## Abstract

The theory of low velocity impact between nano-sized, spherical particles and a rigid plane is developed by assuming fully plastic failure during approach and assuming the Johnson–Kandall–Roberts model of elasticity and adhesion applied during particle recoil. This model predicts initial particle acceleration on approach, followed by either deceleration and eventually instantaneous cessation of particle motion or extreme deformation and possible structural failure of the particle. If the approach motion ceases and the recoil begins, then either the particle is trapped on the surface or it separates. The typical frequency of oscillations of trapped particles is derived. Criteria for the occurrence of the various regimes are derived, together with conditions for when surface adhesion is important.

## 1. Introduction

The analysis of the interaction of macroscopic particles is a classical problem, beginning with Newton in 1687 with his coefficient of restitution (equalling the ratio between the final and initial relative velocities), being extended by Hertz in 1882 with his derivation of the elastic deformation equations, and perhaps reaching its zenith in the 1950s with the first derivation by physicists at the University of Cambridge of Newton's coefficient of restitution for quasi-static impacts (Tabor 1948) by using new results from the plasticity theory. This theoretical result has now been confirmed experimentally, with respect to velocity (Goldsmith 1960), plastic yield strength (Johnson 1985) and geometric and history dependence of the coefficient of restitution (Weir & Tallon 2005), giving confidence in our knowledge of the key processes occurring in macroscopic quasi-static particle interactions.

By contrast, a new set of behaviours occur at the nano-scale, arising primarily from the importance of surface forces (Chang *et al*. 1987) and from the small number and symmetry of component molecules. In nanotechnology, interactions arising from particle impacts, which can be important in fabricating hydrogen sensors and nano-scale devices (Reichel *et al*. 2006), are calculated using molecular dynamics (MD) simulations (Awasthi *et al*. 2006), which use force potentials, such as Lennard-Jones. Thus, at present, there is almost a complete separation between the parameters used in MD simulations of nano-particles and the corresponding parameters used in macroscopic particle impacts.

Adhesion is important at the macroscopic scale for materials with very low elasticity, such as gelatine (Johnson *et al*. 1971) and rubber (Greenwood & Johnson 1981; Maugis & Barquins 1989), which can conform to an external surface, and considerable research has been completed on such systems. Two complementary concepts have been developed, centred on the Cambridge and Moscow schools of tribology, which assume, respectively, geometrical interactions and action at a distance. This latter approach leads to difficult equations, although useful approximations to this concept have been obtained by Dugdale (Maugis 1992).

The desire to relate surface behaviour to surface properties has recently led to a theory of nanotribology (Johnson 1997), which describes the adhesion of elastic bodies to rough surfaces through a finite number of nano-scale asperities. A good agreement has been obtained for these elastic and surface energy problems by considering both the Johnson–Kandall–Roberts (JKR) and Deraguin–Muller–Toporov (Deraguin *et al.* 1975) theories. This theory has also been applied to atomic-force microscope (AFM) tips (Sarid 1991). Numerical methodologies, such as finite elements, also offer useful insights into regime development arising from impact problems (Thornton & Ning 1998; Mesarovic & Fleck 1999; Wu *et al*. 2005), and have shown that the hardness or the fully plastic average pressure varies with deformation and is not constant (Jackson & Green 2005).

Any discussion of impacts at the nano-scale involves a number of assumptions. The need to simplify such processes follows because it is not possible at present to complete accurate fundamental calculations, using methods such as time-dependent density functional theory (Burke *et al*. 2006; Walker *et al*. 2006), for the impacts of particles much greater than a nanometre in radius. MD methods can be applied to particles of many nanometres in radius and provide many useful insights into impact processes, but their use of artificial interaction potentials, while useful, introduces essentially arbitrary elements into the analysis. Compounding these difficulties, validation of theoretical predictions through direct experimentation appears to be impossible at present, because it is not possible to image the dynamic impact of nano-particles. Given these difficulties, we present a very simple model of impact at the nano-scale, in order to identify some of the major behaviours that may develop during such impacts.

This paper assumes that the idealization of a constant yield stress applies at the nano-scale, as suggested in the idealized plasticity theory (Nadai 1950; Hill 1983) for impacting nano-particles. Subject to this assumption, we then extend the theory of Newton's coefficient of restitution through the inclusion of adhesive forces, using a simplification of the methods of Johnson (1985), by using averaged mean quantities of surface tension and plastic yield over the whole of the region of geometrical overlap. We also assume that the impact occurs between an initially spherical particle and a much harder plane surface. This geometrical assumption is made because, while effective geometries are possible in elasticity, using the methods of Hertz, no such generalizations exist for the plastic regime. In addition, spherical particles avoid issues associated with friction (Bowden & Tabor 1950) and complexities arising from shape (Weir 2007).

Our analysis consists of three steps. First, we consider the approach regime, in which the particle approaches the plane and eventually stops, as the original kinetic energy is consumed in plastic work and stored in surface and elastic energy. Second, we consider the recoil regime, in which the particle begins to move away from the plane. We ignore plastic effects during this second step, which we assume holds until geometrical overlap ceases. The third step consists of pull-off (Bradley 1932) when extensional forces develop, with pull-off occurring when the extensional elastic or plastic forces allow the particle to overcome a repulsive potential barrier. Throughout, we ignore the complexities of a faceted surface (which would typically be considered in any molecular simulation) and assume that these effects can be averaged into an effectively spherical surface.

## 2. The approach regime

Considerable energy dissipation can occur during the approach regime, and we assume that this is characterized by a constant pressure (hardness) *Y* at the interface between the particle and the plane. This is introduced in part due to analytical simplicity and also because there is considerable experimental evidence (Meyers *et al*. 2006) for an approximately constant yield stress at the nano-scale for metals, although, at present, the value of such yield stresses cannot be predicted *a priori*. We ignore energy losses from elastic waves.

We also assume that a constant surface tension (surface free energy) *γ* applies at the interface between the two bodies. Again, there will be uncertainty as to the value of *γ*. If the approaching particle is in the liquid state, then the bulk value for a liquid may be applicable. However, if the approaching particle is a solid, and is partially crystalline, then the surface energy will be facet dependent (Vitos *et al*. 1998). Thus, we will treat *γ* as a free parameter dependent on the geometrical arrangement and material properties. Finally, we establish a simple energy balance between the kinetic energy of the particle, the work done plastically and the surface free energy,(2.1)where *m* is the mass of the approaching particle; *v*_{0} is the initial speed of the particle (relative to the plane); *v* is its present speed; *a* is the radius of overlap between the two bodies; *δ* is the distance of overlap of the sphere with the colliding surface; *γ* is the surface energy per unit area of each surface; and the plane is large and hard with respect to the approaching particle.

If *R* is the initial radius of the sphere and *δ* the maximum vertical distance of overlap, then we have(2.2)and(2.3)where the quadratic term −*δ*^{2} has been ignored, and then, from (2.3),(2.4)

Introducing the non-dimensional parameters *L*, *T* and *α* via(2.5)(2.6)and(2.7)allows (2.1) to be rewritten as(2.8)and so we have, from (2.8),(2.9)Hence, on first contact, the particle will accelerate, attaining its maximum velocity at *L*=*α*, after which the particle will decelerate and stop.

The solution of (2.8) is (Gradshetyn & Ryzhik 1980)(2.10)and the key properties of this expression are given in table 1.

Table 1 shows that (subject to the model being valid) all particles will reach a maximum in approach velocity, equal to times the initial velocity, after a non-dimensional time of arctan (*α*). Then, after another time interval at least equal to this, the particle stops. Since there is no limit on the value of *α*, it is quite possible that the maximum approach velocity far exceeds the initial velocity.

The force *F* on the particle can be found by differentiating (2.1) with respect to time, and hence from table 1, the force at the end of the motion, *F*_{0}, is(2.11)

The force is positive (directed away from the particle) during the initial approach period while *L*<*α*, but, after this, the force is negative and directed towards the particle. Interestingly, if the initial approach speed *v*_{0} tends to zero, then the force at the end of the motion is equal in magnitude, but opposite in direction, to the force at first contact. The force at the end of the motion is always negative at the end of the approach regime (provided the particle does stop before the overlap approaches the particle radius).

The maximum value of the radius of overlap, *a*_{max}, from (2.3), (2.5) and table 1, is(2.12)

Three implicit assumptions have been made in deriving (2.10). First, we have assumed that the deformation process becomes plastic very rapidly. This is not necessarily the case, as, for example, when an AFM tip gently taps a solid surface, which can be considered to be essentially an elastic and adhesive process. Second, we have assumed that the deformation is not so large at the time of the maximum velocity of approach, such that *δ* equals or exceeds *R*. If this happens, it is possible that the adhesive forces will dominate and could spread the particle into a pancake shape over the surface, and so the particle could transform approximately into a thin film. The extent of the particle overlap at the time of the maximum speed of approach, from (2.5), (2.7) and table 1, is(2.13)which is independent of the particle radius. Hence, the thin film regime can occur when the adhesive energy overwhelms the energy dissipation from plastic work.

Third, we have also assumed that the particle does not shatter or is destroyed in the collision. A possible estimate for this is to assume that *δ* exceeds *R* before the particle has stopped moving inwards. This is avoided from (2.5), (2.7) and table 1, when(2.14)and so two requirements must be met. First, there must be sufficient capacity for plastic work to consume the work performed by the adhesive forces (i.e. *RY*>2*γ*). If this is not the case, then pancaking of the particle or thin film formation may result. Second, when this is not the case, then the kinetic energy of the initial approach must not be more than about the excess of the capacity of the plastic work minus twice the surface work term. This may correspond to the destruction of the particle, as occurs from fragmentation, for example.

The expressions in (2.14) suggest introducing the energy scales(2.15)where *E* is the elastic parameter for the particle. From (2.15) and (2.7), we have(2.16)

The transition between small and large *α* values can be considered to occur when *α*=1, which implies the existence of a transitional radius, *R*_{γ}. From (2.7),(2.17)where *ρ* is the density of the particle. Particles whose radii are much smaller (larger) than *R*_{γ} correspond to large (small) values of *α*, in which case we expect then that the role of surface tension will be very important (unimportant).

The maximum distance of overlap, *δ*_{max}, for small *α* is(2.18)which is independent of *R*.

The maximum distance of overlap, *δ*_{max}, for large *α* is(2.19)which is independent of *v*_{0} (because then the surface adhesion produces a maximum approach velocity much greater than the initial velocity).

Finally, a possible important effect that can arise at the nano-scale is the force arising from free charge. One upper bound to the force acting on the particle is simply *πR*^{2}*Y*, for low velocity impacts. Another bound to the force acting on a particle is the pull-on force of 4*πR*_{γ}. If the impacting bodies are charged by a single electron, the maximum electrostatic force *F*_{e} between the bodies will be of the order of *e*^{2}/(4*πϵ*_{0}*R*^{2}), where *e* is the electron's charge of 1.6×10^{−19} C and *ϵ*_{0} is the permittivity of free space (8.8×10^{−12}). Equating forces and solving for the transition radius in both the cases above yields length scales at the atomic scale. Hence, we expect that effects from charge may not be very important for the low velocity impact of nano-scale-sized particles.

## 3. The recoil and pull-off regimes

Once the impacting particle has come to rest, the particle will be under compression and elastic forces will tend to separate the impacting surfaces, while surface tension will tend to keep the two surfaces together. The resulting recoil step depends on the relative magnitude of these two competing forces. Plastic work is not considered during the recoil regime.

We shall now proceed by assuming that the JKR theory applies to the recoil and pull-off regimes (Johnson & Greenwood 1997). The JKR theory arises from the Hertz theory, by assuming that the pressure field at the interface between the two impacting bodies is described by two potential functions, one arising from elasticity, involving a modified Young's modulus parameter *E*, and the other from adhesion, involving the surface energy *γ*. The total force *F* between the two bodies is(3.1)

Here, is the radius of curvature of the particle in the region of overlap, after plastic deformation has ended. Typically, the particle will be flattened from the impact, so that we expect that >*R*. The relationship between the radius *a* of overlap in the surface separating the particle and the plane and the position of the particle, *δ*, is defined as(3.2)where we use the same position symbol as in §2, but a different definition to (2.3). The total potential *U*_{E} is(3.3)

If is the initial (constant) potential energy of the system, before recoil begins, the equation for the subsequent motion is(3.4)

We proceed by non-dimensionalizing these expressions, and defining(3.5)(3.6)and(3.7)where *y*, *τ* and *K* are all non-dimensional. Substituting (3.5) into (3.3) yields(3.8)and so the zeros of *U*_{E} occur at(3.9)and the turning points of *U*_{E} occur at(3.10)

The JKR potential then is repulsive for *y* less than 1/6 (*y* is always positive, being a power of the area of overlap), after which the potential becomes attractive until *y*=1, and thereafter is repulsive. The repulsive part of the JKR potential between *y*=0 and 1/6 is not used in this paper, because we have used a plastic (i.e. not a JKR) model for approach, and with separation at *y*=1/6, we avoid the short-range repulsive part of the JKR potential. It is possible, of course, to smooth out the potential so that it approaches zero smoothly from below as *y* tends towards zero, but, in doing this, the relationship connecting *a* and , which is critical to our analysis, will be lost. An earlier analysis (Mesarovic & Johnson 2000) of separating elastic–plastic spheres has led to the concept of a decohesion map, which divides parameter space according to which physical mechanism is operating.

At the beginning of the recoil, the particle will begin with zero velocity and its subsequent motion will then depend on its initial condition. If the value of *U*_{E} at *y*=1/6 is *U*_{m}, and the initial value of is greater than *U*_{m}, then the particle will gain velocity so long as *y* is greater or equal to 1, and then it will begin to slow and eventually separate when *y*=1/6, with a kinetic energy of . If <*U*_{m} but greater than *U*_{E}(*y*=1), then the particle will oscillate between the two largest values of *y* for which *U*_{E}=. If <*U*_{E}(*y*=1), then the motion is impossible and the particle will remain at its point of maximum approach, given by the plastic model above.

The dynamic problem in (3.4) can now be analysed using (3.6) and (3.7),(3.11)

The minimum period of the oscillations about *y*=1 can be determined by expanding about this point,(3.12)(3.13)for a positive but small *ϵ*, which gives, from (3.11),(3.14)as the non-dimensional period for infinitesimally small amplitude oscillations about *y*=1. The corresponding dimensional period is obtained from (3.6).

The general period for trapped particles is obtained by selecting the endpoints (*y*_{1}, *y*_{2}) of the motion, adjusting *K* so that the kinetic energy is zero at these endpoints,(3.15)where(3.16)and so from (3.11), the period of these trapped motions is (the negative sign is omitted, because the integral in (3.11) runs between *y*_{2} and *y*_{1})(3.17)and the corresponding dimensional period is obtained from (3.6).

A plot of these periods (divided by ), the corresponding values of *K* and *y*_{1}, as a function of *y*_{2}, is given in figure 1.

When the initial conditions allow the particle to separate, then the lower limit of the integral in (3.11) is fixed at *y*=1/6, and the dependence on *y*_{2} of the time of overlap, the value of *K* and the kinetic energy at separation is given in figure 1. In contrast to the trapped regime, where the period of oscillation increases with increasing *y*_{2}, the separating regime has a decreasing time of overlap as the initial extent of overlap increases (from (3.11), the time of overlap depends asymptotically on the inverse one-third power of *y*_{2}). There is a rapid increase in *K* with *y*_{2} (from (3.11), *K* depends asymptotically on the 10/3 power of *y*_{2}).

The major remaining task is to link the parameters in the approach regime with the recoil regime. We perform this by defining *F*_{00} as the force acting on the particle at the beginning of the recoil, so that from (3.1) and (3.5), the maximum amount of overlap of the particle at the beginning of the recoil is given by(3.18)

From (2.11), the force is always directed towards the particle, and so will be positive at the beginning of the recoil, and, clearly, *F*_{00}=−*F*_{0}, and so the initial condition on *y*_{2} is determined from (2.11).

The critical value of *y*_{2}, of approximately 1.36983, separates the capture and escaping regimes. From (3.18), this critical value of *y*_{2} corresponds roughly to(3.19)and(3.20)

We proceed by assuming that, at the end of the approach regime, and at the beginning of the recoil regime, the force and the radius of overlap are continuous. The continuity of force fixes the new radius of curvature as(3.21)while from (2.12),(3.22)

From (3.22), (3.21), (2.11) and (3.18), the initial value *y*_{2} can be simplified to(3.23)and so the critical value of 1.36983 that separates the escape and capture regimes can be substituted into (3.23) to provide constraints on the energy scales.

In the limit of small *α* with respect to unity, the square of the coefficient of restitution can be rewritten as(3.24)where *v*_{f} is the final escape velocity and hence for a large initial velocity, or for a small surface tension, the coefficient of restitution is inversely proportional to the one-quarter power of *v*_{0} and is independent of *γ*, as has been found earlier (Johnson 1985); for very large values of *α*, the coefficient of restitution is zero, due to the particle being trapped on the surface.

## 4. Discussion and conclusions

We now discuss an estimate of the parameters considered previously. We consider a particle of 4 nm in radius, *R*=4×10^{−9} m, density *ρ*=3000 kg m^{−3}, elastic parameter *E*=10^{11} Pa, hardness *Y*=10^{9} Pa, initial speed *v*_{0}=30 m s^{−1} and surface tension *γ*=0.5 N m^{−1}. The corresponding mass *m* of the particle is 8×10^{−22} kg. The random thermal speed for a particle of this size, corresponding to a temperature of 300 K, is approximately 3 m s^{−1}.

We noted previously that free surface charge is likely to be unimportant for these parameter values, because the corresponding length scales are atomic. Of course, for particles of atomic size, the forces between free charge can become important, and the motion will approach the atomic analogue of Rutherford scattering, rather than the macroscopic approach used in this paper.

From (2.17), the size of *R*_{γ} is 23 nm, and, as we are considering a particle much smaller than this, surface tension effects will be very important for these parameter values. From (2.12), the maximum penetration distance *δ*_{max} is then independent of velocity and equals 2 nm, which corresponds to an extreme deformation, but is still less than the radius of 4 nm. The surface forces increase the initial velocity by a remarkable factor of 6 and the time taken to achieve this is approximately 10^{−11} s; another similar time interval is needed to decelerate the particle to zero speed. These times are independent of the initial speed.

The corresponding energy scales are as follows: _{Y}=2×10^{−16} N m; _{m}=3.6×10^{−19} N m; _{γ}=5×10^{−17} N m; and _{E}=6.4×10^{−15} N m. The non-dimensional numbers *α*=5.8 and *a*_{max}/*R*=1 show that the particle is highly deformed, well beyond the small deformation assumptions in this paper. The value of *y*_{2} at the beginning of the recoil is only 1.04, which is much less than 1.37 for escape, so the particle will be trapped on the surface and the coefficient of restitution will be zero for such high values of surface energy. The region of overlap is highly flattened, with /*R*=3.6. The oscillations on recoil have a period of approximately 8×10^{−12} s and analogous oscillations appear in corresponding MD simulations (Awasthi *et al*. 2006).

The particle should acquire a kinetic energy on escape of at least *U*_{E}(*y*=1/6), and so there will be a minimum speed of particle escape. Hence, in this model, non-zero values of the coefficient of restitution are always bounded away from zero. For the specific parameter values under discussion, the energy of the barrier at *y*=1/6 is approximately 3.4×10^{−18} N m, and so the minimum velocity of escape is approximately 100 m s^{−1}. This significant velocity arises owing to the nano-scale of the particles. Specifically, for large values of *α* and , the minimum velocity of escape *v*_{min} is approximately(4.1)which will be small, typically, owing to the presence of *Y* in the denominator, unless *R* is sufficiently small, which happens in the case under discussion.

Finally, this paper has assumed that a constant yield stress applies throughout the approach regime. For very low velocity impacts, especially those which just produce plasticity (approx. 0.1 m s^{−1}, say), this is known to be a poor approximation (Kharaz & Gorham 2000). The fully plastic yield typically does not occur until impact velocities are approximately 100 times greater, or approximately 10 m s^{−1}. The estimates above show that such velocities will be exceeded often at the nano-scale, so that the fully plastic assumption made here should be justifiable. An increase in material hardness is known to occur for length scales smaller than the micrometre range, due to the Hall–Petch effect, but whether this trend increases or decreases at the nano-scale is still unresolved (Meyers *et al*. 2006).

Our approach has allowed the derivation of explicit expressions for impact behaviour, which is perhaps the main achievement of this paper. Nevertheless, these high velocities are associated with high surface energies, which, in turn produce very significant deformations that can violate the small deformation assumptions made, especially in (2.3) for the expression connecting the overlap distance and the overlap radius. Consequently, further work will be needed to link the results derived here with fully numerical approaches.

## Acknowledgments

The authors are grateful to Shaun Hendy and Aruna Awasthi for their helpful discussions.

## Footnotes

- Received October 29, 2007.
- Accepted January 23, 2008.

- © 2008 The Royal Society