Oriented Attachment of Cubic or Spherical BaTiO3 Nanocrystals by

Article pubs.acs.org/JPCC

Oriented Attachment of Cubic or Spherical BaTiO3 Nanocrystals by van der Waals Torque 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: It has been believed that dipole−dipole interaction is the reason for the oriented attachment (OA) of nanocrystals if they have electric or magnetic dipoles. In this study, the role of van der Waals torque (Casimir torque) originated in dielectric anisotropy of a nanocrystal is numerically studied for the first time on OA of cubic or spherical BaTiO3 nanocrystals. The present numerical simulations of OA have indicated that the alignment of the crystal axes is mainly due to van der Waals torque when the crystal size is smaller than 5 nm both in aqueous and organic solutions with and without electric charge screening by ions, respectively. For larger crystals, the electric dipole−dipole interaction dominates. The mesocrystal formation of 5 nm BaTiO3 nanocrystals in aqueous solution is due to van der Waals torque.

1. INTRODUCTION When nanocrystals are aggregated in liquid, the crystal axes of nanocrystals are sometimes aligned in an aggregate. Such an aggregate is called mesocrystal.1−6 The attachment of nanocrystals with alignment of their crystal axes is called oriented attachment (OA), which was first reported by Penn and Banfield7,8 in 1998. OA has been observed for various species of nanocrystals: BaTiO3, CaCO3, CeO2, ZnO, Fe3O4, TiO2, Au, Ag, Pt, and so on.1−26 Some of the nanocrystals have electric or magnetic dipoles. For such nanocrystals, it has long been believed that the OA is due to dipole−dipole interaction.27−36 However, there are some other nanocrystals undergoing OA without any dipole. Although some organic ligand on the surface of nanocrystals may lead to OA, it is still under debate for the mechanism of OA for such nanocrystals.37−39 Between two anisotropic bodies, van der Waals torque acts when the optical axes of the two bodies are not aligned.40−51 van der Waals torque is expressed as follows between two parallel plates that are both optically uniaxial.40,47 A S τ(h , θ ) = − θ 2 2 sin(2θ ) (1) 64π l where l is the distance between the surfaces of two bodies, θ is the angle between the optical axes of the two bodies, Aθ is the Hamaker torque constant, and S is the cross-sectional area of a plate. The Hamaker torque constant depends on the materials as follows. ∞

A θ = −2πkBT ∑ ′ n=0

where kB is the Boltzmann constant, T is temperature, the prime in the sum means that the n = 0 term has to be multiplied by 1/2, ε1∥ and ε1⊥ are the component of the dielectric tensor for a plate labeled 1 along the principal optical axis and that perpendicular to it, respectively, ε2∥ and ε2⊥ are those for the other plate labeled 2, and ε3 is the dielectric function of the medium (water, mesitylene, air, etc.). In eqs 1 and 2, the retardation effects have been neglected. The retardation effects are explained as follows.52 The change in the charge distribution due to the internal electronic (and nuclear) motions of a substance causes the propagation of electromagnetic field into the surrounding space with the speed of light. The field reaches the other substance and causes the polarization of the substance. The oscillation of the polarization of the substance causes reradiation of the electromagnetic field into the surrounding space. The field propagates with the speed of light and returns back to the first substance. It results in the lowering in the free energy, which causes the attractive van der Waals force, as well as the van der Waals torque. The propagation time is ∼l/c, where l is the distance between the two substances and c is the speed of light. When the propagation time is longer than the characteristic time for the internal motion, the lowering in the free energy is reduced, which is called the retardation effect. In general, the effect weakens the attractive van der Waals force as well as the van der Waals torque. As the characteristic time for the internal motion is longer than 10−17 s, the distance between the two substances should be smaller than about 3 nm to neglect the retardation effects.52 This holds under the conditions of the present

(ε2 − ε2 ⊥)(ε1 − ε1 ⊥)ε32 (ε12⊥ − ε32)(ε22⊥ − ε32)

⎛ (ε − ε3)(ε2 ⊥ − ε3) ⎞ × ln⎜1 − 1 ⊥ ⎟ (ε1 ⊥ + ε3)(ε2 ⊥ + ε3) ⎠ ⎝ © 2015 American Chemical Society

Received: July 15, 2015 Revised: September 27, 2015 Published: October 5, 2015

(2) 24597

DOI: 10.1021/acs.jpcc.5b06798 J. Phys. Chem. C 2015, 119, 24597−24605

Article

The Journal of Physical Chemistry C

accelerate the formation of BaTiO 3 phase at a high concentration of oleic acid. Oleic acid adsorbed on the surface of BaTiO3 nanocrystals. The films of self-assembled BaTiO3 nanocrystals were fabricated by the dip-coating method. In the dip-coating, a substrate was dipped into the mesitylene solution in which BaTiO3 nanocrystals of 15−20 nm in size were dispersed. Then, the substrate was withdrawn from the solution at a controlled speed of 0.01 μm/s. The thickness of the coated film was about 300−600 nm. The substrates used in the experiment were Pt-coated Si, Pt-coated MgO, and Pt-coated SiO2. The concentration of BaTiO3 nanocrystals in the solution was 6 × 10−3 g/cm3. The operation temperature was about 25 °C. The as-assembled structures were irradiated by UV light for 2 h and dried at 200 °C for 1.5 h to remove residual organics. Then, BaTiO3 nanocrystal assemblies were calcined at 400 °C for 1 h and sintered at 850 °C for 1 h in an O2 atmosphere. The microstructures of these assemblies were observed by fieldemission scanning electron microscopy and high-resolution transmission electron microscopy. FFT images of the BaTiO3 nanocrystal assemblies indicated 4-fold symmetric patterns. It means that the assemblies of BaTiO3 nanocrystals on a variety of substrates were highly ordered. Next, the condition of the experiment by Dang et al.18−20 on the mesocrystal formation of BaTiO3 nanocrystals in aqueous solution is briefly reviewed. Argon gas was dissolved in distilled water by blowing the gas into water for 30 min, and BaCl2 was dissolved in it. TiCl4 was added to the solution, which resulted in the formation of a Ti-based sol suspension. Then 5 M NaOH aqueous solution was added into the mixed suspension at room temperature for peptization. The atomic ratio of Ti to Ba was 1. The initial pH of the suspension was controlled at 14, which corresponds to the Debye length of 0.29 nm. The suspension was irradiated with an ultrasonic horn in open air at 80 °C for 20 min. The ultrasonic intensity was 150 W/cm2. After the synthesis, the precipitate was centrifugally separated and washed with deionized water twice, then dried at 100 °C for 2 h in vacuum. The characterization of BaTiO3 particles was conducted by X-ray diffractometry (XRD), scanning electron microscopy (SEM), transmission electron microscope (TEM), and the dynamic light scattering (DLS) method. Spherical aggregates of BaTiO3 nanocrystals were obtained in the experiment. The size of an aggregate was in the range of 100−400 nm, while the diameter of each (nearly) spherical BaTiO3 nanocrystal was 5−10 nm. The selected-area electrondiffraction (SAED) pattern of an aggregate was the same as that from a single crystal. It means that the aggregate is a mesocrystal.

numerical simulations because the minimum distance between two substances is smaller than 1 nm. In eqs 1 and 2, Matsubara’s technique is used to calculate the free energy that the integration with respect to frequency is replaced by the sum for the Matsubara frequencies.40,45,52,53 The torque is the partial derivative of the free energy with respect to the angle multiplied by −1.40,45 Each component of dielectric tensor and the dielectric function in eq 2 are calculated at imaginary angular frequencies of the electromagnetic wave iξn = i(2πkBT/ℏ)n, where ξn is the Matsubara frequencies, ℏ is the Planck constant divided by 2π, and n is zero or a positive integer. Each dielectric function at an imaginary angular frequency is calculated by eq 3 according to a Kramers−Kronig relation.52,54,55 ⎛ 2 ε(iξ) = ε0⎜1 + π ⎝

∫0

∞

xεr″(x)dx ⎞ ⎟ x2 + ξ2 ⎠

(3)

where ε0 is the dielectric permittivity of a vacuum (=8.85 × 10−12 C/(V m)), ε″r (x) = ε″(x)/ε0, and ε″(ω) is the imaginary part of the dielectric function (ε(ω) = ε′(ω) + iε″(ω)). The imaginary part (ε″(ω)) of the dielectric function is related to the phase lag in the dielectric response of the material as in eq 4.52 δ(ω) = tan−1[ε″(ω)/ε′(ω)]

(4)

where δ(ω) is a phase difference between the electric field (E⃗ (ω)) and the displacement vector (D⃗ (ω)) (or equivalently the polarization P⃗ (ω)). At room temperature, the Hamaker torque constant has been estimated as Aθ = 200kBT for BaTiO3 with tetragonal crystal structure (optically uniaxial), and Aθ = 0.68kBT for calcite (CaCO3) with trigonal crystal structure (optically uniaxial) when the medium between the plates is water.40 The dependence of the Hamaker torque constant on the medium is, however, not so important (a few tens of percent).47 For cubic crystal structure, a crystal is optically isotropic and the Hamaker torque constant is zero. Thus, nonzero Hamaker torque constant for BaTiO3 usually means that there is also electric dipole. For calcite, on the other hand, there may be no electric dipole, but the Hamaker torque constant is nonzero. It has already been suggested that OA of nanocrystals may be due to van der Waals torque.56,57 However, there has been no numerical study on it. In the present paper, numerical simulations of OA of (nearly) cubic or spherical nanoparticles have been performed for the first time taking into account the effect of van der Waals torque. The present numerical simulations have been performed under conditions of two different experiments: in organic solvent without any electric charge screening and in aqueous solution with strong electric charge screening by ions with the Debye length of only 0.3 nm. For the former, electric dipole− dipole interaction works even if two particles are considerably apart from each other. For the latter, electric dipole−dipole interaction works only when two particles are close to each other. The former is the experiment of self-assembly of BaTiO3 nanocrystals in mesitylene by Mimura et al.25,26 The latter is the mesocrystal formation of BaTiO3 nanocrystals in aqueous solution at pH 14 by Dang et al.18−20 The condition of the experiment by Mimura et al. 26 is as follows: BaTiO 3 nanocrystals with (nearly) cubic shape were prepared by hydrothermal methods with a water-soluble titanium complex, tert-butylamine, and oleic acid.23 tert-Butylamine was used to

2. MODEL First, the model used in the numerical simulations for organic solvent (mesitylene) under the experimental condition of selfassembly of BaTiO3 nanocrystals by Mimura et al.26 is discussed. In dip-coating, it has been suggested that capillary force acting on nanocrystals on the liquid surface plays an important role. However, the thickness of the self-assembly was about 30× that of a single nanocrystal size. It suggests that, not only OA at the liquid surface, but also OA in the bulk liquid, should be important for the formation of the self-assembly.58 Thus, in the present study, numerical simulations have been performed for OA of BaTiO3 nanocrystals in the bulk liquid without any capillary force. Furthermore, only a collision of two BaTiO3 nanocrystals have been numerically simulated without 24598


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of the two cubic nanocrystals are in parallel, although the electric dipoles (z-axes of BaTiO3 nanocrystals) are initially not aligned (Figure 1). The last term in the right side of eq 5 is the random force causing the Brownian motion.59,60 The first term in the right side of eq 6 is the torque of dipole−dipole interaction which makes a pair of dipoles in antiparallel.55 The second term is the van der Waals torque discussed in the Introduction, which makes a pair of z-axes of BaTiO3 nanocrystals in parallel or antiparallel. The last term is the random torque that causes the rotational Brownian motion.61 In order to make the initial condition controllable in numerical simulations, the random force and torque are included after a collision of particles takes place. It should be noted that the magnitudes of the random force and torque depend on the choice of the time step (Δt) in the numerical simulations according to the present model in eqs 5 and 6. Thus, the results of the present numerical simulations are only qualitatively valid. In the present numerical simulations, Δt = 1 × 10−13(s) = 0.1 (ps) was used, and eqs 5 and 6 were numerically integrated with the Euler method.62,63 Now, we discuss the dipole moment (p) of a BaTiO3 nanocrystal. It has been widely known that the crystal structure of a BaTiO3 single crystal becomes cubic when the crystal size is smaller than a critical one at room temperature, while the crystal structure of a macroscopic BaTiO3 single crystal is tetragonal (the size effect).64,65 When the crystal structure is cubic, there is no spontaneous polarization due to its highly symmetric structure.66 For the tetragonal crystal structure, on the other hand, there is spontaneous polarization and nonzero dipole moment. According to the numerical calculations of the free energy of a BaTiO3 single crystal taking into account the depolarization energy, the crystal structure could be tetragonal, even for a BaTiO3 nanocrystal of 5 nm when there is strong adsorbate-induced charge screening, which sufficiently reduces the depolarization energy.65 The depolarization energy is the energy associated with the depolarization field, which is the electric field formed by the surface charges in the opposite direction to the spontaneous polarization. The depolarization energy is the main reason for the size effect.65 The numerical calculations have indicated that the polarization at the crystal surface should be as small as (PS/P) < 4 × 10−3 by the adsorbate-induced charge screening for 5 nm BaTiO3 nanocrystal to be tetragonal, where PS is the polarization at the crystal surface and P is the bulk polarization for a macroscopic single crystal of BaTiO3 (P = 0.16 (C/m2)). The dipole moment (p) of a BaTiO3 nanocrystal, which is macroscopically cubic, is expressed as follows.65

considering any aggregates (a fraction of the self-assembly) as a first step to understand the self-assembly formation. The equations of translational and rotational motion of a nanocrystal are given by eqs 5 and 6, respectively. m1

∂(Vdipole + VvdW + Vbridge + Vsteric + Vdepletion) dz1 =− ∂z1 dt + 2

2

I1

d θ1 dt 2

6πd1μkBT rn cos φr Δt

=

p sin(θ1 − θ2) 4πεl 3 +

−

Aθ S 64π 2l 2

(5)

sin(2(θ1 − θ2))

2πd3μkBT r′n cos ϕ′r Δt

(6)

where m1 is the mass of particle (nanocrystal) 1, z1 is the position (z component) of particle 1, t is time, Vdipole is the potential of electric dipole−dipole interaction, VvdW is the potential of van der Waals interaction, Vbridge is the potential of the bridging interaction by oleic acid adsorbed on the surface of nanocrystals, Vsteric is the potential of steric repulsion due to oleic acid on the nanocrystal surface, Vdepletion is the potential for the depletion force originated in the osmotic pressure due to large molecules (oleic acid) dispersed in the solution when large molecules are excluded in the narrow region between the two nanocrystals, d1 is the size of particle (nanocrystal) 1, μ is the viscosity of the solution, Δt is the time step in the numerical integration of the differential equation (the equation of motion), rn is a random number between −1 and 1, φr is a random number between 0 and 2π, I1 is the moment of inertia for particle (nanocrystal) 1 (=m1d21/6 for a cubic nanocrystal), θ1 and θ2 are the angle between the electric dipole and x-axis for particle 1 and 2, respectively (Figure 1), p is the electric

p=

PSd 2 (1 − e−κd) κ

(7)

where d is the size of a nanocrystal and κ is the Debye−Hückel parameter due to the presence of mobile charge carrier in the nanocrystal. The inverse of the Debye−Hückel parameter is equivalent to the thickness of the space-charge layer, which is assumed as 1/κ = 10 nm.65 For (PS/P) = 3 × 10−3 and PS = 5 × 10−4 C/m2, the electric dipole moment is 15 D for d = 5 nm and 518 D for d = 20 nm (1 D = 3.3356 × 10−30 (C m)), which have been assumed in the present numerical simulations. In the self-assemblies in the experiment of Mimura et al.,26 the distance between the BaTiO3 nanocrystals is about 1 nm before drying at high temperature, because oleic acid was present on the crystal surface.21 Thus, in the present numerical

Figure 1. Configuration of a collision between two nanocrystals.

dipole moment of a BaTiO3 nanocrystal, ε is the dielectric constant of the medium (=2.24ε0 for mesitylene), l is the distance between the surfaces of the two particles (nanocrystals), Aθ is the Hamaker torque constant (=200 kBT for BaTiO3),40 S is the cross-sectional area of a particle (nanocrystal; =d2 for a cubic nanocrystal), r′n is a random number between −1 and 1, and ϕr′ is a random number between 0 and 2π. Here, it has been assumed that the surfaces 24599


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by Dang et al.18−20 is briefly discussed. The model has been described in refs 34−36. The only modification is the inclusion of the van der Waals torque (eq 1) in the equations of the rotational motion of nanocrystals. In this case, the minimum distance between the surfaces of nanocrystals is set as lmin = 0.15 nm because there is no organic ligand such as oleic acid on the crystal surface. A BaTiO3 nanocrystal is assumed to be spherical with the diameter of d = 5 nm and with p = 10 D, which is calculated from PS = 5 × 10−4 C/m2 multiplied by the volume. In the equation of translational motion, the following effects are taken into account; electric dipole−dipole interaction with charge screening in an aqueous solution of pH 14 with the Debye length of 0.29 nm, double-layer interaction between charged particles at zeta potential of −40 mV with the charge screening, hydrogen bond due to −OH on the surfaces of BaTiO3 nanocrystals with the charge screening, and the random force causing the Brownian motion. In the equation of the rotational motion, the following effects are taken into account; electric dipole−dipole interaction with the charge screening, the friction due to viscosity of the solution, the dynamic friction when the two particles are attached, the random torque causing the rotational Brownian motion, and the van der Waals torque.

simulations for organic solvent (mesitylene) under the condition of the experiment by Mimura et al.,26 the minimum distance between the surfaces of nanocrystals is set as lmin = 1 nm. The potentials in eq 5 are given as follows.55,67−70 Vdipole =

p2 cos(θ1 − θ2) 4πεl 3

VvdW = −

(8)

d1d 2 d1d 2 A⎡ ⎢ 2 + 2 12 ⎢⎣ l + d1l + d 2l l + d1l + d 2l + d1d 2

⎞⎤ ⎛ l 2 + d1l + d 2l ⎟⎥ + 2 ln⎜ 2 ⎝ l + d1l + d 2l + d1d 2 ⎠⎥⎦

(9)

Vbridge = −SEvdW Γ(LC − l)/Lseg Vsteric =

(10)

πd1d 2 π 2kBT Γ(l0)3 (d1 + d 2) 6Nb2 ⎤ ⎡ 9 1 1 (1 − u6)⎥ × ⎢− ln(u) − (1 − u) + (1 − u3) − ⎦ ⎣ 5 3 30 (11)

⎡1 3 2 Vdepletion = − kBTβϕ⎢ q2 + − q(q2 − 1)−1/2 (1 − x) ⎣3 2 3 ⎤ 2 1/2 × ln(q + (q − 1) ) − (1 − x)x ⎥ ⎦

3. RESULTS AND DISCUSSIONS First, each potential for the transvers motion of a BaTiO3 nanocrystal in organic solvent (mesitylene) in eq 5 has been numerically calculated as a function of the distance between the surfaces of nanocrystals in order to see the dominant potential (Figure 2). The macroscopic shape of a nanocrystal is (nearly) cubic, but the crystal structure is assumed as tetragonal (there is electric dipole given by eq 7). In Figure 2, the absolute value of each potential is shown with logarithmic vertical axis. For dipole−dipole interaction, the potential is shown for the two dipoles in antiparallel. The potential for depletion force is not shown in Figure 2 because it is not important. Only the potential of the steric repulsion is positive, and the other potentials are negative (attractive). The relative amplitude of the potential of the dipole−dipole interaction to the other potentials increases as the nanocrystal size increases. In other words, the role of the electric dipole−dipole interaction is more important for larger BaTiO3 nanocrystals (Figures 2 and 3). The amplitude of the torque for the rotational motion is, however, not directly related to the amplitude of the potential (force) for the transverse motion shown in Figures 2 and 3. In Figure 3, the absolute value of each force for the transverse motion of a BaTiO3 nanocrystal is shown as a function of the distance between the surfaces of nanocrystals with logarithmic vertical axis in order to see the dominant force. Only the steric repulsion is positive, and the other forces are negative (attractive). Each force is just the gradient of each potential. The dominant attractive force at the distance of l = 1 nm is the bridging force due to oleic acid for the crystal size of 5 nm, while for the crystal size of 20 nm, the dipole−dipole interaction is also important. Next, the result of the numerical simulation of a collision between two BaTiO3 nanocrystals is discussed under the conditions of the experiment of Mimura et al.26 on the selfassemblies of nanocrystals by dip coating in mesitylene (Figure 4). The macroscopic shape of a nanocrystal is (nearly) cubic, but the crystal structure is assumed as tetragonal. The two nanocrystals of d = 20 nm collide at t = 5.6 μs when the attachment takes place due to the bridging force (Figure 4a). At

(12)

−20

where A is the Hamaker constant (=5 × 10 (J)), d1 and d2 are the sizes of particles (nanocrystal) 1 and 2, respectively, EvdW is the binding energy per segment of oleic acid for van der Waals interaction (= 0.4 × 10−20 (J)), Γ is the surface concentration of oleic acid on the nanocrystal surface (assumed as 3.3 × 1018 (m−2)), LC is the contour length of oleic acid (CH3(CH2)7CHCH(CH2)7COOH) (=2.77 (nm)), Lseg is the segment length of oleic acid (=0.154 (nm)), N is the socalled degree of polymerization or, equivalently, the number of “Kuhn monomers” each with a characteristic length b (N = 2 and b = 1 (nm) are assumed for oleic acid),69,71 l0 is the equilibrium thickness of the polymer bush (l0 = Γ1/3Nb5/3 is assumed), u is defined as u = l/(2l0), β is defined as β = d/aA, d is the size of a nanocrystal, aA is the length of a macromolecular solute (oleic acid) along the chain (assumed as 2.772 (nm)), ϕ is the volume concentration of macromolecules (oleic acid; assumed as 1 × 10−3), q is defined as q = aA/aB, aB is the width of a macromolecular solute, and x is defined as x = (aA − l)/aA when l < aB (x = (aA − aB)/aA is assumed when l ≥ aB, which means that there is no depletion force for l ≥ aB). The bridging force (Vbridge) is taken into account only when l < LC. The steric repulsion (Vsteric) is taken into account only when l < 2l0. The initial velocity of each nanocrystal is assumed as that for the Brownian motion as follows.36,72 dz1 dt

dz 2 dt

=− t=0

= t=0

3m2kBT m1(m1 + m2)

3m1kBT m2(m1 + m2)

(13)

(14)

where m2 is the mass of particle (nanocrystal) 2. Next, the model used in the numerical simulations for aqueous solution with ions under the condition of the experiment of mesocrystal formation of BaTiO3 nanocrystals 24600


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Figure 2. Absolute value of the potential energy for the transverse motion of a BaTiO3 nanocrystal in organic solvent (mesitylene) as a function of the distance between the surfaces of two BaTiO3 nanocrystals is shown in order to see the dominant potential. The potential energy for the electric dipole−dipole interaction is calculated for dipoles in antiparallel. Only the potential energy of steric repulsion is positive (the other potential energies are negative (attractive)). (a) For d = 5 nm with p = 15 D. (b) For d = 20 nm with p = 518 D.

Figure 3. Absolute value of the force for the transverse motion of a BaTiO3 nanocrystal in organic solvent (mesitylene) is shown in order to see the dominant force. The force for the electric dipole−dipole interaction is calculated for dipoles in antiparallel. Only the force of the steric repulsion is positive (the other forces are negative (attractive)): (a) for d = 5 nm with p = 15 D; (b) for d = 20 nm with p = 518 D.

C/m2), which is larger than the upper bond for 5 nm BaTiO3 nanocrystals of tetragonal crystal structure, the electric dipoles are aligned in parallel due to van der Waals torque according to the present numerical simulations. It confirms the above conclusion for d = 5 nm. For d = 20 nm, the electric dipole moment could be much smaller than 518 D. Without any electric dipole, it takes more than 150 μs for c-axes aligned by van der Waals torque for d = 20 nm according to numerical simulations. The typical time between successive collisions of nanocrystals could be in the order of 20 μs.34 Thus, the alignment of c-axes solely by van der Waals torque is difficult for d = 20 nm. In conclusion, for d = 5 nm, c-axes (electric dipoles) of the two BaTiO3 nanocrystals are aligned in parallel or antiparallel mainly due to the van der Waals torque. On the other hand, for d = 20 nm, OA is due to dipole−dipole interaction, and the electric dipoles is aligned only in antiparallel. Finally, the result of the numerical simulation in aqueous solution with ions under the condition of the experiment of mesocrystal formation of BaTiO3 nanocrystals is discussed. The macroscopic shape of a nanocrystal as well as an aggregate of nanocrystals is (nearly) spherical, but the crystal structure is tetragonal. For a collision between a nanocrystal of d = 5 nm (n = 1) and an aggregate consisting of 65 nanocrystals (n = 65, the diameter is about 25 nm), the attachment takes place at t = 1.7 μs when the initial distance is about 10 μm (Figure 6a). At the attachment, the electric dipoles of nanocrystals quickly rotate to

and after the attachment, the two electric dipoles rotate to be in antiparallel (the relative angle of −π) (Figure 4b). It takes about 20 μs for the total rotation. The rotation is mostly due to the electric dipole−dipole interaction (Figure 4c). The result of the numerical simulation for the nanocrystal size of d = 5 nm is shown in Figure 5. The two BaTiO3 nanocrystals collide at t = 3.5 μs when attachment takes place due to the bridging force of oleic acid (Figure 5a). Contrary to the case of d = 20 nm, the two electric dipoles more quickly rotate to be in parallel (the relative angle of 0) at the attachment (Figure 5b). About 6% of the total rotation of π/4 (rad) occurs before the attachment, and the rest of the rotation (about 94%) does in about 16 ns after the attachment. It is solely due to the van der Waals torque because the dipole− dipole interaction only makes the two dipoles in antiparallel. The random fluctuation in the relative angle of dipoles in Figure 5b is due to the rotational Brownian motion, which is much more significant compared to the case of d = 20 nm, because the moment of inertia for d = 5 nm is 3 orders of magnitude smaller than that for d = 20 nm, while the random torque is only 1 order of magnitude smaller. As described in the model, the rotational Brownian motion is taken into account after the collision takes place in order to fix the initial condition. Even if the electric dipole moment for d = 5 nm is as large as 19 D (corresponding to (PS/P) = 4 × 10−3 and PS = 6.4 × 10−4 24601


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Figure 5. Result of the numerical simulation for organic solvent (mesitylene) for d = 5 nm with 15 D. The other conditions are the same as those in Figure 4; (a) position of nanocrystals and (b) relative angle of electric dipoles.

confirms the above conclusion in aqueous solution with ions with the Debye length of 0.29 nm. In refs 34−36, the authors suggested that the mesocrystal formation under the experimental condition of Dang et al.18−20 could be due to dipole− dipole interaction. However, the present numerical simulations taking into account the effect of the van der Waals torque have indicated that it is due to the van der Waals torque. For d = 20 nm under this condition, the electric dipoles are aligned in antiparallel due to dipole−dipole interaction, as in the case of Figure 4. It is concluded that for BaTiO3 nanocrystals of d = 20 nm OA is due to dipole−dipole interaction even with charge screening in aqueous solution with ions.

Figure 4. Result of the numerical simulation of a collision between two BaTiO3 nanocrystals (d = 20 nm with p = 518 D) in organic solvent (mesitylene) as a function of time under the condition of the experiment of Mimura et al.26 The initial relative angle of the electric dipoles is −π/4; (a) position of nanocrystals; (b) relative angle of electric dipoles; and (c) torque.

4. CONCLUSION Numerical simulations of OA of nanocrystals have been performed for the first time taking into account the van der Waals torque between (nearly) cubic or spherical BaTiO3 nanocrystals for two cases; in aqueous solution and in organic solvent (mesitylene) with and without charge screening by ions, respectively. In the numerical simulations, mainly three kinds of torques are taken into account; the electric dipole−dipole interaction, random torque causing the rotational Brownian motion, and the van der Waals torque which originates in dielectric anisotropy of a crystal. It has been shown that for BaTiO3 nanocrystals smaller than d = 5 nm OA is mainly due to the van der Waals torque in both aqueous and organic

be in parallel solely due to the van der Waals torque (Figure 6b). About 5% of the total rotation of π/4 (rad) occurs before the attachment, and the rest of the rotation (about 95%) does in about 19 ns after the attachment. The attachment in this case is due to hydrogen bond between −OH on the surfaces of BaTiO3 nanocrystals (Figure 6c). Even if the electric dipole moment is as large as 15 D, which is even larger than the upper bound for tetragonal crystal structure, electric dipoles are aligned in parallel due to van der Waals torque according to the present numerical simulations. It 24602


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BaTiO3 nanocrystals in aqueous solution in the experiment of Dang et al.18−20 is due to van der Waals torque.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Phone: +81-52-736-7438. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors would like to thank Ken-ichi Mimura (AIST) for useful comments.

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REFERENCES

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Figure 6. Result of the numerical simulation for a collision of a nanocrystal (n = 1) and an aggregate (n = 65) of BaTiO3 nanocrystals in aqueous solution of pH 14 under the condition of the experiment of Dang et al.18−20 The size of a spherical nanocrystal is d = 5 nm with 10 D. The diameter of the aggregate is about 25 nm. The initial relative angle of electric dipoles is π/4; (a) position of nanoparticles; (b) relative angle of electric dipoles; and (c) potential energy.

solutions. For larger BaTiO3 nanocrystals (d = 20 nm), OA is mainly due to electric dipole−dipole interaction in both aqueous and organic solutions. It suggests that mechanism of OA is not significantly influenced by the electric charge screening by ions. The role of electric dipole−dipole interaction becomes more important as the crystal size increases. The rotational Brownian motion becomes more important as the crystal size decreases. The mesocrystal formation of 5 nm 24603


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