Calculation of Normal Contact Forces between ... - ACS Publications

May 20, 2013 - In recent years, many efforts have been devoted to the direct measurement of normal and tangential forces between particles or between ...
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Calculation of Normal Contact Forces between Silica Nanospheres Weifu Sun,† Qinghua Zeng,‡ Aibing Yu,*,† and Kevin Kendall§ †

Laboratory for Simulation and Modeling of Particulate Systems, School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia ‡ School of Computing, Engineering and Mathematics, University of Western Sydney, Penrith, NSW 2751, Australia § School of Chemical Engineering, University of Birmingham, Edgbaston, Birmingham, B15 2TT, U.K. ABSTRACT: In this work, interaction forces between two silica nanospheres after contact, including the van der Waals (vdW) attraction, Born repulsion, and mechanical contact forces are studied by molecular dynamics (MD) simulations. The effects of interaction path (approach or departure), initial relative velocity, and relative orientations of two nanospheres are first examined. The results show that the interparticle forces are, to a large degree, independent of these variables. Then, emphasis is given to other important variables. At a small contact deformation, the size dependence of the vdW attraction and Born repulsion qualitatively agrees with the prediction based on the conventional theories, but this becomes vague upon further deformation due to the gradually flattened shape of deformed particles. An alternative approach is provided to calculate the interparticle vdW attraction and Born repulsion forces. Moreover, the MD simulations show that the Hertz model still holds to describe the mechanical contact force at low compression, which is obtained by subtracting the vdW attraction and Born repulsion forces from the total normal force. Comparisons with the Johnson−Kendall−Roberts (JKR) and Derjaguin−Muller−Toporov (DMT) models, in terms of force-displacement relationships and contact radius, show that the two models can be used to provide the first approximation, but there is some deviation from the MD simulated results. The origins of the quantitative difference are analyzed. New equations are formulated to estimate the interaction forces between silica nanospheres, which should be useful in the dynamic simulation of silica nanoparticle systems.

1. INTRODUCTION Interparticle forces including the van der Waals (vdW) attraction, Born repulsion, and contact forces play a central role in many systems, such as microelectro-mechanical systems,1 and in a variety of natural phenomena and industrial processes, such as aggregation,2,3 adsorption,4 dispersion,5 packing,6 self-assembly,7 and flow8 of granular materials. A large variety of nanoparticles with special shapes and properties have been synthesized in the past years.9,10 Self-assembly of such nanoparticles offers many opportunities for generating a spectrum of structures and functional materials.11 It is known that one important factor that governs the behaviors of these nanoparticle systems is the interaction forces between particles.12 In the past, the interparticle forces upon contact deformation have been studied by theoretical models13−17 or atomic force microscopy (AFM).18−24 To date, the most wellknown theoretical models are the Johnson−Kendall−Roberts (JKR)13 and Derjaguin−Muller−Toporov (DMT)14 models. Both models are based on an earlier analysis by Hertz,25 who considered two elastic bodies in contact under an external load but ignored interparticle attractive forces. The two models can be regarded as the specific limiting cases of a more general description, the Maugis-Dugdale model.15 All these models have been tested against the AFM measurements (e.g., the JKR model by Carpick et al.,26 the Maugis−Dugdale model by Lantz et al.,27 and the DMT model by Enachescu et al.).18 However, © XXXX American Chemical Society

since these theoretical models are derived based on continuum contact mechanics, some assumptions and/or approximations inherent in these theoretical models, such as the surface energy approximation, the geometry relationship between displacement and contact radius, and the breakdown within the last few nanometers of the bifurcation boundary of the JKR model, may lead to their failure at the nanoscale. Thus, it is necessary to test these models and study the contact forces between nanoparticles at the atomic level. In recent years, many efforts have been devoted to the direct measurement of normal and tangential forces between particles or between a tip and particle. For example, adhesion and rolling friction between silica microspheres were directly measured by AFM,19 and later a three-dimensional model was developed to calculate the normal and frictional forces between spherical silica particles from the measured deflections of an AFM cantilever.20 However, the accuracy of AFM measurements, especially in the lateral direction, is critically dependent on factors such as the sensitive spring constant, precise calibration of the cantilever, and methods of handling. Moreover, there are other sources of error due to contamination, water meniscus formation, drift, laser interference, and the coupling between Received: March 21, 2013 Revised: May 11, 2013

A

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Figure 1. Typical sequential snapshots of two nanospheres made of neutral silica engaged in a head-on collision, captured at the times of (a) 77 ps, (b) 85 ps, (c) 122.5 ps, (d) 128.5 ps, (e) 136 ps, (f) 144 ps, (g) 149.5 ps, (h) 257.5 ps, (i) total kinetic and potential energies, and (j) LJ potential evolution as a function of time. The red spheres denote oxygen atoms while (the larger) yellow ones silicon atoms. For this case, R = 2.006 nm, d = 4.0 nm, and Vr,0 = 25 m/s. Lennard−Jones (LJ) potentials between nonbonded atoms, the COMPASS force field also encompasses valence interactions, given by

vertical and lateral deflections. How to generate accurate measurements under different conditions is still a challenge in the research community. Computer simulation can be a powerful tool toward the quantification of interaction forces between nanoparticles due to its recent success under various conditions.28−31 In particular, by use of molecular dynamics (MD) simulation, the noncontact forces between silica nanoparticles, including the vdW attraction and Born repulsion, have been quantified.31 In this work, interparticle forces between two contacting silica nanospheres, including the vdW attraction, Born repulsion along with contact forces, will be examined by the same MD simulation techniques as reported before.31 The aim is to further understand the nature of these forces at the atomic scale and to formulate equations for estimating these forces for general application in discrete particle simulation. This paper is organized as follows: Simulation Method and Conditions describe the MD simulation method and procedures. Results and Discussion presents the numerical results and detailed analysis of internanoparticle forces, where the validity of the JKR and DMT models is also tested. The main findings are summarized in the last section.

Evalence = Eb + Eθ + Eφ + Eχ + Eb , b ′ + Eb , θ + Eb , φ + Eθ , θ ′ + Eθ , φ + Eb , θ , φ

(1)

where the terms on the right represent bond stretch (b), angle bending (θ), angle torsion (φ), out-of-plane angle inversion (χ), and the crosscoupling terms, including combinations of two or three valence terms, respectively. While the simulation method has been detailed in the previous study,31 the major simulation procedures are outlined below: (1) Silica nanospheres with different radii ranging from 1.0 to 4.0 nm are first carved out of bulk silica, followed by a relaxation process, using the NVT ensemble (i.e., constant number of atoms, constant volume, and constant temperature) at 298.0 K. (2) Then, two identical silica nanospheres are placed at a certain distance, followed by MD simulations using the NVE ensemble (i.e., constant number of atoms, constant volume, and constant energy), and a time-step of 1 fs (1.0 × 10−15 s) at an initial temperature of 298.0 K. The two fully relaxed silica nanospheres are allowed to move toward each other at an equal but opposite initial velocity. (3) Energies, forces, and other particle information are recorded every 100 steps in an output trajectory file and then correlated to the surface separation distance (d) between the two nanospheres. A typical sequence of a pair of colliding nanoparticles is shown in Figure 1 (panels a−h). The two nanospheres of radius 2.006 nm are given an initial relative velocity of Vr,0 = 25 m/s. The nanospheres initially move toward each other (Figure 1, panels a and b) and reach a proximity (Figure 1c) where jump-to-contact phenomenon can be

2. SIMULATION METHOD AND CONDITIONS The MD simulations are performed using the COMPASS force field,32 available in Materials Studio as reported before.31 Apart from the B

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Figure 2. Typical sequential snapshots of two nanospheres made of neutral silica engaged in a head-on collision, captured at the following times: (a) 7.0 ps, (b) 7.6 ps, (c) 8.0 ps, (d) 8.99 ps, (e) 9.6 ps, (f) 10.07 ps, (g) 10.36 ps, (h) 11.15 ps, (i) total kinetic and potential energies, and (j) individual LJ potential evolution as a function of time. For this case, R = 2.006 nm, d = 4.0 nm, and Vr,0 = 500 m/s. finally become almost flat, too. Since the ensemble used is NVE, the total energy always keeps unchanged. It is worth mentioning that although the kinetic energy of the system increases compared to the initial kinetic energy, the relative velocity of the two nanospheres gradually decreases to almost zero (Figure 1h). Figure 2i shows the energy evolution of the system at a high initial relative velocity of 500 m/s. The kinetic energy first remains almost constant and then decreases sharply to a minimum (close to zero) due to the short-range repulsive force, followed by a sharp increase, and finally becomes flat. As shown in Figure 2j, the individual LJ potential is separated from the total potential energy and the magnitude of the LJ potential first increases, followed by a restoring process back to its previous value. During this process, due to the dislocation of interacting atoms from two surfaces, the LJ potential fluctuates slightly. However, the LJ potential does not change much compared to the change of potential energy. That is, most of the decrease or increase of kinetic energy converts to or from the valence energy. Although interparticle potentials are independent of ensembles used (NVE or NVT) as discussed before,31 the controlling method of temperature was not introduced explicitly. It is worth mentioning that temperature and the average kinetic energy are related through the equipartition theorem: the relationship between temperature and the average kinetic energy is given by33

observed (reflected by an abrupt increase in relative velocity). Then, they collide with each other (Figure 1d). Figure 1 (panels e, f, and g) demonstrate the enduring contact process of detaching and attaching. That is, the two nanoparticles may attract or repel each other until they reach an equilibrium state shown in Figure 1h (surface separation d = 0.16 nm). However, if the initial relative velocity is high, the two nanoparticles will rebound after the first contact, as shown in Figure 2, panels a−h. In Figures 1 and 2, Vr is the relative velocity of the two nanospheres, with Vr > 0 showing the approach toward each other and Vr < 0 indicating the departure from each other. δn = −d + 2Δd (valid when r < 2R) is the normal displacement, where r is the center-to-center separation distance, R the particle radius, and Δd ≤ max{Rsurf i } − R, is the radial distance of surface atom i, as defined in our where Rsurf i previous study.31 δn = 0 corresponds to the point where d = 2Δd, and at this point, mechanical contact force begins to arise, so δn > 0 indicates contact and δn < 0 noncontact. As discussed in Sections 3.1 and 3.2, when d is about 0.2−0.4 nm for different particle radii, the surface atoms will be subject to rearrangement and/or retraction due to the repulsive force, and hence, the mechanical force begins to arise. Furthermore, once surface separation d decreases to about 0.125−0.15 nm, both vdW attraction and Born repulsion will experience a turning point and become “flat” to some degree with slight fluctuation. Figures 1 and 2 (panels i and j) demonstrate the energy dissipation mechanism of the systems at initial relative velocities of 25 and 500 m/ s, respectively. Note that the total energy includes the kinetic energy (KE) and potential energy, and the potential energy is the sum of the valence energy and LJ potential energy, as mentioned at the beginning of this section. As shown in Figure 1 (i and j), the kinetic energy of the system first increases sharply due to the attractive force, followed by fluctuations, and finally becomes almost flat with small fluctuations, when two particles reach an almost equilibrium state; meanwhile, the magnitudes of both the total potential energy and the individual LJ potential energy first increase drastically, followed by fluctuations, and

N

∑ i

Pi2 2m

= ⟨K ⟩ =

Nf kBT 2

(2)

where Pi is the momentum, Nf is the number of degrees of freedom, kB is the Boltzmann constant, and T is the thermodynamic temperature. Note that no matter what ensemble is used (e.g., NVT or NVE ensemble), in essence, both have the same dissipation mechanism, and temperature is controlled by the equipartition theorem. Moreover, temperature is a thermodynamic quantity and is largely meaningful only at equilibrium, but the interaction between two nanospheres is a C

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Figure 3. (a) Dependence of dmin on the product of RVr,0 and (b) critical initial relative velocity as a function of particle radius R. Note that R and Vr,0 are in units of nanometers and meters per second, respectively (the same applied to the whole paper).

LJ LJ Figure 4. (a) Minimum LJ interaction energy ELJ Min as a function of R and (b) normalized minimum LJ interaction energy, E /|EMin|, as a function of normalized separation distance d/σSi−O. The data for bulk are taken from the curve of Figure 16e.

nonequilibrium dynamic process in which the NVE ensemble is usually used.34 Hence, NVE is mainly used in the present MD study. The interaction energy, EInter, between two nanoparticles takes the form,30

EInter = E12 − E1 − E2 1

2

may not be in real contact because of the strong Born repulsion force; at a large Vr,0, the inertial force can overcome the repulsion force, leading to the contact deformation between the two particles. Therefore, to study the contact forces between silica nanospheres, a critical initial relative velocity is needed to initiate a real contact between two nanospheres. That is, an initial relative velocity larger than the critical one must be applied to create contact deformation. In the previous work,31 it has demonstrated that the minimum gap, dmin, where relative velocity becomes almost zero and starts to change direction (from positive to negative), linearly depends on both the initial relative velocity, Vr,o, and particle radius, R. On the basis of the MD results, the linear relationship between dmin and the product of the initial relative velocity and particle radius (i.e., RVr,o) is shown in Figure 3a, where dmin = 0.125−4.1 × 10−4 RVr,0. When Vr,0 = 0, the intercept at the Y axis is around 0.125 nm, which is in reasonable agreement with the equilibrium separation of 0.165−0.2 nm.31,35 When dmin = 0, RVr,0 = 304.9, which can be used to estimate the critical initial relative velocity. As shown in Figure 3b, the region above this curve means the two particles can be in real contact. It follows that the smaller the particle size is, the greater impact speed is needed to initiate such a contact, and when particle radius is large enough, the relative speed needed to initiate interparticle contact becomes almost zero. This is consistent with the common sense that for macroparticles, they can have some deformation once they are in contact, but for nanoparticles, an external energy/load must be input to make them contact or compact.

(3)

12

where E , E , and E are the potential energies of particle 1, particle 2, and the two particles, respectively. By differentiating the interaction potential energy with respect to surface separation d between nanoparticles, the particle interaction force can be obtained. That is, F Inter = −

∂EInter ∂d

(4)

This treatment can also be applied to the individual potential contributions (e.g., the vdW attraction or Born repulsion).

3. RESULTS AND DISCUSSION Under the present simulation conditions, the potential of collision between two nanoparticles and the resulting deformation both depend on the initial relative velocity and particle size. As an extension of the processes described in Section 2 and the previous work,31 the relationship between the minimum gap dmin and the product of initial relative velocity, Vr,0, and particle radius, R, will first be discussed to generate some background for the discussion of interparticle forces in this section. In the system considered, two particles are set to move toward each other at a given initial relative velocity, Vr,o, with their kinetic energy damped mainly by converting into valence energy as treated in Section 2. At a small Vr,o, two nanoparticles D

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Figure 5. The vdW attraction and Born repulsion forces between silica nanospheres of radius 3.063 nm for the approach and departure processes as a function of the surface separation, d = r − 2R, where r is the center-to-center separation, R is the nanoparticle radius at Vr,0 of (a) 25 m/s and (b) 500 m/s. They are corresponding to Figures 1 and 2, respectively.

As described before,31 the interparticle LJ potential varies with separation, d, and there is a minimum in the interparticle LJ potential at the so-called equilibrium separation at around d = 0.2 nm. The relationship between the minimum LJ LJ and particle radius R is further interaction energy EMin investigated as part of the present study. As shown in Figure 4a, the magnitude of ELJ Min linearly increases with R, described by −20 − 1.505 × 10−19R. The plot of the ELJ Min(J) = 3.355 × 10 LJ LJ normalized E /|EMin| for different particle sizes as shown in Figure 4b are almost consistent with each other for nanospheres, although there is some discrepancy for bulk (theoretically obtained when R is infinitely large). It can also be observed that there is an energy barrier between the equilibrium separation (d = 0.2 nm) and d = 0, evidence to confirm that an external energy must be input to make nanoparticles contact. Therefore, to effect the contact between nanoparticles and deformation, large initial relative velocities are used for most of the reported simulations in this work. Moreover, there is a minimum gap dmin or maximum penetration, although it is not explicitly discussed in the force analysis below. Note that the initial relative velocities used in the present study are much higher than what was usually employed in an experiment, although they may occasionally be found in some special studies.38 They are used here in order to generate contact and deformation, so that the resulting interparticle forces can be quantified. Such high relative velocities have also been employed in other MD simulations in the literature.28,36,37 As indicated in our previous study,31 however, they do not affect the resulting relationship between forces and deformation, the final outcome of the present study. 3.1. vdW Attraction and Born Repulsion Forces. In the previous work,31 the effects of several simulation parameters, including interaction path, initial relative velocity, and relative orientations of two particles on interparticle potentials, have been studied, showing that in most cases, interparticle potentials are independent of these parameters. Their effects on the interparticle vdW attraction and Born repulsion forces after two particles’ contact are also examined as part of the present study. The two forces in the approach and departure process are separately obtained, as shown in Figure 5. The results show that interaction path has almost no impact on the interparticle forces before contact (Figure 5a), but there are small differences between the two curves after contact (Figure 5b). However, in general, the differences are small and can be

ignored. That is, the interparticle vdW attraction and Born repulsion forces are mostly independent of the interaction path. Note that when d becomes negative, for convenience of discussion, the concept of penetration depth is often used instead of surface separation. It can be observed from Figure 5 that when d approaches zero, both the attraction and repulsion forces are close to a finite value, but not infinity; moreover, their magnitudes do not increase sharply, as predicted by the Hamaker approach, but become flat to some degree. This discrepancy at close separation has been pointed out in the AFM measurement.24 Hence, the conventional Hamaker approach is not adequate at close separation. This can be attributed to the fact that on one hand, the continuum Hamaker approach treats the particle as a rigid (incompressible) body, implying that in close contact the particle will not deform and the forces will increase sharply and infinitely and on the other hand, the reality is the surface atoms bear flexibility and space of movement since the structure is not only “soft” but also discrete, therefore the forces will not increase so drastically as predicted by the continuum model. As mentioned in our previous work,31 due to the different bonding state and chemical environment, with a decrease in particle size, the surface atoms represent an increased contribution and tend to affect the behavior of particles. During a head-on dynamic impact, the surface atoms may be subject to some structural change. Figure 6 shows the influence of relative orientations of two particles on interparticle forces. Prior to the application of an initial velocity and simulations,

Figure 6. vdW attraction and Born repulsion forces between two silica nanospheres of 3.063 nm in radius with different relative orientations at Vr,0 = 500 m/s. E

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Figure 7. Effect of initial relative velocity, Vr,0, on (a) the vdW attraction and Born repulsion forces between silica nanospheres of radius 3.063 nm and (b) the LJ potentials (i.e., the sum of the vdW attraction and Born repulsion potentials, as a function of surface separation d).

Figure 8. The vdW attraction and Born repulsion forces between silica nanospheres of radius in the range of 0.975 to 4.120 nm obtained in the approach process at Vr,0 = 500 m/s. The solid lines represent proposed formulas that correspond to the symbol colors.

be ignored. In the following, interparticle forces were collected from the approach process at Vr,0 = 500 m/s. Figure 8 shows the interparticle forces between silica nanospheres of different radii, ranging from 0.975 to 4.120 nm, in the approach process at a high initial relative velocity. It is evident that the vdW attraction and Born repulsion forces depend on particle size. It can be observed that their magnitudes increase with penetration depth. When d > 0.15 nm, their magnitudes increase sharply with the decrease in d: for a given d, the magnitudes of interparticle forces increase with particle size. Upon further increase in penetration depth, both forces become scattered, leading to less size dependence. As will be further discussed in Section 3.2, when two particles are less than 1 atomic layer apart, the two particles are almost in contact. Therefore, the magnitudes of both forces increase steeply. However, upon further penetration, due to the dislocations of contacting atoms on the opposing surfaces, the shape of particles becomes increasingly flattened, hence, the interacting surface becomes less curved, and size dependence becomes less evident. In the previous work,31 modified equations have been proposed for the vdW attraction and Born repulsion forces at d ≥ 0 nm. In the present work, at d ≤ 0.15 nm, the forces FvdW Modified or FBorn Modified at d = 0.15 nm calculated according to the modified equations are used as a reference (i.e., the ratio of or Born vdW or Born FvdW MD, d ≤ 0.15 nm/FModified, d = 0.15 nm is employed) and two formulas have been proposed to describe the vdW attraction and Born repulsion forces, given by

one nanoparticle is kept stationary while the other one is rotated clockwise, along one of its vertical central axes by an angle of 0°, 90°, 180°, and 270°, so different configurations of two particles are achieved. In fact, changing relative orientations herein is equivalent to altering interacting surfaces of two particles and hence surface roughness. The results show that at close contact but before large flattening, there is a small difference among the four curves. However, both the vdW attraction and Born repulsion forces are largely independent of relative orientations of two particles (i.e., surface roughness). Generally, an initial relative velocity, Vr,0, is needed to study contact forces. However, whether the external load can affect simulation results or not is yet unknown. In this part, a series of initial relative velocities are employed to examine the effect of Vr,0 on interparticle forces (Figure 7). The results show that different initial relative velocities give rise to different penetration depths: the higher the initial relative velocity, the larger degree of the flattening of the two nanospheres (Figure 7a). Moreover, both the vdW attraction and Born repulsion forces are largely independent of Vr,0 prior to contact but become scattered and vary slightly due to the dynamic effect after contact. This could be ascribed to the displacements of contacting atoms on the opposing surfaces,39 such as the thermal vibrations and/or dislocations of interacting atoms from the two opposing surfaces. These can be reflected from our simulation results (Figure 18): with an increase in contact deformation, the contact radii determined from the MD simulations become gradually larger than the Hertz prediction. However, to quantify interparticle forces, the effect of Vr,0 could F

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vdW vdW 1.2 FMD, d ≤ 0.15nm / FModified, d = 0.15nm = 1 + 0.20(1 − d /0.15)

(5)

and Born Born 1.2 FMD, d ≤ 0.15nm / FModified, d = 0.15nm = 1 + 0.40(1 − d /0.15)

(6)

The combination of eqs 5 or 6 and the previously obtained modified equations, which are summarized in eqs 7 and 8, allows for the calculation of the corresponding vdW interaction force or the Born repulsion force between two silica nanospheres over the whole range of separation distance. As shown in Figure 8, the so-calculated interparticle vdW attraction and Born repulsion forces can match the MD simulated results reasonably well.

vdW FMD

⎧ −4d 0.6 3.16(rms/ R )0.18 vdW )e FH , d ≥ 0.4nm ⎪(1 + 26e ⎪ 3.2(0.4 − d)0.65 vdW FModified, d = 0.4nm , ⎪e = ⎨ 0.15 ≤ d < 0.4nm ⎪ ⎪[1 + 0.2(1 − d /0.15)1.2 ]F vdW Modified, d = 0.15nm , ⎪ ⎩ d < 0.15nm (7)

and

Born FMD

⎧ −6d 0.5 2.09(rms/ R )0.10 Born FH , )e ⎪(1 + 701.6e ⎪ d ≥ 0.4nm ⎪ ⎪ 4.65(0.4 − d)0.75F Born Modified, d = 0.4nm , = ⎨e ⎪ 0.15 ≤ d < 0.4nm ⎪ Born ⎪[1 + 0.4(1 − d /0.15)1.2 ]FModified, d = 0.15nm , ⎪ ⎩ d < 0.15nm

FvdW H

Figure 9. (a) Interparticle vdW attraction and (b) Born repulsion forces as a function of d between silica spheres of different radii R. Note that “negative” represents attraction forces here and lines 1 to 6 correspond to particle radii of 1 to 5000 nm, respectively.

bond stretch, angle bending, angle torsion, angle inversion, and there is always a force constant to describe these terms in their individual expressions.41 These valence interactions should be related to macro-mechanical properties, leading to mechanical contact force related to contact deformation. In the following, this contact force Fc, as a collected outcome, is calculated by

(8)

FBorn H

where and are the vdW attraction and Born repulsion forces calculated from Hamaker equations, respectively, and rms = 0.064 nm is the average surface roughness.31 In formulating the above equations, an attempt has been made to ensure that the above equations are continuous and can be applied to nanospheres with a radius much larger than 4 nm. Using eqs 7 and 8, the corresponding interparticle forces between two silica nanospheres over a wide range of separation distance can be calculated as displayed in Figure 9. It can be observed that both vdW attraction and Born repulsion forces increase with the decrease in separation distance, and the larger the particle size, the higher the magnitudes of the forces. This is consistent with the general understanding of the two opposing forces. In accordance with the continuum JKR and DMT models, the contribution of intermolecular forces to the total contact forces between macroparticles (i.e., adhesion forces) is usually treated by the surface energy approximations.40 This treatment is known to be effective for macroparticles. However, at the nanoscale, the intermolecular force and contact force are quantitatively comparable; an error may be introduced using surface energy approximations. Therefore, eqs 7 and 8 can provide an alternative to estimate the interparticle vdW attraction and Born repulsion forces for silica nanospheres. 3.2. Contact Force. As mentioned in Section 2, apart from the LJ interactions in the forcefield COMPASS, the computation still encompasses valence interactions such as

Fc = Fn − FvdW − FBorn

(9)

where the total normal force Fn is evaluated by accumulating the horizontal forces that all atoms from one nanoparticle exerting on the other nanoparticle during the head-on dynamic impact, FvdW and FBorn, are obtained by differentiating interaction potential energies with respect to surface separation d, according to eq 4. Note that in the present work, the mechanical contact forces arise when surface separation d is about 0.2−0.4 nm, which is 0.5−1.0 times one diameter of atom (σSi−O = 0.407 nm). This can be explained in two aspects: (i) the surface atoms are in contact before the geometrically defined contact takes place, a reason why we need to introduce Δd when discussing Figures 1 and 2 and (ii) the repulsion force between atoms may cause the rearrangement and/or retraction of surface atoms and hence deformation on the two contacting surfaces.42 This leads to the fact that until surface separation d decreases to about 0.125− 0.15 nm (Figures 3 and 8), both vdW attraction and Born repulsion forces experience a turning point where both become less steep and begin to fluctuate slightly. It is again important to examine the effects of the interaction path, relative orientations of two nanospheres, and initial relative velocity on the contact forces. The results are shown in Figures 10−12, respectively. Since the hysteresis phenomenon is often experimentally observed in the force−displacement G

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measurements,43,44 the effect of the interaction path is examined (Figure 10). The results show that there is a small

The effect of initial relative velocity on contact force is also studied (Figure 12). It can be seen that the contact forces

Figure 10. Contact forces Fc between silica nanospheres of radii 2.006 and 3.063 nm obtained in the approach and departure processes as a function of normal displacement δn at Vr,0 = 500 m/s.

Figure 12. Contact forces Fc between silica nanospheres of radius 3.063 nm obtained when Vr,0 = 300 and 500 m/s, respectively.

hysteresis in the contact forces obtained in the approach and departure processes. This is different from the cases of the vdW attraction and Born repulsion forces, which are almost independent of the interaction path. However, this work is to evaluate the validity of the Hertz model in simulating nanoparticle dynamic behaviors using discrete element simulations. The effect of the interaction path on the contact force is usually ignored when formulating an equation to calculate the contact force. In fact, as recently demonstrated by Zheng et al.,29 the hysteresis effect can implicitly be taken into account in implementing such an equation in a discrete element simulation. Surface roughness is known to influence adhesion and friction forces.19,20 To study the effect of surface roughness, contact forces between two nanospheres with different relative orientations are calculated. The results in Figure 11 show that

obtained in the approach process at initial relative velocities of 300 and 500 m/s are almost consistent with each other. Similarly, those obtained in the departure process also agree well with each other. Therefore, within the velocity range considered here, the effect of initial relative velocity on contact forces can be largely ignored. A high KE impact (up to 10 ev/ atom)45 may create a high pressure inside the nanoparticles and hence affect the apparent contact properties. However, the KE in our simulation is relatively small (less than 0.04 eV/atom), hence the effect of compressive stress on mechanical properties can be ignored within the range of impact velocity considered in this study. It is also worth mentioning that the initial relative impact velocity can be related to KE (in eV/atom) by KE = 5.2 × 10−9 mV2r,0, where m is the atomic mass in amu (atomic mass unit) and Vr,0 initial relative impact velocity in m/s. Furthermore, as discussed in Section 2, the average kinetic energy is related to the temperature of a system, according to the equipartition theorem. Strictly speaking, kinetic energies are related to the absolute rather than relative velocities. However, the initial relative velocity is largely proportional to the KE in our case. Therefore, a similar conclusion may be drawn that temperature has a negligible impact on contact forces. In the previous work,31 the effect of temperature on the LJ force (i.e., the total of vdW attraction and Born repulsion forces) has been studied, showing that LJ force is largely independent of temperature. In the present work, to directly examine the effect of temperature, several MD simulations based on a small silica nanosphere of radius 2.006 nm are performed. The results in Figure 13 show that the contact forces obtained at temperatures ranging from 1 to 300 K are almost identical, indicating that contact forces are indeed independent of temperature. With the background established based on the results above, the size dependence of contact forces between silica nanospheres can now be investigated. The contact force predicted in accordance with the Hertz model25 is given by 4 Fc = E∗ R∗ δn3/2 (10) 3

Figure 11. Contact forces Fc between silica nanospheres of radius 3.063 nm with different relative orientations at Vr,0 = 500 m/s.

the contact force is independent of relative orientations between two silica particles. The reasons for it are (i) the surface roughness arises only from atomic discrete structure as discussed in the previous work,31 which is only about 0.065 nm and can be ignored, and (ii) the nanospheres produced in the present study are very symmetrical, so different orientations should not change the surface atoms much.

where δn is the normal displacement, 1/E* = (1 − ν21)/E1 + (1 − ν22)/E2 and 1/R* = 1/R1 + 1/R2, where R1 and R2 (R1 = R2 = R in the present study) are the radii of particles 1 and 2, respectively. For silica, the experimental value of Young’s H

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Figure 15. Fc/(8R2E*/3) as a function of strain δn/(2R), obtained in the approach process from MD simulations for different sized silica nanospheres. Solid line represents the Hertz prediction.

Figure 13. Contact forces Fc between silica nanospheres of radius 2.006 nm at different temperatures at Vr,0 = 100 m/s. The contact force obtained by using the NVE ensemble at Vr,o = 500 m/s is also displayed for comparison.

radii in the range of 0.975 to 4.120 nm. It can be observed that for the case of 0.975 nm, the simulated contact force is slightly smaller than those predicted by the Hertz model, but for the other three cases, the contact forces are comparable to those predicted from the Hertz model. These findings agree well with those in the literature:47 Young’s modulus of nanoparticles decreases with particle size, especially for R < 1 nm. However, when particle size is greater than 2 nm in radius, this trend is not obvious. This may be because on one hand, the deviation from the bulk Young’s modulus is less than 10% or even smaller with an increase in particle size,47 and on the other hand, in reality, the coordination number of surface atoms may increase or even become greater than the bulk material upon deformation;48 hence, the dynamic effect may be important, which actually was ignored in the reported work.47 Nonetheless, the contact forces at low compression are in reasonable agreement with those predicted by the Hertz model using Young’s modulus of silica bulk. But at high compression, contact forces obtained from the MD simulations can deviate from the Hertz model, implying that the Hertz model only holds at low compression where the contact force can be considered to be elastic. A similar phenomenon has also been observed from finite element analysis because of the viscoelastic or even plastic effect.29 As for the deviation from the Hertzian prediction at high compression, it is also observed in experiments49,50 or

modulus E(=E1 = E2) is 72.5 GPa, and Poisson ratio ν(= ν1 = ν2) is 0.1746(i.e., E* = 37.3 GPa). To test the accuracy of the force field COMPASS for silica, the Young’s modulus of silica bulk is calculated by MD simulations on a simulation cell of L × M × M (L = 8.0 nm, M = 6.0 nm) with the periodic boundary condition (PBC) implemented, as schematically shown in Figure 14a. After geometry optimization, the MD simulation is first conducted using a NVT ensemble at 298.0 K, running for at least 50.0 ps after equilibration. The fully relaxed structure in the final frame is then exported for MD simulations using a NPT ensemble (i.e., constant number of atoms, constant pressure, and constant temperature) at 298.0 K for at least 50.0 ps, following geometry optimization. A series of external pressures are applied to establish a compressive stress−strain curve (Figure 14b). With an increase in stress, the strain first increases linearly and then continues increasing gradually. The slope of the initial linear part, according to Hooke’s law, is used to derive Young’s modulus. The resulting value of 73.0 GPa agrees well with the experimental result of 72.5 GPa,46 with only a 0.69% deviation. This agreement indicates the effectiveness of the COMPASS force field for silica. Figure 15 shows the size dependence of the normalized contact forces on strain between silica nanospheres of different

Figure 14. (a) Simulation model used to measure Young’s modulus of silica counterpart bulk. The yellow atoms represent silicon atoms (larger ones), while the red ones denote oxygen atoms. The arrows indicate the direction of applied external pressure and the solid blue lines represent PBC. (b) Compressive stress−strain curve of silica bulk. I

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computer simulations,48,51 although they are based on silicon nanoparticles. This phenomenon is usually interpreted as a result of high internal pressure49 or the creation of new phase with a higher Young’s modulus.48 If the rate of loading is sufficiently slow for the stress to be in the statistical equilibrium with the external loads at all times, the dynamic effect could be ignored. In practice, without a high external load, the large deformation of nanoparticle seldom happens and is usually ignored. Therefore, this work only concentrates on the contact force at relatively low compression. 3.3. Comparison with the JKR and DMT Models. As mentioned in Section 1, the JKR13 and DMT14 models are the most well-known theoretical models to treat the contact between two particles. Both models are developed based on the Hertz model,25 which considers two particles’ contact in external load but ignores interparticle attractive forces. In the JKR model, it is assumed that surface forces act only within the contact area.16 The total normal force comprises a deformation contribution, according to the Hertz model, and an adhesion component due to the surface energy, governed by the following two equations,16,17 δn =

Fn =

2 aJKR

R*

−

3 4E∗aJKR

3R∗

2 3 −

πaJKR W E*

(11)

3 8πWE∗aJKR

where the work of adhesion per unit area W is given by W = γ1 + γ2 − γ12

(12) 35

Figure 16. Sequential snapshots of simulation system used to measure the work of adhesion, W, between two silica planar surfaces at different times of (a) 0.25 ps, (b) 10.75 ps, (c) 13.0 ps, and (d) 20 ps and (e) the LJ potential as a function of d between two silica cubes, with the inset showing the corresponding LJ force.

(13)

where γ1, γ2, and γ12 are the surface energies of the two solid surfaces 1 and 2, and the interfacial energy, respectively. For the same materials, γ1 = γ2 = γ, γ12 = 0, W = 2γ.13,35 On the other hand, in the DMT model, it is assumed that the attractive forces act only along the contact area.16 The total normal force consists of the contact force and adhesion force, given by16 Fn =

4 ∗ ∗ 3/2 E R δn − 2πR∗W 3

of about 0.059 J/m2, according to WBulk = ELJ Min/(2A), where A is contact area of 3.600 × 10−17 m2. This value is in reasonable agreement with experimental measurements which range in 0.05−0.08 J/m2 for bulk materials.52,53 Note that as observed from the inset of Figure 16e, the interparticle LJ force between two planar surfaces at close separation also varies with surface separation, but after d < 0.20 nm, the slope of the LJ force becomes less steep to some degree. By applying the yielded work of adhesion per unit area to eqs 11 and 12 or 14, the total normal forces can be predicted. As shown in Figure 17, the results are in reasonable agreement with the MD simulations. However, there is some discrepancy between the MD simulations and those predicted by the JKR and DMT models. In general, the MD results are larger than those predicted by the JKR models, and the DMT predictions are closer to the MD simulated results. The MD data are also included in the figure. They are not so smooth, different from the calculated results, according to the JKR and DMT models. This is because the two models are theoretical, but the MD data are obtained from numerical experiments which, although reproducible, suffer from “errors” related to different conditions or settings for simulations for different points. There are various reasons for the observed discrepancy in Figure 17. But it is difficult to identify them clearly because the MD simulation and the JKR and DMT models are developed based on different assumptions and at different time and length scales. The MD simulation is conducted at the atomic or molecular scale, while the other two models are continuum-

(14)

The relationship between normal displacement and contact radius in the DMT model is the same as that in the Hertz model (i.e., aDMT = aHertz), given by16 δn =

2 aDMT R∗

(15)

The validity of the JKR and DMT models, when applied to silica nanospheres, is examined in this work. In this connection, the work of adhesion of bulk is first evaluated using the MD simulations (Figure 16). Two silica cubes with dimensions of 60.0 × 60.0 × 60.0 Å are used to approximately represent two bulks whose surfaces are usually planar. MD simulations are performed on a simulation cell of 240 × 60 × 60 Å with PBC applied in the two horizontal directions, involving two silica cubes, initially separated by a distance of 60 Å, followed by a head-on collision as done for nanospheres. The results show that the trends of the interparticle LJ potential and force are similar to those between two nanospheres, and with a decrease in surface separation, both the potential and force first decrease to a maximum negative value and then increase gradually. The corresponding maximum negative value of the LJ potential is about 4.245 × 10−18 J, thus giving rise to the work of adhesion J

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Figure 17. Comparative studies of total normal forces Fn between silica nanospheres of different radii ranging from (a) 0.975, (b) 2.006, (c) 3.063 to (d) 4.120 nm obtained from the MD simulations, JKR, and DMT models, respectively. Note that different scales are used in this figure.

Another reason for the discrepancy between the MD, JKR, and DMT results lies in the calculation of the contact radii. With the silica nanosphere of radius 4.120 nm as an example, the contact area AMD and contact radius aMD are respectively given by the following equations in the MD simulation:

based and at a much larger scale, effected with different assumptions which may be material dependent. To find the origin of the discrepancy, the following discussion is made by directly comparing the terms in the models and based on the MD simulations. With the DMT model as an example, the two terms in the right of eq 14 have direct physical interpretations: the first term corresponds to the contribution of contact force, while the second term corresponds to the contribution of intermolecular force (i.e., the LJ force). In the model, the first term is the outcome of the Hertz model which gives results comparable to the MD simulation, particularly when the deformation is small; but the contribution of intermolecular force is reflected by the adhesion force, which is constant and evaluated by surface energy approximation.40 In reality, the LJ force varies with contact deformation or penetration depth, as demonstrated in Figure 16e. But this variation is not taken into account in the DMT model, leading to some differences. This consideration also applies to the JKR model, although it is not so explicit. It should be noted that there seems to be controversy about treating the adhesion force as a constant in the continuum models in the literature. The two spherical surfaces of macroparticles at a small separation usually can be treated as planar surfaces according to the Derjaguin approximation,55 since it satisfies R ≫ d, but for the curved spherical nanoparticles, particle size and separation distance are usually comparable in magnitude so that the Derjaguin approximation fails. This problem is hidden for macroparticles because for these particles, the magnitude of mechanical contact force is usually much larger than the intermolecular LJ forces.

′ )/2 AMD = (Stotal − Stotal

(16)

aMD =

(17)

AMD /π

where Stotal and S′total are the calculated total vdW surface areas of two silica nanospheres before and after contact, respectively. Meanwhile, the contact radii aJKR and aDMT calculated from the JKR and DMT models using eqs 11 and 15 are plotted against δn. As observed from Figure 18, since the JKR model is developed for soft materials with high surface energy, the contact radii calculated from the JKR models is not zero even before δn = 0 nm, due to the attractive force. At a small normal displacement, the predictions from the JKR and DMT models are in reasonable agreement with the MD simulated results. However, with further increase in penetration depth, the contact radii deviate from the continuum models, and the MD radius is higher. The contact radius changes by a factor of less than 1.5, which is comparable to that reported,54 and this difference is usually ascribed to the surface roughness arisen from the atomic discrete structure.

4. CONCLUSIONS MD simulations have been employed to study normal forces between two contacting silica nanospheres, including the vdW K

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel: +61-2-9385-4429. Fax: +61-29385-5956. Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS The work is financially supported by the Australian Research Council (ARC).

■

Figure 18. Contact radii a between silica nanospheres of radius R = 4.120 nm based on JKR and DMT (or Hertz) models and MDsimulated results with an error bar at an initial relative velocity of Vr,0 = 500 m/s, with different relative orientations.

REFERENCES

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attraction, Born repulsion, and mechanical contact forces. It is shown that two nanoparticles approaching each other may have contact with each other, depending on the external energy input. For a given particle size, the minimum gap achieved decreases with the increase of their initial relative velocity. The critical initial relative velocity needed to initiate two particles’ contact is inversely proportional to particle size. The larger the particle radius is, the smaller the critical initial relative velocity. The MD simulations confirm our common understanding that macroparticles can readily collide with each other but may not be so for nanoparticles. The vdW attraction and Born repulsion forces between two silica nanospheres are, to a large degree, independent of interaction path, relative orientations, initial relative velocity, and temperature. Both forces vary in a regular pattern at small deformation and demonstrate obvious size dependence, but at large deformation both forces become flat and scattered. Such behavior cannot be described by the conventional models. On the basis of the MD results, two equations have been put forward to estimate the vdW attraction and Born repulsion forces. The MD simulations demonstrate that the Hertz model can largely describe the mechanical contact force between nanospheres at low compression. However, Young’s modulus may become slightly smaller for very small nanospheres (R < 1 nm). When the mechanical contact, vdW attraction, and Born repulsion forces are coupled together, the interactions among atoms are complicated. The JKR and DMT can be used as the first approximation. The present equations can also be used as an alternative, which, obtained at the atomic scale, should be more reliable for silica nanospheres. The differences between the MD and JKR or DMT results are demonstrated to result from the treatments of surface energy approximation and contact radius in the JKR or DMT model, which should be used with care in the general application. In the past two decades or so, discrete particle simulation has increasingly been used to study the dynamics of particle systems.8 However, such studies are found to be difficult for nanoparticles because the interparticle forces cannot properly be determined. The MD approach proposed offers a method to overcome this difficulty. In fact, the equations formulated in this study can be used in the dynamics simulation of silica nanoparticles. Their application in the study of the packing and flow of silica nanospheres will be reported in the future. L

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M

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