solution of pH 14 is so strong that the dipole
dipole interaction works only when the two particles nearly attach to each other. It may result in the effective dipole moment for an aggregate (mesocrystal) being equal to that of primary particle (nanocrystal) because the interaction between two colliding

ð4Þ

where r is the distance between the centers of mass of each particle and is given by eq 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ r ¼ ðx1
x2 Þ2 þ ðy1
y2 Þ2 þ ðz1
z2 Þ2 where (x1,y1,z1) and (x2,y2,z2) are the positions of the center of mass for particles 1 and 2, respectively. In eq 4, lmin is the minimum distance between the surfaces of the two particles and is assumed to be 0.15 (nm) in the present study. a1 and a2 are the radii of particles 1 and 2, respectively. Nanoparticles of metal oxides often have their surface charge depending on pH due to the shift in the following equilibrium reactions.36 S
OH2 þ a S
OH a S
O


ð6Þ

where S denotes a solid surface. When the surface of a nanoparticle is electrically charged, an equivalent number of oppositely charged counterions gather near the surface in aqueous electrolyte solution. The surface charge on a nanoparticle and the 320

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associated counterion charge together constitute the electrical double layer. When a nanoparticle moves in the liquid, a part of the associated counterions does not follow the nanoparticle. Then, a plane of shear is defined as the boundary between the fixed and mobile parts of the electric double layer. The electrical potential at the shear plane is called the zeta potential of the nanoparticle. The zeta potential (ζ) of BaTiO3 nanoparticles at pH 14 seems to be nonzero according to some experimental measurements. (It is probably ζ =
40 mV).37
42 Therefore, the surface of a BaTiO3 nanoparticle should be charged at pH 14. When two charged particles approach each other in an electrolyte solution, their electrical double layers overlap, and a repulsive force acts between them in the case of similarly charged particles. This interaction is called double-layer interaction. The potential energy of the double layer interaction between two particles is given by eq 7.36 Vdouble ¼

128πa1 a2 ni kB Tγ1 γ2 e
kl ða1 þ a2 Þk2

ð7Þ

where γ1 and γ2 are given by eqs 8 and 9 as a function of the zeta potential ζ1 and ζ2 of particles 1 and 2, respectively. expðy0 =2Þ
1 γ¼ expðy0 =2Þ þ 1 y0 ¼

el ζ kB T

Figure 2. Result of numerical simulation for a collision between a primary particle (n = 1) and an aggregate (n = 523) when p = 18.7(D) = 6.25  10
29(C 3 m) = 5  10
4(C/m2)V1, where V1 is the volume of a primary particle in cubic meters. n denotes the number of constituent primary particles in each particle. (a) Positions of the two particles as a function of time and (b) relative angle (θ1
θ2) of the two dipoles as a function of time. The initial relative angle is π/4. 0 means that they are in parallel; π means that they are in antiparallel. (c) Potential energies of dipole
dipole interaction, double-layer interaction, and attractive interaction as a function of time.

ð8Þ ð9Þ

In the present study, ζ1 = ζ2 =
4  10
2 (V) has been assumed.37
42 In the experiment of Dang et al.,9
11 BaTiO3 nanocrystals aggregated to form mesocrystals. For the aggregation, there should be some attractive force between BaTiO3 nanoparticles. In the present model, the attractive Coulomb potential is assumed between two BaTiO3 nanoparticles considering the hydrogen bond between particles due to S
OH in eq 6. It is crudely assumed that the hydrogen bond may be expressed by the Coulomb interaction between a positive charge of +0.3el at H atom and a negative charge of
0.3el at O atom.43 Vattractive

ð0:3el NÞ2 e
kl ¼ 4πεl

position of particle 2. The mass of a particle was estimated assuming the density of BaTiO3 nanoparticles as 6 (g/cm3). The initial velocities of two particles are assumed as follows considering the velocity of the Brownian motion.46  rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi! dz1  1 3kB T 3kB T m2 ¼
ð12Þ þ  2 m1 m2 m1 þ m2 dt  t¼0

 dz2   dt 

ð10Þ

where N is the number of hydroxyl groups on the surface of a BaTiO3 nanocrystal (primary particle). In the present numerical simulations, N is an unknown parameter and assumed as N = 8. The equation of motion for particle 1 is given by eq 11. m1

∂ðVdipole þ Vdouble þ Vattractive Þ dz1 þ ¼
∂z1 dt

t¼0

1 ¼ 2

rffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi! 3kB T 3kB T m1 þ m1 m2 m1 þ m2

ð13Þ

The x and y components of the initial velocity are assumed to be zero. The equation of motion for the direction of the electric dipole moment of particle 1 is given by eq 14.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 12πa1 μkB T rn cos jr Δt

I1

ð11Þ where m1 is the mass of particle 1, μ is the viscosity of the aqueous solution, Δt is the time step in the numerical integration of the differential equation (the equation of motion), rn is the random number between
1 and 1, and jr is the random number between 0 and 2π. The last term in the right-hand side of eq 11 is the random force causing the Brownian motion.44,45 In the present numerical simulations, the term was included after the collision of the two particles to ensure the occurrence of the collision. With regard to the x and y components of the position of particle 1, similar equations to eq 11 are used as well as for the

μd a1 Nf θ_1 d2 θ1 P2 sinðθ1
θ2 Þe
kl ¼
8πμa31 θ_1
2 3 jθ_1 j dt 4πεl rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 16πa31 μkB T 0 þ rn cos ϕr 0 Δt

ð14Þ

where I1 is the moment of inertia for particle 1, μd is the dynamic friction coefficient, Nf is the total load (the total force perpendicular to the surface of particle 2 when particle 1 pushes particle 2 at l = lmin; when l > lmin, the third term in the right-hand side of eq 14 is neglected), rn0 is a random number between
1 and 1, and jr0 is a random number between 0 and π. The first term of 321

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Figure 3. Result of numerical simulation for a collision between two aggregates (n = 523). The dipole moment is the same as that for Figure 2. (a) Positions of the two particles as a function of time. (b) Relative angle (θ1
θ2) of the two dipoles as a function of time. The initial relative angle is π.

Figure 4. Result of numerical simulation for a collision between a primary particle (n = 1) and an aggregate (n = 523) when p = 37.5(D) = 1.25  10
28(C 3 m) = 1  10
3 (C/m2)V1. (a) Positions of the two particles as a function of time. (b) Relative angle (θ1
θ2) of the two dipoles as a function of time. The initial relative angle is the same as that for Figure 2 (π/4).

the right-hand side of eq 14 is the dipole
dipole interaction.35 The second term is the friction due to viscosity of the liquid.47 In the present numerical simulations, μ = 100 mPa s was assumed.12 The third term is the dynamic friction when the two particles are attached each other (μd = 0.1 was assumed).48,49 The last term is a random torque that causes the rotational Brownian motion of a particle.50 The last term was included only after a collision took place to fix the initial condition of the collision. For particle 2, a similar equation is employed.

interaction potential decreases to a negative value. In the present case, the double-layer interaction plays a minor role. In Figure 3, the result for a collision between two aggregates has been shown. In this case, two particles reflect each other (Figure 3a) even when the two dipoles are initially in antiparallel (Figure 3b). This is because the amplitude of the potential of repulsive double-layer interaction in this case is ∼5.5 times larger than that for the case of Figure 2 according to eq 7. The results in Figures 2 and 3 fulfill the requirements based on the previous study.12 It suggests that the dipole
dipole interaction model reproduces the experimental results of Dang et al.9
11 When both aggregates are larger than n = 80 (diameter of 26.7 nm), they never attach to each other because of the repulsive double-layer interaction according to the present numerical simulations. When one of two aggregates is smaller than n = 80, however, they attach to each other if the initial dipole
dipole interaction is sufficiently strongly attractive. In the future, the particle (aggregate) size distribution should be numerically calculated under this condition for attachment instead of that only primary particle attaches in the previous paper.12 Next, the result of numerical simulations with a larger dipole moment is discussed. In Figure 4, the result for p = 37.5(D) = 1.25  10
28(C 3 m) = 1  10
3(C/m2)V1 has been shown for a collision between a primary particle and an aggregate (n = 523). The dipole moment in this case is double that for Figures 2 and 3. In this case, a primary particle is reflected from the other particles because of the repulsive dipole
dipole interaction, although the initial condition is the same as that in Figure 2. Therefore, it is concluded that the spontaneous polarization of 5 nm BaTiO3 may be smaller than that of a macroscopic BaTiO3 crystal (0.16 C/m2) to reproduce the experimental results of Dang et al.9
11 For a smaller dipole moment than that for Figures 2 and 3, two dipoles are hardly aligned in antiparallel, although a primary particle attaches with an aggregate. The relative angle (θ1
θ2) of the two dipoles fluctuates near π due to the rotational Brownian motion. For example, when the dipole moment is 1/5 of that for Figures 2 and 3, the deviation of the relative angle (θ1
θ2) from π is as much as 20%. Therefore, it is concluded that the dipole moment is not so much smaller than that for Figures 2 and 3 to form a mesocrystal. It is difficult, however, to discuss the amplitude of dipole moment quantitatively because the exact attractive potential for BaTiO3 nanocrystals is unknown at present.

3. RESULTS AND DISCUSSIONS In the present paper, two cases in collisions between two particles are shown. One is a collision between a primary particle (diameter of 5 nm) and an aggregate (diameter of 50 nm). The aggregate consists of 523 primary particles. The other is a collision between two aggregates of the same size (diameter of 50 nm). According to the previous study by the authors,12 a primary particle should attach with an aggregate with their crystal axis aligned under the experimental condition of Dang et al.9
11 An aggregate should not attach with the other aggregate. This requirement is a result of the study on the experimental data of the particle size distributions by numerical simulations.12 In the present study, numerical simulations of collisions of two particles for the two cases have been performed for various values of the electric dipole moment (p) to see whether the dipole
dipole interaction model explains the above requirements. In Figure 2, a result of the numerical simulations of a collision between a primary particle and an aggregate (n = 523) has been shown for p = 18.7(D) = 6.25  10
29(C 3 m) = 5  10
4 (C/m2)V1, where V1 is the volume of a primary particle in m3. The spontaneous polarization of 5  10
4 (C/m2) corresponds to 0.3% of the spontaneous polarization of a macroscopic BaTiO3 crystal (0.16 (C/m2)).51
53 In Figure 2a, it is seen that two particles attach each other at t = 4.3 μs, even though the initial dipole
dipole interaction is repulsive (Figure 2c). From Figure 2b, it is seen that the electric dipole rotates after the attachment and that the two dipoles are aligned in antiparallel after t = ∼6.3 μs. (The relative angle (θ1
θ2) becomes π.) There is a small fluctuation in the direction of the dipoles due to the rotational Brownian motion. From Figure 2c, it is seen that the reason for the attachment at t = 4.3 μs is the stronger attractive potential than the repulsive dipole
dipole potential. As the dipoles rotate to align in antiparallel, the dipole
dipole 322

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Stronger attractive force (hydrogen bond) may result in the attachment of larger aggregates. In other words, larger N in eq 10 may result in the attachment of larger aggregates. It will be a future study to see the influence of the strength of attractive force on mesocrystal formation. Now we will discuss the possibility of three body (or many body) collisions. Under the present condition of the experiment,9
11 the time between successive collisions of primary particles is ∼20 μs. It is estimated by the following equation.54 τ¼

1 1 ¼ pffiffiffi z 2σ vn ̅ particle

ð15Þ

where τ is the time between successive collisions of primary particles, z is the collision frequency of a primary particle, σ is the collision cross section, ν̅ is the mean velocity of a primary particle, and nparticle is the number density of particles in the aqueous solution. Under the experimental condition,9
11 σ ≈ πd2, where d is the diameter of a primary particle (d = 5 nm), ν̅ = (3kBT/m)1/2 ≈ 4.4 (m/s), where kB is the Boltzmann constant, T is the temperature of the aqueous solution (T = 353 (K)), and m is the mass of a primary particle (m = 7.5  10
22 (kg)), and nparticle ≈ 1020 (m
3). Therefore, τ ≈ 20 μs. The time for the rotation of the dipole moment to be aligned after the attachment of a primary particle is 0.5 to 2 μs according to the present numerical simulations. This is more than an order of magnitude smaller than the time between successive collisions. Therefore, under the present condition, three (or many) body collisions may be neglected. In future studies, however, three (or many) body collisions should be investigated to see whether mesocrystal is formed under such collisions. In conclusion, the results in Figures 2 and 3 are consistent with the requirements by the previous paper12 to reproduce the experimental data of the particle size distributions. It suggests that 5 nm BaTiO3 nanocrystals may have some spontaneous polarization. Then, is it consistent with the previous experimental reports13
19 that the crystal structure of BaTiO3 nanocrystals is cubic for crystal size smaller than 10 nm? One possibility is the difference in the amount of impurities or defects between BaTiO3 nanocrystals synthesized in the previous studies13
19 and those in the experiment of Dang et al.9
11 Another possibility is that the lattice structure of a nanocrystal deviates from the ideal cubic due to the surface effect.55 It may possibly result in some spontaneous polarization for nonmetallic nanocrystals.56 Some condition of the termination of the crystal surface may also result in some spontaneous polarization of a nanocrystal.57

4. CONCLUSIONS The electric dipole
dipole interaction model has been studied for collisions between two particles by numerical simulations under the experimental condition of Dang et al.9
11 If 5 nm BaTiO3 nanocrystal has a spontaneous polarization smaller than that of a macroscopic BaTiO3 crystal, then the dipole
dipole interaction model has resulted in attachment of only small particles to other particles, which has been suggested to reproduce the experimental data of the particle size distributions.12 Furthermore, the model yields the alignment of two dipoles in antiparallel. The results have suggested that 5 nm BaTiO3 nanocrystals synthesized under ultrasound by Dang et al.9
11 may have some spontaneous polarization, although it has been experimentally reported that the crystal structure of BaTiO3 naocrystals is cubic for smaller crystal size than 10 nm.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected].

’ LIST OF SYMBOLS I1 = moment of inertia for particle 1 N = number of hydroxyl groups on the surface of a BaTiO3 nanocrystal (primary particle) NA = Avogadro’s number (6.0  1023 (mol
1)) Nf = total load (force) perpendicular to particle 2 from particle 1 when two particles are attached (at l = lmin) T = temperature Vattractive = potential energy of the attractive force between the surfaces of two particles (eq 10) Vdipole = potential energy of the dipole
dipole interaction (eq 2) Vdouble = potential energy of the double-layer interaction (eq 7) V1 = volume of a primary particle (5 nm BaTiO3 nanocrystal) a1 = radius of particle 1 a2 = radius of particle 2 d = diameter of a primary particle el = electron charge (1.6  10
19 (C)) kB = Boltzmann constant (1.38  10
23 (J/K)) l = distance between the surfaces of the two particles (eq 4) lmin = minimum distance between the surfaces of the two particles m = mass of a primary particle m1 = mass of particle 1 m2 = mass of particle 2 n = number of constituent primary particles in an aggregate ni = number concentration of ion in the aqueous solution nparticle = number density of particles in the aqueous solution p = electric dipole moment r = distance between the centers of mass of each particle (eq 5) rn = random number between
1 and 1 rn0 = random number between
1 and 1 t = time ν̅ = mean velocity of a primary particle y0 = quantity determined by temperature and zeta potential of a particle (eq 9) z = collision frequency of a primary particle (x1,y1z1) = position of the center of mass for particle 1 (x2,y2,z2) = position of the center of mass for particle 2 Δt = time step in the numerical integration of the differential equation (the equation of motion) γ = quantity determined by temperature and zeta potential of a particle (eq 8) γ1 = γ for particle 1 γ2 = γ for particle 2 ε = permittivity of the aqueous solution ε0 = permittivity of vacuum ζ = zeta potential ζ1 = zeta potential for particle 1 ζ2 = zeta potential for particle 2 θ1 = angle of the dipole of the particle 1 relative to x axis θ2 = angle of the dipole of the particle 2 relative to x axis · θ1 = (dθ1/dt) k = Debye
H€uckel parameter (eq 3) μ = viscosity of the aqueous solution μd = dynamic friction coefficient σ = collision cross section τ = time between successive collisions of primary particles 323

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jr = random number between 0 and 2π jr0 = random number between 0 and π

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