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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 © 2011 American Institute of Physics Downloaded 22 Feb 2011 to 128.6.231.3. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 22 Feb 2011 to 128.6.231.3. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 22 Feb 2011 to 128.6.231.3. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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. Downloaded 22 Feb 2011 to 128.6.231.3. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 22 Feb 2011 to 128.6.231.3. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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 Downloaded 22 Feb 2011 to 128.6.231.3. Redistribution subject to AIP license or copyright; see http://jcp.aip.org/about/rights_and_permissions 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. 1 M. S. Woolridge, Prog. Energy Combust. Sci. 24, 63 (1998). Hawa and M. R. Zachariah, Phys. Rev. B 71, 165434 (2005). 3 L. Ravi and S. L. Girshick, Phys. Rev. E 79, 026408 (2009). 4 U. Kortshagen and U. Bhandarkar, Phys. Rev. E 60, 887 (1999). 5 X. Y. Li and B. E. Logan, Environ. Sci. Technol. 31, 1237 (1997). 6 P. J. Lu, J. C. Conrad, H. M. Wyss, A. B. Schofield, and D. A. Weitz, Phys. Rev. Lett. 96, 028306 (2006). 7 W. B. Russel, D. A. Saville, and W. 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