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Cite this: Phys. Chem. Chem. Phys., 2014, 16, 5846
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Interaction forces between a spherical nanoparticle and a flat surface† Weifu Sun Due to the breakdown of Derjaguin approximation at the nanoscale level apart from the neglect of the atomic discrete structure, the underestimated number density of atoms, and surface effects, the continuum Hamaker model does not hold to describe interactions between a spherical nanoparticle and a flat surface. In this work, the interaction forces including van der Waals (vdW) attraction, Born repulsion and mechanical contact forces between a spherical nanoparticle and a flat substrate have been studied using molecular dynamic (MD) simulations. The MD simulated results are compared with the Hamaker
Received 3rd December 2013, Accepted 30th January 2014
approach and it is found that the force ratios for one nanosphere interacting with a flat surface are
DOI: 10.1039/c3cp55082f
separate formulas have been proposed to estimate the vdW attraction and Born repulsion forces between
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a nanosphere and a flat surface. Besides, it is revealed that the mechanical contact forces between a spherical nanoparticle and a flat surface still can be described by the continuum Hertz model.
different from those for two interacting nanospheres, both qualitatively and quantitatively. Thus two
1. Introduction It is known that the continuum models such as the Hamaker approach1 and the Hertz continuum model2 do not generally apply at the nanoscale level due to the neglect of the atomic discrete structure,3,4 surface effects, the underestimated number density of atoms or the neglected intermolecular forces. On the one hand, the continuum Hamaker approach treats the particles as rigid (incompressible) bodies without considering the atoms’ vibration, bond’s torsion, inversion, etc., implying that the particles will not deform and this is contrary to reality; on the other hand, in the Hamaker approach, particles are assumed or treated as a continuum medium with a uniform (density) distribution of atoms. But in essence, the structure of a nanoparticle is not only ‘soft’, but also discrete. Although in the continuum model, the interaction between a spherical particle and a flat surface can be considered as a limiting case of interactions between two spherical particles according to the Derjaguin approximation,5 at the nanoscale, because the size of nanoparticle R is comparable to the surface separation d, thus the Derjaguin approximation breaks down since it does not satisfy R c d. What is more, one should bear in mind that even at the macroscale, the formulas governing interactions between one sphere and a flat surface are different from those
School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia. E-mail: [email protected], [email protected] † Electronic supplementary information (ESI) available. See DOI: 10.1039/ c3cp55082f
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describing interactions between two spheres. To date, the interaction forces between a spherical nanoparticle and a flat surface is still unknown, although some effort has been made to model atomic force microscopy (AFM) measurements using a tip and a substrate, in which attention is paid to the contact force only through indentations.6,7 Nowadays, a full quantitative analysis of inter-nanoparticle forces is still challenging for experimental techniques at the nanoscale with particle size less than 10 nm.8 The accuracy of the AFM measurement, especially in the lateral direction, is critically dependent on factors such as sensitive spring constant, precision calibration of the cantilever, etc. In contrast, molecular dynamics (MD) simulation can circumvent these demerits and has proven to be an effective tool for calculating interaction forces between nanoparticles,9–12 but whether these modifying formulas can be extended to interactions between a nanosphere and a flat surface or not is still unexplored. Here efforts will be devoted to formulate interaction forces between a spherical nanoparticle and a flat substrate using a similar previously-established procedure10–12 with a view to further testing the continuum models, such as the Hamaker approach and the Hertz model. A silicon/diamond structure is of great interest in a variety of applications13 such as anodes for lithium-ion cells,14 solar cells,15 chemical sensors,16 biosensors,17 drug delivery,18 super-capacitors and hydrogen storage,19 catalysis,20 nanoelectronics,21,22 silicon metal-oxide-semiconductor field-effect transistors23 and microchips in semiconductors, integrated circuits and field-emission devices,24 and also micro/nano electromechanical systems (MEMS/NEMS) building upon the existing silicon infrastructure to create micro/nanoscale machines.25–27
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With the continuing development of silicon-based ultra large scale integrated (ULSI) circuits, an exponential growth of the number of devices on a chip has been projected by Moore’s law. However, the reduction in the device size results in increased parasitic leakage currents and power dissipation at the device level. To overcome these adverse effects, to enhance the operation speed and to lower the power consumption of chips, silicon-on-diamond (SOD), an alternative concept of silicon-on-insulator, becomes an ever-increasing promising candidate, in which the thermally insulating silica has been replaced by highly thermally conductive diamond. SOD is a substrate engineered to address the major challenges of siliconbased ULSI technology. Experimentally, a thin, single crystalline Si layer is deposited on a highly oriented diamond layer that serves as an electrical insulator, a heat spreader and a supporting substrate.28,29 Therefore, the analysis of the interface properties such as interfacial structure30 and interface forces is important for the design of MEMS/NEMS devices, and MD simulations will be performed in this work to study interfacial interactions between a silicon nanosphere and a diamond flat substrate.
2. Simulation conditions and method The simulations are performed in a similar way as done before.10,11 The interatomic interaction parameters are taken from the COMPASS force field,31 which encompasses a LennardJones n-m potential (LJn-m, n = 9 and m = 6) function between non-bonded atoms, given by32 Eij ¼
m X m sij n n sij eij n m n m r rij ij i;j
(1)
where atoms i and j are separated by a distance rij, sij is the collision diameter of atoms i and j at which the force becomes zero, and e is the potential well depth of interaction between atoms. The cut-off distance used in the MD simulations is set to be as large as 100 nm to completely avoid the errors introduced by the cutoff distance. From the perspective of quantum mechanics, the electron correlations are governed by solving more ¨dinger accurate and rigorous relationships such as the Schro equation;33 but in MD simulations, this interaction is represented by the short-ranged repulsion, also called the Born repulsion which originates from the strong repulsive forces between atoms as their electron shells interpenetrate each other.34 In the COMPASS force field, the Born repulsion is embodied in the Lennard-Jones (9-6) potential function by the 9-powered part.31 Apart from the LJ potentials between non-bonded atoms, COMPASS also encompasses valence interactions such as bond stretching, angle bending, angle torsion, angle inversion, etc., between bonded atoms, taking the form of Evalence = Eb + Ey + Ej + Ew + Eb,b + Eb,b 0 + Eb,y + Eb,j + Eb,j + Eb,y,j (2)
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where the terms on the right represent bond stretching (b), angle bending (y), angle torsion (j), out-of-plane angle inversion (w), and the cross-coupling terms including combinations of two or three valence terms, respectively. The values of si and sj for silicon and diamond-like carbon are 4.45 and 3.854 Å, respectively,31,35 the values of ei and ej for silicon and diamond-like carbon are 0.198 and 0.062 kcal mol1; therefore the Hamaker constant (A = p2C/v2, where C = 3es6 is the vdW attraction interaction parameter, v = 4p(s/2)3/3 is the atom volume estimated in a way inherent in the Hamaker approach) can be, respectively, estimated to be about 1.4848 1019 and 4.65 1020 J, which is close to experimental values of 1.9 or 2.17 1019 J.36 Therefore, the Hamaker constant of diamond and silicon interacting across air/vacuum is about ASi–C E (ASiADiamond)0.5 E 8.3 1020 J. For a pair of dissimilar atoms, sij is governed by the Waldman– Hagler combination rule37 in terms of the sSi and sC values of the individual silicon and carbon atoms, that is sSi–C = [(sSi6 + sC6)/2]1/6, thus giving rise to sSi–C = 4.204 Å. The simulated system consisting of one silicon nanosphere and the diamond substrate is first fully relaxed using the NVT (i.e., constant number of atoms, constant volume and constant temperature) ensemble at 300 K. Then, the fully relaxed silicon nanosphere is allowed to move towards the diamond substrate at an initial atomic velocity using a NVE ensemble (i.e., constant number of atoms, constant volume and constant energy) at an initial temperature of 300 K. The bottom layer of the diamond substrate is fixed during the MD simulations. Numerical integration is performed using the velocity Verlet integration algorithm with a time step Dt = 1.0 1015 s (1 fs). Frames are output every 100 steps into the trajectory file. The forces, coordinates, velocities of atoms and energies, etc. as a function of the shortest surface-to-surface separation distance (d) between the two components are recorded in an output trajectory file. Note that in our case, in order to ensure a normal impact (that is central collision), 5.0 Å ps1 (i.e., 500 m s1) along the Y-axis of each atom in the nanosphere was specified while 0 Å ps1 was specified in the other two directions. The silicon nanosphere of 1.0, 2.0, 3.0 and 4.0 nm radius contains 220, 1707, 5707 and 13 407 atoms, respectively. The dimensions of a diamond flat substrate are 10 nm length, 10 nm width and 2 nm thickness, containing 36 742 carbon atoms. After running the MD simulations using the NVE ensemble, the properties of atoms such as coordinates, forces and velocities along each direction (i.e., X-, Y- or Z-axis) can be separately obtained from the trajectory files. A typical sequence of the silicon nanosphere colliding with the diamond substrate is shown in Fig. 1. The nanosphere moves towards the substrate and reaches a relative position as shown in Fig. 1a. Then, the nanosphere continues moving towards the substrate (Fig. 1b) and comes in close proximity (Fig. 1c) and achieves contact deformation (Fig. 1d). Fig. 1e and f indicates the relative positions of the two components in the departure process. Note that there is no atom exchange or dissociation observed during the present collision process at
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Fig. 1 Snapshots of simulation of a typical head-on collision between the silicon nanosphere of 2.0 nm radius and the diamond substrate with dimensions of 10 2 10 nm3 obtained at (a) 1.20, (b) 1.40, (c) 1.60, (d) 1.70, (e) 1.90 and (f) 2.10 ps. The initial shortest surface-to-surface separation d (i.e. the separation of surfaces along the line of centers of the two bodies) between the silicon nanosphere and the diamond substrate is 8.0 nm, and the initial relative velocity is 500 m s1. Vr 4 0 denotes they approach towards each other and Vr o 0 represents they depart from each other. d = r R0 h/ 2 where r is the centre-to-centre separation, R0 is the nanoparticle radius and h is the thickness of the substrate.
Vr,o = 500 m s1, maybe because both the silicon atoms or carbon atoms are connected by the strong covalent bonds in their individual crystalline structure. The interaction potential energies and their individual potential contributions (e.g., vdW attraction, Born repulsion etc.) can be calculated as follows.10–12 The interaction energy between the two bodies takes the form, Einter = E12 E1 E2
(3)
where E12, E1, and E2 are the total energies of the sum of two bodies, body 1 and body 2, respectively. By differentiating interparticle potentials with respect to the separation distance between two bodies, the forces can be obtained as, F Inter ¼
@E Inter @d
(4)
It is worth mentioning that since the process simulated here, i.e., head-on collision, is a non-equilibrium dynamic process, there is always a change in temperature when two particles get close no matter which ensemble (NVE or NVT) is used. But it is demonstrated that the temperature and the ensemble have little effect on the MD simulated results, which can be ignored.10,11 This also implies that a small disturbance, for example, stemming from thermal vibrations within nanoparticles, does not affect the simulated results. Another concern is about the many-body effect among atoms, which has been at least partly considered in formulating the internanoparticle forces since the possible interactions among atoms in the nanospheres have been accounted through the
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COMPASS force field (e.g., valence terms including bending, torsion, inversion, etc.); on the other hand, it was reported that many-body effects can be ignored for a shape-isotropic nanocluster;38 thus MD simulations could provide accurate results of interaction forces.
3. Results and discussion The interaction forces between nanospheres can be calculated by the MD simulation. In this work, an initial relative velocity of 500 m s1 will be adopted based on the following considerations. On the one hand, when two particles are closely put together, there will exist an equilibration separation distance between them, usually around 0.16–0.20 nm,10,39 due to the two opposing forces of vdW attraction and Born repulsion forces. The use of such high relative velocity is to overcome the short range repulsion force and generate a significant contact deformation, which is needed to produce results about the contact forces between nanospheres. On the other hand, as per the continuum theory,40 there is one critical impact velocity below which the inelastic deformation can be ignored, i.e., when the impact speed n r Y/(rC0) E 500 m s1, inelastic deformation could not happen, where Y is the yield stress of 8.46 GPa,12 r is the density of 2320 kg m3 and C0 (C0 = (E0/r)0.5) is a characteristic of materials equal to 7469.5 m s1 for silicon.40 For silicon, E0 is Young’s modulus of 130 GPa.41 Recently, the MD simulations demonstrate that the validity of this continuum theory still holds at the nanoscale.12 Therefore, while speeding
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up the dynamic process and effecting contact deformation, also in order to minimize the dynamic effect, 500 m s1 is employed in the present simulations. Besides, since the interaction potentials are largely independent of the interaction path (e.g., approach or departure), the relative orientations of two nanospheres and the initial relative velocity below the critical one, interaction potentials obtained in the approach at Vr,o = 500 m s1 are used for analysis. In addition, for crystalline solids such as silicon and diamond, there are almost no surface atoms that fluctuate out of the cut-off radius R0, satisfying Rcore + dMax r R0, where Rcore is the radius of the particle core and dMax is the maximum surface thickness (ESI,† Table S-1 for nomenclature).12 Therefore, for the sake of convenient discussion, the cut-obtained particle size R0 is adopted rather than the defined particle size in the present work. 3.1
Size dependence of interparticle potentials
Fig. 2 shows the evolution of interparticle potentials due to different particle sizes of silicon nanospheres ranging from 1.0 to 4.0 nm in radius obtained from the approach process at an initial impact velocity of 500 m s1. It can be observed that both interparticle vdW attraction and Born repulsion potentials vary with particle size in a regular fashion. For a given separation, both the interparticle vdW attraction and Born repulsion potentials increase with an increase in the particle size. We now introduce the feature which distinguishes the concept of quasi-static loading at the ultrasmall nanoscale from that at the macroscale. At the macroscale, especially in AFM measurements, normally quasi-static loading implies a small speed of 1 m s1 or even less. But at the nanoscale level, because the intermolecular forces have a more pronounced
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effect on nanoparticles than macroparticles, the speed involved in the molecule–nanoparticle system (for instance, NEMS devices) is usually one or two orders of magnitude higher than that involved in macroparticle systems.42,43 Therefore, along with the synergistic thermal effect, the instantaneous velocity of the atom can be up to 0.5 Å ps1 (i.e., 50 m s1) as observed from the MD simulations.10,44 Thus the initial impact velocity of only 25 m s1 is almost equivalent to no external load applied. Under this condition, the interparticle potential only slightly fluctuates, and can be largely considered to reach an equilibrium with a minimum potential.11 Of note, the use of a high impact speed may have its pros and cons. On the one hand, at a high impact velocity, the two interacting particles may not have enough time to reach the minimum potential position while at a low impact velocity, the two particles may have time to adjust their relative orientations to reach a minimum potential. Since the interparticle potentials such as vdW attractive and Born repulsive potentials are given as a function of relative positions of atoms contained in each particle, strictly speaking, different impact velocities can have an influence on the interparticle potentials at close separation. However, in the previous work based on silica nanospheres, the results show that the inter-particle potentials are almost independent of the impact velocity.11 This may indicate that the effect of the slight change of atom’s positions at close separation on interaction potentials is limited. On the other hand, since the mechanical properties (i.e., Young’s modulus) of crystalline silicon is strongly dependent on the crystallographic orientations,41 in order to avoid the uncertainty of Young’s modulus when compared with the Hertz model, the head-on impact along the (100) direction is usually adopted in both computer simulation and experiment.45,46 From this point of view, the use of high speed can ensure the head-on collision and decrease the effect of non-central collision to a minimum. Another benefit is that at high impact velocity certain deformation of the particle can be created while at low impact velocity no obvious deformation may be observed. 3.2 Quantification of interparticle vdW attraction and Born repulsion forces For two solid spherical particles of radii R1 and R2, the vdW attraction interaction energy between them at a distance d apart (the separation of surfaces along the line of centers of the particles) is given by47 vdW EHamaker
Fig. 2 Interparticle potentials as a function of the shortest surface-tosurface separation between silicon nanospheres of different radii ranging from 1.0 to 4.0 nm and the diamond substrate. The solid dots represent interparticle LJ potentials. The inset highlights the individual contribution to the total LJ potential. The open dots represent interparticle Born repulsion potentials while the cross-centered dots denote interparticle vdW attraction potentials. The initial relative velocity employed here is 500 m s1.
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" # A 2l 2l r02 ð1 þ lÞ2 ¼ þ þ ln 6 r02 ð1 þ lÞ2 r02 ð1 lÞ2 r02 ð1 lÞ2 (5)
Here A is the Hamaker coefficient, 1.48 1019 and 4.65 1020 J for crystalline silicon and diamond, respectively. The Hamaker constant ASi–Diamond for two interacting silicon and diamond interface is ASi–Diamond = (ASiADiamond)1/2 = 8.30 1020 J, which will be used in eqn (5).
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Integrating the repulsive part (n-powered part) of LJn-m potential over two spherical colloidal particles yields47 n6 s ðn 8Þ! 1 Born EHamaker ¼ 4A R1 ðn 2Þ!r0 " r02 ðn 5Þðl 1Þr0 ðn 6Þ l2 ðn 5Þl þ 1 ðr0 1 þ lÞn5
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þ
þ
r02 þ ðn 5Þðl 1Þr0 ðn 6Þ½l2 ðn 5Þl þ 1 ðr0 þ 1 lÞn5 r02 þ ðn 5Þðl þ 1Þr0 þ ðn 6Þ½l2 þ ðn 5Þl þ 1 ðr0 þ 1 þ lÞn5
r02 ðn 5Þðl þ 1Þr0 þ ðn 6Þ½l2 þ ðn 5Þl þ 1 þ ðr0 1 lÞn5
#
(6) here l = R2/R1, r0 = (R1 + R2 + d)/R1 = 1 + l + d/R1, is the center-tocenter separation made dimensionless on R1. Since the formula for Born repulsion is also derived from the mathematic integral, in the following text, this approach is also called ‘‘Hamaker approach’’. Since the interaction between one sphere and one flat substrate can be considered as one of the limiting cases of interactions between two spheres where the radius of planar substrate approaches infinity, thus when l - +N, eqn (5) and (6) can be reduced to and simplified as A 1 1 d=R1 vdW lim EHamaker ¼ þ þ ln (7) l!þ1 6 d=R1 2 þ d=R1 2 þ d=R1 Born ¼ 4A lim EHamaker
l!þ1
s R1
3
"
1 6 2d=R1 6 2d=R1 þ 7! ð2 þ d=R1 Þ4 ðd=R1 Þ4
eqn (7) will be reduced to the most familiar formula of eqn (9) at the macroscale, given by39 vdW lim EHamaker ¼
l!þ1 Rd
Eqn (7) and (8) can be used to accurately calculate the interaction energies between the spherical particle of radius R and the planar substrate at the macroscale. In practice, when R c d,
(9)
Next, a comparison between MD simulated results and those calculated from eqn (7) and (8) is made. And the force ratios as a function of surface separation are shown in Fig. 3. The ratio first increases sharply from almost zero to a peak at around 0.35 nm, followed by a gradual decrease with the surface separation and approaching zero at an infinite large separation. This trend is quite different from the case of two interacting nanospheres in which the force ratio for vdW attraction first increases sharply from almost zero to a peak at around 0.4 nm, then decreases drastically and finally becomes constant asymptotically (Fig. S-3a, ESI†). In addition, another difference is that the force ratios of the former case are less than one order of magnitude (Fig. 3) while those of the latter case of two interacting nanospheres are up to several ten times at close separation (Fig. S-3, ESI†). As shown in Fig. 3, at d Z 0.35 nm, the ratio is a function of the particle size and the surface separation distance. Likewise, in the present case, assuming the ratio at d = 0.35 nm, kd=0.35nm , as a function of the particle size and independent vdW of surface separation, i.e., kd=0.35nm = f (R), the relationship vdW between kd=0.35nm and R can be described by the expression vdW as shown in the inset of Fig. 3a = 9.45 exp[0.04(R R0)0.65] kd=0.35nm vdW
#
(8)
AR 6d
(10)
where R0 = 1.0 nm and R is the particle radius. The normalized force ratio, the ratio at d = 0.35 nm, i.e., kd=0.35nm , is given as a vdW function of surface separation,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i Hamaker d¼0:35nm h MD FvdW ¼ 0:983 expð0:05 d 0:35Þ FvdW kvdW (11)
Fig. 3 The ratio of interparticle forces between the silicon nanosphere and the diamond substrate obtained from MD simulations to those from the Hamaker approach: (a) the vdW attraction and (b) the Born repulsion force. The insets show the ratio kd=0.35nm and kd=0.35nm as a function of vdW Born particle radius R.
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Therefore the interparticle vdW attraction forces can be interacting nanospheres, but not from the case of one nanoestimated based on MD simulations by the following expres- sphere interacting with the substrate. This is because the former case is directly compared with the initial form of sions (as shown in Fig. 4a): 8 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii vdW > 9:29 exp 0:04ðR R0 Þ0:65 0:05 d 0:35 FHamaker ; d 0:35 nm > > > > < vdW FMD ¼ e3:63ð0:35dÞ0:65 F vdW (12) Modified; d¼0:35 nm ; 0:15 d o 0:35 nm > > > > > : 1 þ 0:45ð1 d=0:15Þ0:2 ; d o 0:15 nm Similarly, the ratios of Born repulsion forces (Fig. 3b) depend largely on surface separation and the particle size. The size-dependent force ratio at d = 0.35 nm, i.e., kd=0.35nm , Born is used as a reference, kd=0.35nm = 6.43 exp[0.028(R R0)0.60] Born
(13)
The normalized force ratio kd=0.35nm is given as a function of Born surface separation and can be approximately represented by the following formula (as shown in Fig. 4b): Hamaker (FMD )/kd=0.35nm = [1.04 exp(0.075(d 0.35)0.10)] Born/FBorn Born (14)
Therefore the interparticle Born repulsion forces can be obtained by the following modified expression from the Hamaker approach,
Born FMD
equations from the Hamaker approach, that is, eqn (5) and (6), whereas the latter is compared with eqn (7) and (8), which are derived from the Hamaker approach by solving the limit mathematically. During this process, one more origin of deviation from the Hamaker approach may be introduced and hence compounds the difficulties in clearly distinguishing each origin of the deviation. When formulating inter-nanoparticles between two interacting nanospheres, an effort has been made to demonstrate that the resulting equations may be applied to nanoparticles with a radius larger than 4 nm. This is achieved by the use of eqn (S-3) or (S-4) (ESI†) for vdW attraction or Born repulsion, respectively, so that when R is large enough, eqn (S-1) or (S-2) (ESI†) produces the same results as those given by the Hamaker approach, i.e., eqn (5) and (6). But there is no compelling reason to clearly claim a size limit above which the Hamaker model still works. In practice, it depends
8 h i 0:60 0:10 Born > ; d 0:35 nm FHamaker > > 6:69 exp 0:028ðR R0 Þ 0:075ðd 0:35Þ > > > < 0:60 Born ¼ e8:07ð0:35dÞ FModified; d¼0:35 nm ; 0:15 d o 0:35 nm > > > > h i > > : 1 þ 0:695ð1 d=0:15Þ0:5 F Born Modified; d¼0:15 nm ; d o 0:15 nm
Although the Hamaker approach breaks down at the nanoscale, nevertheless, it is interesting to inquire into the estimated size limit above which the Hamaker model still works; one perhaps can seek such information from eqn (S-3) and (S-4) (ESI†), i.e., the proposed equations for the case of two
(15)
on the accuracy required. For example, if for the purpose of engineering applications, it may satisfy the requirement by just taking that the Hamaker approach still applies, say, when R 4 1 mm as the proposed equations tend to become much closer to the Hamaker equations.
Fig. 4 Interparticle vdW attraction and Born repulsion forces between the silicon nanosphere and the diamond flat substrate.
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The combination of eqn (7) and (12), or eqn (8) and (15), allows the calculation of the corresponding vdW attraction or Born repulsion forces between the nanosphere and the flat surface. As shown in Fig. 4, the calculated results can match the MD simulated results reasonably well. Note that, at close separation and especially after contact deformation, the force–displacement curve is not so smooth but displays a small fluctuation. This can reflect the steplike structure of two interacting surfaces due to the atomic discrete structure. Unlike the continuum models such as the Hamaker approach, in which the interacting bodies are treated as continuous media.39 Such treatment would result in a smooth force–displacement curve. However, the assumption of continuous media is not held at the molecular level in the case of nanoparticles. The nature of the discrete atomic structure in particular on the surface could not be ignored and would be reflected from the small fluctuations. With further increase in the deformation, dislocation of atoms may happen, which may lead to a small ‘jump’ in force–displacement.12 However, this phenomenon cannot be necessarily observed herein as the impact velocity herein is limited to 500 m s1 and the strain of silicon is below 0.1 (Fig. 5). Of note, there exist some difficulties in direct comparison with experimental results in terms of measured vdW forces. The main barrier lies in the size of silicon nanospheres. Due to the limiting computation efficiency, the size of nanoparticles in MD simulations is limited to below 5.0 nm radius. Therefore, a compromise between the size of the modeled system and the computation efficiency is made. In contrast, experimental studies on silicon nanospheres involve the particles ranging from 20 to 140 nm radius.46,48 What is more, most of them focus on contact forces. However, with the advancement of technology, this gap may be bridged in the future. Nevertheless, the present simulated results closely resemble the experimental ones in a qualitative way. Consider the case of the silicon nanosphere of 4.0 nm radius for example. The interparticle LJ force,
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i.e., the total of vdW attraction and Born repulsion forces, first decreases to a minimum with decreasing surface separation, then increases gradually at close separation and even changes the sign and becomes the repulsive force after contact deformation (ESI,† Fig. S-5b). This trend is similar to those measured in AFM experiments.49,50 In practice, the interparticle forces can be affected by physical properties such as density, shape, unequal particle size, and even possible ‘viscosity’ when interacting across one third medium, etc. The effect of density, the medium interacting across or unequal size can be taken into account by tailing the Hamaker constant or the particle radius in the Hamaker equation. For example, different nanoparticles have different densities according to A = p2Cr1r2, where r1 and r2 are the number densities of atoms of particles 1 and 2, respectively; thus the Hamaker constant A can be correspondingly modified using the corresponding value of the number density of atoms; when two particles interact across different media, normally the viscosity is different. In this case, the Hamaker constant can be adjusted by the following equation. Two nanoparticles interacting across a pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi
medium A132 A11 A33 A22 A33 where A11, A22 and A33 are the Hamaker constants of particles 1 and 2 and medium 3, respectively; for two nanoparticles with different radii R1 and R2, the R in the Hamaker equation can be replaced by the pffiffiffiffiffiffiffiffiffiffiffi geometrical mean R ¼ R1 R2 , the validity of which has been confirmed by the MD simulations.10 Moreover, the idealized spherical particles are often used in theoretical models or the ideal spherical shape is often assumed in dealing with particles in experiments, such as estimation of the surface area, although in practice most of particles deviate from the ideal spherical shape. Therefore, attention is paid to the spherical nanoparticles in this work. However, ellipsoidal particles are some of the most common non-spherical particles, and the effect of shape may be approximately considered by adjusting the aspect ratio and the particle size. The work based on ellipsoidal nanoparticles has been carried out and will be reported elsewhere soon. 3.3
Mechanical contact force
Apart from the LJ interactions in the forcefield COMPASS, the computation still encompasses valence interactions such as bond stretching, angle bending, angle torsion, and angle inversion, and there is always a force constant to describe these terms in their individual expressions.32 These valence interactions should be related to macro-mechanical properties, leading to the mechanical contact force related to contact deformation. Therefore, in addition to the interparticle vdW attraction and Born repulsion forces, the mechanical contact force Fc will arise upon the contact deformation of two nanospheres, which can be calculated by11 Fc = Fn FvdW FBorn pffiffiffi
Fig. 5 F c 8 2R2 E 3 as a function of strain dn/(2R) obtained in the approach process from MD simulations between the diamond substrate and silicon nanospheres of different radii ranging from 1.0 to 4.0 nm. Solid line represents the Hertz prediction using Young’s moduli of their counterpart bulks.
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(16)
where the total normal force Fn was evaluated by accumulating the vertical forces (Y direction) that all atoms from the substrate exerted on the nanoparticle during the central collision process, FvdW and FBorn were obtained by differentiating interaction potentials with respect to surface separation d according
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to eqn (4). In detail, the forces of one particular atom along the X, Y, and Z directions at different simulation times can be the output, then the total normal force of the nanoparticle along one particular direction such as the Y axis, i.e., Fn, can be obtained by accumulating the vertical forces of all the atoms along the Y axis. Contact between solid spheres is usually described by the models of JKR51 or DMT,52 both of which are based on an earlier analysis by Hertz,2 who considered two elastic bodies in contact under an external load but ignored interparticle attractive forces. Normal contact forces between spherical particles 1 and 2 are given by the form,53,54 4 pffiffiffiffiffiffi Fn ¼ E R dn3=2 3
(17)
where 1/E* = (1 n12)/E1 + (1 n22)/E2, 1/R* = 1/R1 + 1/R2 in which E1 and E2 are Young’s moduli of particles 1 and 2, respectively, herein Young’s moduli of silicon41 and diamond55 are 130 and 1050 GPa, respectively, n is the Poisson ratio of 0.2841 and 0.0855 along the crystal plane [100], respectively, R1 and R2 are the radii of particles 1 and 2, respectively, thus E* is about 124.45 GPa, dn is relative movement, and dn = 0 corresponds to the first non-zero of contact force Fc. For contact forces between the elastic spherical particle of radius R and the flat surface, eqn (17) can be reduced to and given by pffiffiffi 4 pffiffiffiffi 8 2 2 3=2 Fn ¼ E Rdn3=2 ¼ E R e (18) 3 3 where e = dn/(2R). Note that since Young’s modulus of diamond is far greater than that of silicon, the deformation of diamond during impact can be ignored and considered as ‘rigid’, and silicon suffers from almost all the appreciable deformation. In addition, the MD simulations have proven that the COMPASS force field can accurately describe the mechanical properties of silicon materials such as Young’s modulus (ESI,† Fig. S-6),12 which can lend support to our current simulated results. The calculated results of contact forces Fc using eqn (16) show that the first non-zero value of mechanical contact forces arise when surface separation is about d = 0.16–0.18 nm. This value is consistent with the observed minima of interparticle LJ potentials from Fig. 2. The minimum point of the LJ potential (i.e., the zero point of LJ force) corresponds to the point where the occurrence of the mechanical contact force initiates, confirming that due to the intermolecular repulsive forces, the surface atoms are prone to be subject to ‘‘deformation’’ and hence the mechanical contact force arises when two surfaces are less than equilibrium separation distance apart. Fig. 5 shows the normalized mechanical contact forces between the diamond substrate and silicon nanospheres of different radii. These contact forces are in reasonable accordance with those predicted by the Hertz theory using Young’s moduli of diamond and silicon bulk and this confirms that the herein deformation is almost completely elastic. As observed from ESI,† Fig. S-5b, upon decreasing the surface separation, the interacting LJ forces (i.e., the total of vdW attraction and Born repulsive forces) first become
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adhesive, then become gradually repulsive around d E 0.16 nm. Before contact, the interparticle force is dominated by the LJ force (mainly the vdW attractive force) while there is almost no contact force. But after contact deformation, apart from the contribution of the LJ force, the contact force also arises. Thus both of them should be coupled together to represent the total interaction forces between the silicon nanoparticle and the diamond substrate using eqn (12), (15), and (18) over the whole surface separation and even after contact deformation. In JKR and DMT models, the contribution of the intermolecular force is reflected by the adhesive force, which is constant and evaluated by surface energy approximation.56 Detailed discussion and comparison with JKR and DMT models have been treated in one recent MD simulations,11 indicating that JKR and DMT models still can be used as the first approximation. Thus attention herein is paid to test the applicability of the Hertz model to the case of the nanosphere interacting with the flat substrate, but without giving emphasis to the comparison with the continuum models anymore.
4. Conclusions In this study, the interactions between a spherical nanoparticle and a flat surface including vdW attraction, Born repulsion and mechanical contact forces have been studied and formulated based on fully atomistic MD simulations. It is found that different from two interacting nanospheres, the force ratios between the nanosphere and the flat surface decrease gradually after one peak and tend to approach zero without exhibiting an asymptotically constant ratio at large enough separation. Thus two separate modified equations to the Hamaker approach have been proposed for interparticle vdW attraction and Born repulsion forces, respectively. Moreover, the results confirm that the continuum Hertz model still holds to describe mechanical contact force between the nanosphere and the flat surface. Finally, the behaviors of the nanoparticle system, such as aggregation, packing, flow, etc., are of importance to a variety of practical applications. The proposed formulas for inter-nanoparticle forces can serve as input parameters in the so-called cost-effective discrete element method (DEM) to study these behaviors in future.
Acknowledgements The authors gratefully acknowledge the support from the University of New South Wales.
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Cite this: Phys. Chem. Chem. Phys., 2014, 16, 5846
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Interaction forces between a spherical nanoparticle and a flat surface† Weifu Sun Due to the breakdown of Derjaguin approximation at the nanoscale level apart from the neglect of the atomic discrete structure, the underestimated number density of atoms, and surface effects, the continuum Hamaker model does not hold to describe interactions between a spherical nanoparticle and a flat surface. In this work, the interaction forces including van der Waals (vdW) attraction, Born repulsion and mechanical contact forces between a spherical nanoparticle and a flat substrate have been studied using molecular dynamic (MD) simulations. The MD simulated results are compared with the Hamaker
Received 3rd December 2013, Accepted 30th January 2014
approach and it is found that the force ratios for one nanosphere interacting with a flat surface are
DOI: 10.1039/c3cp55082f
separate formulas have been proposed to estimate the vdW attraction and Born repulsion forces between
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a nanosphere and a flat surface. Besides, it is revealed that the mechanical contact forces between a spherical nanoparticle and a flat surface still can be described by the continuum Hertz model.
different from those for two interacting nanospheres, both qualitatively and quantitatively. Thus two
1. Introduction It is known that the continuum models such as the Hamaker approach1 and the Hertz continuum model2 do not generally apply at the nanoscale level due to the neglect of the atomic discrete structure,3,4 surface effects, the underestimated number density of atoms or the neglected intermolecular forces. On the one hand, the continuum Hamaker approach treats the particles as rigid (incompressible) bodies without considering the atoms’ vibration, bond’s torsion, inversion, etc., implying that the particles will not deform and this is contrary to reality; on the other hand, in the Hamaker approach, particles are assumed or treated as a continuum medium with a uniform (density) distribution of atoms. But in essence, the structure of a nanoparticle is not only ‘soft’, but also discrete. Although in the continuum model, the interaction between a spherical particle and a flat surface can be considered as a limiting case of interactions between two spherical particles according to the Derjaguin approximation,5 at the nanoscale, because the size of nanoparticle R is comparable to the surface separation d, thus the Derjaguin approximation breaks down since it does not satisfy R c d. What is more, one should bear in mind that even at the macroscale, the formulas governing interactions between one sphere and a flat surface are different from those
School of Materials Science and Engineering, The University of New South Wales, Sydney, NSW 2052, Australia. E-mail: [email protected], [email protected] † Electronic supplementary information (ESI) available. See DOI: 10.1039/ c3cp55082f
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describing interactions between two spheres. To date, the interaction forces between a spherical nanoparticle and a flat surface is still unknown, although some effort has been made to model atomic force microscopy (AFM) measurements using a tip and a substrate, in which attention is paid to the contact force only through indentations.6,7 Nowadays, a full quantitative analysis of inter-nanoparticle forces is still challenging for experimental techniques at the nanoscale with particle size less than 10 nm.8 The accuracy of the AFM measurement, especially in the lateral direction, is critically dependent on factors such as sensitive spring constant, precision calibration of the cantilever, etc. In contrast, molecular dynamics (MD) simulation can circumvent these demerits and has proven to be an effective tool for calculating interaction forces between nanoparticles,9–12 but whether these modifying formulas can be extended to interactions between a nanosphere and a flat surface or not is still unexplored. Here efforts will be devoted to formulate interaction forces between a spherical nanoparticle and a flat substrate using a similar previously-established procedure10–12 with a view to further testing the continuum models, such as the Hamaker approach and the Hertz model. A silicon/diamond structure is of great interest in a variety of applications13 such as anodes for lithium-ion cells,14 solar cells,15 chemical sensors,16 biosensors,17 drug delivery,18 super-capacitors and hydrogen storage,19 catalysis,20 nanoelectronics,21,22 silicon metal-oxide-semiconductor field-effect transistors23 and microchips in semiconductors, integrated circuits and field-emission devices,24 and also micro/nano electromechanical systems (MEMS/NEMS) building upon the existing silicon infrastructure to create micro/nanoscale machines.25–27
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With the continuing development of silicon-based ultra large scale integrated (ULSI) circuits, an exponential growth of the number of devices on a chip has been projected by Moore’s law. However, the reduction in the device size results in increased parasitic leakage currents and power dissipation at the device level. To overcome these adverse effects, to enhance the operation speed and to lower the power consumption of chips, silicon-on-diamond (SOD), an alternative concept of silicon-on-insulator, becomes an ever-increasing promising candidate, in which the thermally insulating silica has been replaced by highly thermally conductive diamond. SOD is a substrate engineered to address the major challenges of siliconbased ULSI technology. Experimentally, a thin, single crystalline Si layer is deposited on a highly oriented diamond layer that serves as an electrical insulator, a heat spreader and a supporting substrate.28,29 Therefore, the analysis of the interface properties such as interfacial structure30 and interface forces is important for the design of MEMS/NEMS devices, and MD simulations will be performed in this work to study interfacial interactions between a silicon nanosphere and a diamond flat substrate.
2. Simulation conditions and method The simulations are performed in a similar way as done before.10,11 The interatomic interaction parameters are taken from the COMPASS force field,31 which encompasses a LennardJones n-m potential (LJn-m, n = 9 and m = 6) function between non-bonded atoms, given by32 Eij ¼
m X m sij n n sij eij n m n m r rij ij i;j
(1)
where atoms i and j are separated by a distance rij, sij is the collision diameter of atoms i and j at which the force becomes zero, and e is the potential well depth of interaction between atoms. The cut-off distance used in the MD simulations is set to be as large as 100 nm to completely avoid the errors introduced by the cutoff distance. From the perspective of quantum mechanics, the electron correlations are governed by solving more ¨dinger accurate and rigorous relationships such as the Schro equation;33 but in MD simulations, this interaction is represented by the short-ranged repulsion, also called the Born repulsion which originates from the strong repulsive forces between atoms as their electron shells interpenetrate each other.34 In the COMPASS force field, the Born repulsion is embodied in the Lennard-Jones (9-6) potential function by the 9-powered part.31 Apart from the LJ potentials between non-bonded atoms, COMPASS also encompasses valence interactions such as bond stretching, angle bending, angle torsion, angle inversion, etc., between bonded atoms, taking the form of Evalence = Eb + Ey + Ej + Ew + Eb,b + Eb,b 0 + Eb,y + Eb,j + Eb,j + Eb,y,j (2)
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where the terms on the right represent bond stretching (b), angle bending (y), angle torsion (j), out-of-plane angle inversion (w), and the cross-coupling terms including combinations of two or three valence terms, respectively. The values of si and sj for silicon and diamond-like carbon are 4.45 and 3.854 Å, respectively,31,35 the values of ei and ej for silicon and diamond-like carbon are 0.198 and 0.062 kcal mol1; therefore the Hamaker constant (A = p2C/v2, where C = 3es6 is the vdW attraction interaction parameter, v = 4p(s/2)3/3 is the atom volume estimated in a way inherent in the Hamaker approach) can be, respectively, estimated to be about 1.4848 1019 and 4.65 1020 J, which is close to experimental values of 1.9 or 2.17 1019 J.36 Therefore, the Hamaker constant of diamond and silicon interacting across air/vacuum is about ASi–C E (ASiADiamond)0.5 E 8.3 1020 J. For a pair of dissimilar atoms, sij is governed by the Waldman– Hagler combination rule37 in terms of the sSi and sC values of the individual silicon and carbon atoms, that is sSi–C = [(sSi6 + sC6)/2]1/6, thus giving rise to sSi–C = 4.204 Å. The simulated system consisting of one silicon nanosphere and the diamond substrate is first fully relaxed using the NVT (i.e., constant number of atoms, constant volume and constant temperature) ensemble at 300 K. Then, the fully relaxed silicon nanosphere is allowed to move towards the diamond substrate at an initial atomic velocity using a NVE ensemble (i.e., constant number of atoms, constant volume and constant energy) at an initial temperature of 300 K. The bottom layer of the diamond substrate is fixed during the MD simulations. Numerical integration is performed using the velocity Verlet integration algorithm with a time step Dt = 1.0 1015 s (1 fs). Frames are output every 100 steps into the trajectory file. The forces, coordinates, velocities of atoms and energies, etc. as a function of the shortest surface-to-surface separation distance (d) between the two components are recorded in an output trajectory file. Note that in our case, in order to ensure a normal impact (that is central collision), 5.0 Å ps1 (i.e., 500 m s1) along the Y-axis of each atom in the nanosphere was specified while 0 Å ps1 was specified in the other two directions. The silicon nanosphere of 1.0, 2.0, 3.0 and 4.0 nm radius contains 220, 1707, 5707 and 13 407 atoms, respectively. The dimensions of a diamond flat substrate are 10 nm length, 10 nm width and 2 nm thickness, containing 36 742 carbon atoms. After running the MD simulations using the NVE ensemble, the properties of atoms such as coordinates, forces and velocities along each direction (i.e., X-, Y- or Z-axis) can be separately obtained from the trajectory files. A typical sequence of the silicon nanosphere colliding with the diamond substrate is shown in Fig. 1. The nanosphere moves towards the substrate and reaches a relative position as shown in Fig. 1a. Then, the nanosphere continues moving towards the substrate (Fig. 1b) and comes in close proximity (Fig. 1c) and achieves contact deformation (Fig. 1d). Fig. 1e and f indicates the relative positions of the two components in the departure process. Note that there is no atom exchange or dissociation observed during the present collision process at
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Fig. 1 Snapshots of simulation of a typical head-on collision between the silicon nanosphere of 2.0 nm radius and the diamond substrate with dimensions of 10 2 10 nm3 obtained at (a) 1.20, (b) 1.40, (c) 1.60, (d) 1.70, (e) 1.90 and (f) 2.10 ps. The initial shortest surface-to-surface separation d (i.e. the separation of surfaces along the line of centers of the two bodies) between the silicon nanosphere and the diamond substrate is 8.0 nm, and the initial relative velocity is 500 m s1. Vr 4 0 denotes they approach towards each other and Vr o 0 represents they depart from each other. d = r R0 h/ 2 where r is the centre-to-centre separation, R0 is the nanoparticle radius and h is the thickness of the substrate.
Vr,o = 500 m s1, maybe because both the silicon atoms or carbon atoms are connected by the strong covalent bonds in their individual crystalline structure. The interaction potential energies and their individual potential contributions (e.g., vdW attraction, Born repulsion etc.) can be calculated as follows.10–12 The interaction energy between the two bodies takes the form, Einter = E12 E1 E2
(3)
where E12, E1, and E2 are the total energies of the sum of two bodies, body 1 and body 2, respectively. By differentiating interparticle potentials with respect to the separation distance between two bodies, the forces can be obtained as, F Inter ¼
@E Inter @d
(4)
It is worth mentioning that since the process simulated here, i.e., head-on collision, is a non-equilibrium dynamic process, there is always a change in temperature when two particles get close no matter which ensemble (NVE or NVT) is used. But it is demonstrated that the temperature and the ensemble have little effect on the MD simulated results, which can be ignored.10,11 This also implies that a small disturbance, for example, stemming from thermal vibrations within nanoparticles, does not affect the simulated results. Another concern is about the many-body effect among atoms, which has been at least partly considered in formulating the internanoparticle forces since the possible interactions among atoms in the nanospheres have been accounted through the
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COMPASS force field (e.g., valence terms including bending, torsion, inversion, etc.); on the other hand, it was reported that many-body effects can be ignored for a shape-isotropic nanocluster;38 thus MD simulations could provide accurate results of interaction forces.
3. Results and discussion The interaction forces between nanospheres can be calculated by the MD simulation. In this work, an initial relative velocity of 500 m s1 will be adopted based on the following considerations. On the one hand, when two particles are closely put together, there will exist an equilibration separation distance between them, usually around 0.16–0.20 nm,10,39 due to the two opposing forces of vdW attraction and Born repulsion forces. The use of such high relative velocity is to overcome the short range repulsion force and generate a significant contact deformation, which is needed to produce results about the contact forces between nanospheres. On the other hand, as per the continuum theory,40 there is one critical impact velocity below which the inelastic deformation can be ignored, i.e., when the impact speed n r Y/(rC0) E 500 m s1, inelastic deformation could not happen, where Y is the yield stress of 8.46 GPa,12 r is the density of 2320 kg m3 and C0 (C0 = (E0/r)0.5) is a characteristic of materials equal to 7469.5 m s1 for silicon.40 For silicon, E0 is Young’s modulus of 130 GPa.41 Recently, the MD simulations demonstrate that the validity of this continuum theory still holds at the nanoscale.12 Therefore, while speeding
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up the dynamic process and effecting contact deformation, also in order to minimize the dynamic effect, 500 m s1 is employed in the present simulations. Besides, since the interaction potentials are largely independent of the interaction path (e.g., approach or departure), the relative orientations of two nanospheres and the initial relative velocity below the critical one, interaction potentials obtained in the approach at Vr,o = 500 m s1 are used for analysis. In addition, for crystalline solids such as silicon and diamond, there are almost no surface atoms that fluctuate out of the cut-off radius R0, satisfying Rcore + dMax r R0, where Rcore is the radius of the particle core and dMax is the maximum surface thickness (ESI,† Table S-1 for nomenclature).12 Therefore, for the sake of convenient discussion, the cut-obtained particle size R0 is adopted rather than the defined particle size in the present work. 3.1
Size dependence of interparticle potentials
Fig. 2 shows the evolution of interparticle potentials due to different particle sizes of silicon nanospheres ranging from 1.0 to 4.0 nm in radius obtained from the approach process at an initial impact velocity of 500 m s1. It can be observed that both interparticle vdW attraction and Born repulsion potentials vary with particle size in a regular fashion. For a given separation, both the interparticle vdW attraction and Born repulsion potentials increase with an increase in the particle size. We now introduce the feature which distinguishes the concept of quasi-static loading at the ultrasmall nanoscale from that at the macroscale. At the macroscale, especially in AFM measurements, normally quasi-static loading implies a small speed of 1 m s1 or even less. But at the nanoscale level, because the intermolecular forces have a more pronounced
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effect on nanoparticles than macroparticles, the speed involved in the molecule–nanoparticle system (for instance, NEMS devices) is usually one or two orders of magnitude higher than that involved in macroparticle systems.42,43 Therefore, along with the synergistic thermal effect, the instantaneous velocity of the atom can be up to 0.5 Å ps1 (i.e., 50 m s1) as observed from the MD simulations.10,44 Thus the initial impact velocity of only 25 m s1 is almost equivalent to no external load applied. Under this condition, the interparticle potential only slightly fluctuates, and can be largely considered to reach an equilibrium with a minimum potential.11 Of note, the use of a high impact speed may have its pros and cons. On the one hand, at a high impact velocity, the two interacting particles may not have enough time to reach the minimum potential position while at a low impact velocity, the two particles may have time to adjust their relative orientations to reach a minimum potential. Since the interparticle potentials such as vdW attractive and Born repulsive potentials are given as a function of relative positions of atoms contained in each particle, strictly speaking, different impact velocities can have an influence on the interparticle potentials at close separation. However, in the previous work based on silica nanospheres, the results show that the inter-particle potentials are almost independent of the impact velocity.11 This may indicate that the effect of the slight change of atom’s positions at close separation on interaction potentials is limited. On the other hand, since the mechanical properties (i.e., Young’s modulus) of crystalline silicon is strongly dependent on the crystallographic orientations,41 in order to avoid the uncertainty of Young’s modulus when compared with the Hertz model, the head-on impact along the (100) direction is usually adopted in both computer simulation and experiment.45,46 From this point of view, the use of high speed can ensure the head-on collision and decrease the effect of non-central collision to a minimum. Another benefit is that at high impact velocity certain deformation of the particle can be created while at low impact velocity no obvious deformation may be observed. 3.2 Quantification of interparticle vdW attraction and Born repulsion forces For two solid spherical particles of radii R1 and R2, the vdW attraction interaction energy between them at a distance d apart (the separation of surfaces along the line of centers of the particles) is given by47 vdW EHamaker
Fig. 2 Interparticle potentials as a function of the shortest surface-tosurface separation between silicon nanospheres of different radii ranging from 1.0 to 4.0 nm and the diamond substrate. The solid dots represent interparticle LJ potentials. The inset highlights the individual contribution to the total LJ potential. The open dots represent interparticle Born repulsion potentials while the cross-centered dots denote interparticle vdW attraction potentials. The initial relative velocity employed here is 500 m s1.
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" # A 2l 2l r02 ð1 þ lÞ2 ¼ þ þ ln 6 r02 ð1 þ lÞ2 r02 ð1 lÞ2 r02 ð1 lÞ2 (5)
Here A is the Hamaker coefficient, 1.48 1019 and 4.65 1020 J for crystalline silicon and diamond, respectively. The Hamaker constant ASi–Diamond for two interacting silicon and diamond interface is ASi–Diamond = (ASiADiamond)1/2 = 8.30 1020 J, which will be used in eqn (5).
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Integrating the repulsive part (n-powered part) of LJn-m potential over two spherical colloidal particles yields47 n6 s ðn 8Þ! 1 Born EHamaker ¼ 4A R1 ðn 2Þ!r0 " r02 ðn 5Þðl 1Þr0 ðn 6Þ l2 ðn 5Þl þ 1 ðr0 1 þ lÞn5
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þ
þ
r02 þ ðn 5Þðl 1Þr0 ðn 6Þ½l2 ðn 5Þl þ 1 ðr0 þ 1 lÞn5 r02 þ ðn 5Þðl þ 1Þr0 þ ðn 6Þ½l2 þ ðn 5Þl þ 1 ðr0 þ 1 þ lÞn5
r02 ðn 5Þðl þ 1Þr0 þ ðn 6Þ½l2 þ ðn 5Þl þ 1 þ ðr0 1 lÞn5
#
(6) here l = R2/R1, r0 = (R1 + R2 + d)/R1 = 1 + l + d/R1, is the center-tocenter separation made dimensionless on R1. Since the formula for Born repulsion is also derived from the mathematic integral, in the following text, this approach is also called ‘‘Hamaker approach’’. Since the interaction between one sphere and one flat substrate can be considered as one of the limiting cases of interactions between two spheres where the radius of planar substrate approaches infinity, thus when l - +N, eqn (5) and (6) can be reduced to and simplified as A 1 1 d=R1 vdW lim EHamaker ¼ þ þ ln (7) l!þ1 6 d=R1 2 þ d=R1 2 þ d=R1 Born ¼ 4A lim EHamaker
l!þ1
s R1
3
"
1 6 2d=R1 6 2d=R1 þ 7! ð2 þ d=R1 Þ4 ðd=R1 Þ4
eqn (7) will be reduced to the most familiar formula of eqn (9) at the macroscale, given by39 vdW lim EHamaker ¼
l!þ1 Rd
Eqn (7) and (8) can be used to accurately calculate the interaction energies between the spherical particle of radius R and the planar substrate at the macroscale. In practice, when R c d,
(9)
Next, a comparison between MD simulated results and those calculated from eqn (7) and (8) is made. And the force ratios as a function of surface separation are shown in Fig. 3. The ratio first increases sharply from almost zero to a peak at around 0.35 nm, followed by a gradual decrease with the surface separation and approaching zero at an infinite large separation. This trend is quite different from the case of two interacting nanospheres in which the force ratio for vdW attraction first increases sharply from almost zero to a peak at around 0.4 nm, then decreases drastically and finally becomes constant asymptotically (Fig. S-3a, ESI†). In addition, another difference is that the force ratios of the former case are less than one order of magnitude (Fig. 3) while those of the latter case of two interacting nanospheres are up to several ten times at close separation (Fig. S-3, ESI†). As shown in Fig. 3, at d Z 0.35 nm, the ratio is a function of the particle size and the surface separation distance. Likewise, in the present case, assuming the ratio at d = 0.35 nm, kd=0.35nm , as a function of the particle size and independent vdW of surface separation, i.e., kd=0.35nm = f (R), the relationship vdW between kd=0.35nm and R can be described by the expression vdW as shown in the inset of Fig. 3a = 9.45 exp[0.04(R R0)0.65] kd=0.35nm vdW
#
(8)
AR 6d
(10)
where R0 = 1.0 nm and R is the particle radius. The normalized force ratio, the ratio at d = 0.35 nm, i.e., kd=0.35nm , is given as a vdW function of surface separation,
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi i Hamaker d¼0:35nm h MD FvdW ¼ 0:983 expð0:05 d 0:35Þ FvdW kvdW (11)
Fig. 3 The ratio of interparticle forces between the silicon nanosphere and the diamond substrate obtained from MD simulations to those from the Hamaker approach: (a) the vdW attraction and (b) the Born repulsion force. The insets show the ratio kd=0.35nm and kd=0.35nm as a function of vdW Born particle radius R.
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Therefore the interparticle vdW attraction forces can be interacting nanospheres, but not from the case of one nanoestimated based on MD simulations by the following expres- sphere interacting with the substrate. This is because the former case is directly compared with the initial form of sions (as shown in Fig. 4a): 8 h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii vdW > 9:29 exp 0:04ðR R0 Þ0:65 0:05 d 0:35 FHamaker ; d 0:35 nm > > > > < vdW FMD ¼ e3:63ð0:35dÞ0:65 F vdW (12) Modified; d¼0:35 nm ; 0:15 d o 0:35 nm > > > > > : 1 þ 0:45ð1 d=0:15Þ0:2 ; d o 0:15 nm Similarly, the ratios of Born repulsion forces (Fig. 3b) depend largely on surface separation and the particle size. The size-dependent force ratio at d = 0.35 nm, i.e., kd=0.35nm , Born is used as a reference, kd=0.35nm = 6.43 exp[0.028(R R0)0.60] Born
(13)
The normalized force ratio kd=0.35nm is given as a function of Born surface separation and can be approximately represented by the following formula (as shown in Fig. 4b): Hamaker (FMD )/kd=0.35nm = [1.04 exp(0.075(d 0.35)0.10)] Born/FBorn Born (14)
Therefore the interparticle Born repulsion forces can be obtained by the following modified expression from the Hamaker approach,
Born FMD
equations from the Hamaker approach, that is, eqn (5) and (6), whereas the latter is compared with eqn (7) and (8), which are derived from the Hamaker approach by solving the limit mathematically. During this process, one more origin of deviation from the Hamaker approach may be introduced and hence compounds the difficulties in clearly distinguishing each origin of the deviation. When formulating inter-nanoparticles between two interacting nanospheres, an effort has been made to demonstrate that the resulting equations may be applied to nanoparticles with a radius larger than 4 nm. This is achieved by the use of eqn (S-3) or (S-4) (ESI†) for vdW attraction or Born repulsion, respectively, so that when R is large enough, eqn (S-1) or (S-2) (ESI†) produces the same results as those given by the Hamaker approach, i.e., eqn (5) and (6). But there is no compelling reason to clearly claim a size limit above which the Hamaker model still works. In practice, it depends
8 h i 0:60 0:10 Born > ; d 0:35 nm FHamaker > > 6:69 exp 0:028ðR R0 Þ 0:075ðd 0:35Þ > > > < 0:60 Born ¼ e8:07ð0:35dÞ FModified; d¼0:35 nm ; 0:15 d o 0:35 nm > > > > h i > > : 1 þ 0:695ð1 d=0:15Þ0:5 F Born Modified; d¼0:15 nm ; d o 0:15 nm
Although the Hamaker approach breaks down at the nanoscale, nevertheless, it is interesting to inquire into the estimated size limit above which the Hamaker model still works; one perhaps can seek such information from eqn (S-3) and (S-4) (ESI†), i.e., the proposed equations for the case of two
(15)
on the accuracy required. For example, if for the purpose of engineering applications, it may satisfy the requirement by just taking that the Hamaker approach still applies, say, when R 4 1 mm as the proposed equations tend to become much closer to the Hamaker equations.
Fig. 4 Interparticle vdW attraction and Born repulsion forces between the silicon nanosphere and the diamond flat substrate.
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The combination of eqn (7) and (12), or eqn (8) and (15), allows the calculation of the corresponding vdW attraction or Born repulsion forces between the nanosphere and the flat surface. As shown in Fig. 4, the calculated results can match the MD simulated results reasonably well. Note that, at close separation and especially after contact deformation, the force–displacement curve is not so smooth but displays a small fluctuation. This can reflect the steplike structure of two interacting surfaces due to the atomic discrete structure. Unlike the continuum models such as the Hamaker approach, in which the interacting bodies are treated as continuous media.39 Such treatment would result in a smooth force–displacement curve. However, the assumption of continuous media is not held at the molecular level in the case of nanoparticles. The nature of the discrete atomic structure in particular on the surface could not be ignored and would be reflected from the small fluctuations. With further increase in the deformation, dislocation of atoms may happen, which may lead to a small ‘jump’ in force–displacement.12 However, this phenomenon cannot be necessarily observed herein as the impact velocity herein is limited to 500 m s1 and the strain of silicon is below 0.1 (Fig. 5). Of note, there exist some difficulties in direct comparison with experimental results in terms of measured vdW forces. The main barrier lies in the size of silicon nanospheres. Due to the limiting computation efficiency, the size of nanoparticles in MD simulations is limited to below 5.0 nm radius. Therefore, a compromise between the size of the modeled system and the computation efficiency is made. In contrast, experimental studies on silicon nanospheres involve the particles ranging from 20 to 140 nm radius.46,48 What is more, most of them focus on contact forces. However, with the advancement of technology, this gap may be bridged in the future. Nevertheless, the present simulated results closely resemble the experimental ones in a qualitative way. Consider the case of the silicon nanosphere of 4.0 nm radius for example. The interparticle LJ force,
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i.e., the total of vdW attraction and Born repulsion forces, first decreases to a minimum with decreasing surface separation, then increases gradually at close separation and even changes the sign and becomes the repulsive force after contact deformation (ESI,† Fig. S-5b). This trend is similar to those measured in AFM experiments.49,50 In practice, the interparticle forces can be affected by physical properties such as density, shape, unequal particle size, and even possible ‘viscosity’ when interacting across one third medium, etc. The effect of density, the medium interacting across or unequal size can be taken into account by tailing the Hamaker constant or the particle radius in the Hamaker equation. For example, different nanoparticles have different densities according to A = p2Cr1r2, where r1 and r2 are the number densities of atoms of particles 1 and 2, respectively; thus the Hamaker constant A can be correspondingly modified using the corresponding value of the number density of atoms; when two particles interact across different media, normally the viscosity is different. In this case, the Hamaker constant can be adjusted by the following equation. Two nanoparticles interacting across a pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi
medium A132 A11 A33 A22 A33 where A11, A22 and A33 are the Hamaker constants of particles 1 and 2 and medium 3, respectively; for two nanoparticles with different radii R1 and R2, the R in the Hamaker equation can be replaced by the pffiffiffiffiffiffiffiffiffiffiffi geometrical mean R ¼ R1 R2 , the validity of which has been confirmed by the MD simulations.10 Moreover, the idealized spherical particles are often used in theoretical models or the ideal spherical shape is often assumed in dealing with particles in experiments, such as estimation of the surface area, although in practice most of particles deviate from the ideal spherical shape. Therefore, attention is paid to the spherical nanoparticles in this work. However, ellipsoidal particles are some of the most common non-spherical particles, and the effect of shape may be approximately considered by adjusting the aspect ratio and the particle size. The work based on ellipsoidal nanoparticles has been carried out and will be reported elsewhere soon. 3.3
Mechanical contact force
Apart from the LJ interactions in the forcefield COMPASS, the computation still encompasses valence interactions such as bond stretching, angle bending, angle torsion, and angle inversion, and there is always a force constant to describe these terms in their individual expressions.32 These valence interactions should be related to macro-mechanical properties, leading to the mechanical contact force related to contact deformation. Therefore, in addition to the interparticle vdW attraction and Born repulsion forces, the mechanical contact force Fc will arise upon the contact deformation of two nanospheres, which can be calculated by11 Fc = Fn FvdW FBorn pffiffiffi
Fig. 5 F c 8 2R2 E 3 as a function of strain dn/(2R) obtained in the approach process from MD simulations between the diamond substrate and silicon nanospheres of different radii ranging from 1.0 to 4.0 nm. Solid line represents the Hertz prediction using Young’s moduli of their counterpart bulks.
5852 | Phys. Chem. Chem. Phys., 2014, 16, 5846--5854
(16)
where the total normal force Fn was evaluated by accumulating the vertical forces (Y direction) that all atoms from the substrate exerted on the nanoparticle during the central collision process, FvdW and FBorn were obtained by differentiating interaction potentials with respect to surface separation d according
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to eqn (4). In detail, the forces of one particular atom along the X, Y, and Z directions at different simulation times can be the output, then the total normal force of the nanoparticle along one particular direction such as the Y axis, i.e., Fn, can be obtained by accumulating the vertical forces of all the atoms along the Y axis. Contact between solid spheres is usually described by the models of JKR51 or DMT,52 both of which are based on an earlier analysis by Hertz,2 who considered two elastic bodies in contact under an external load but ignored interparticle attractive forces. Normal contact forces between spherical particles 1 and 2 are given by the form,53,54 4 pffiffiffiffiffiffi Fn ¼ E R dn3=2 3
(17)
where 1/E* = (1 n12)/E1 + (1 n22)/E2, 1/R* = 1/R1 + 1/R2 in which E1 and E2 are Young’s moduli of particles 1 and 2, respectively, herein Young’s moduli of silicon41 and diamond55 are 130 and 1050 GPa, respectively, n is the Poisson ratio of 0.2841 and 0.0855 along the crystal plane [100], respectively, R1 and R2 are the radii of particles 1 and 2, respectively, thus E* is about 124.45 GPa, dn is relative movement, and dn = 0 corresponds to the first non-zero of contact force Fc. For contact forces between the elastic spherical particle of radius R and the flat surface, eqn (17) can be reduced to and given by pffiffiffi 4 pffiffiffiffi 8 2 2 3=2 Fn ¼ E Rdn3=2 ¼ E R e (18) 3 3 where e = dn/(2R). Note that since Young’s modulus of diamond is far greater than that of silicon, the deformation of diamond during impact can be ignored and considered as ‘rigid’, and silicon suffers from almost all the appreciable deformation. In addition, the MD simulations have proven that the COMPASS force field can accurately describe the mechanical properties of silicon materials such as Young’s modulus (ESI,† Fig. S-6),12 which can lend support to our current simulated results. The calculated results of contact forces Fc using eqn (16) show that the first non-zero value of mechanical contact forces arise when surface separation is about d = 0.16–0.18 nm. This value is consistent with the observed minima of interparticle LJ potentials from Fig. 2. The minimum point of the LJ potential (i.e., the zero point of LJ force) corresponds to the point where the occurrence of the mechanical contact force initiates, confirming that due to the intermolecular repulsive forces, the surface atoms are prone to be subject to ‘‘deformation’’ and hence the mechanical contact force arises when two surfaces are less than equilibrium separation distance apart. Fig. 5 shows the normalized mechanical contact forces between the diamond substrate and silicon nanospheres of different radii. These contact forces are in reasonable accordance with those predicted by the Hertz theory using Young’s moduli of diamond and silicon bulk and this confirms that the herein deformation is almost completely elastic. As observed from ESI,† Fig. S-5b, upon decreasing the surface separation, the interacting LJ forces (i.e., the total of vdW attraction and Born repulsive forces) first become
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adhesive, then become gradually repulsive around d E 0.16 nm. Before contact, the interparticle force is dominated by the LJ force (mainly the vdW attractive force) while there is almost no contact force. But after contact deformation, apart from the contribution of the LJ force, the contact force also arises. Thus both of them should be coupled together to represent the total interaction forces between the silicon nanoparticle and the diamond substrate using eqn (12), (15), and (18) over the whole surface separation and even after contact deformation. In JKR and DMT models, the contribution of the intermolecular force is reflected by the adhesive force, which is constant and evaluated by surface energy approximation.56 Detailed discussion and comparison with JKR and DMT models have been treated in one recent MD simulations,11 indicating that JKR and DMT models still can be used as the first approximation. Thus attention herein is paid to test the applicability of the Hertz model to the case of the nanosphere interacting with the flat substrate, but without giving emphasis to the comparison with the continuum models anymore.
4. Conclusions In this study, the interactions between a spherical nanoparticle and a flat surface including vdW attraction, Born repulsion and mechanical contact forces have been studied and formulated based on fully atomistic MD simulations. It is found that different from two interacting nanospheres, the force ratios between the nanosphere and the flat surface decrease gradually after one peak and tend to approach zero without exhibiting an asymptotically constant ratio at large enough separation. Thus two separate modified equations to the Hamaker approach have been proposed for interparticle vdW attraction and Born repulsion forces, respectively. Moreover, the results confirm that the continuum Hertz model still holds to describe mechanical contact force between the nanosphere and the flat surface. Finally, the behaviors of the nanoparticle system, such as aggregation, packing, flow, etc., are of importance to a variety of practical applications. The proposed formulas for inter-nanoparticle forces can serve as input parameters in the so-called cost-effective discrete element method (DEM) to study these behaviors in future.
Acknowledgements The authors gratefully acknowledge the support from the University of New South Wales.
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