Effect of van der Waals force cut-off distance on adhesive collision parameters in DEM simulation

    Effect of van der Waals force cut-off distance on adhesive collision parameters in DEM simulation Hamed Abbasfard, Geoffrey Evans, Roberto Moreno-Atanasio PII: DOI: Reference:

S0032-5910(16)30254-6 doi: 10.1016/j.powtec.2016.05.020 PTEC 11660

To appear in:

Powder Technology

Received date: Revised date: Accepted date:

15 December 2015 10 May 2016 14 May 2016

Please cite this article as: Hamed Abbasfard, Geoffrey Evans, Roberto Moreno-Atanasio, Effect of van der Waals force cut-off distance on adhesive collision parameters in DEM simulation, Powder Technology (2016), doi: 10.1016/j.powtec.2016.05.020

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ACCEPTED MANUSCRIPT Effect of van der Waals force cut-off distance on adhesive collision

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parameters in DEM simulation

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Hamed Abbasfard, Geoffrey Evans, Roberto Moreno-Atanasio*

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Chemical Engineering, The University of Newcastle, Callaghan 2308 NSW Australia

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Abstract

Rapid advancement in computer technology makes it possible to perform simulations of large

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particulate systems using Discrete Element Method (DEM). However, it is still a challenge to adjust the DEM simulation parameters, especially the interparticle forces, in order to obtain

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rational results. Many of these forces diverge as the distance between particle surfaces approaches zero. Therefore, a cut-off distance needs to be considered in order to avoid such a

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problem. This work presents the results of the computational analysis of the influence of the cut-off distance, which is required to avoid the divergence of the van der Waals force, on the collision of a single particle against a flat surface. Hence, the effect of the cut-off distance on the coefficient of restitution, collision duration and maximum overlap has been studied. In addition, we have theoretically derived an expression for the minimum velocity under which the particle remains adhered to the surface (critical velocity) as a function of the cut-off distance. The simulation predictions of the critical velocity are within the range of experimental data published in the literature. We demonstrate that the cut-off distance has a profound influence on particle rebound and therefore, a careful selection of this parameter

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[email protected] 1

ACCEPTED MANUSCRIPT should take place when simulating bulk particle behaviour. Given that the hydrophobic force is usually simulated using the same expression as the van der Waals force, the results

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presented here could also be considered in the context of the simulation of the hydrophobic

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force.

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Keywords: DEM simulation; Van der Waals cut-off distance; Critical impact velocity; Contact mechanics

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1. Introduction

Particle technology is a rapidly developing interdisciplinary research area as many natural

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phenomena as well as industrial applications involve the handling and transport of powders and granular solids [1-3]. However, the relationship between single and bulk particle

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properties is still unclear and there is considerable effort by researchers in the field to provide a reliable prediction of powder and granular solid behaviour under different external

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conditions [4, 5]. Such research efforts have led to the development of the computer simulation technique known as Discrete Element Method (DEM) [4-15]. DEM is based on

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Newton‟s second law to model particle motion and contact mechanics laws to determine the elastic force between overlapping particles [6-15]. This technique has allowed the simulations of bulk particle behaviour to be carried out due to the ease in which macroscopic properties directly emerge from the behaviour of individual interacting particles [6-15]. The computational efficiency of any DEM computer code is mainly determined by the efficiency of the contact detection and computation of the forces which act on the particles [16]. These forces can be classified into two categories: interparticle forces and environmental forces. The former refers to the interactions between pairs of particles, whether the particles are contacting or not, while the latter refers to hydrodynamic interactions, gravity and other external fields. Special attention has been paid to the analysis 2

ACCEPTED MANUSCRIPT of the influence of interparticle interactions on bulk behaviour as the dynamics of most macroscopic systems are controlled by the nature and strength of these forces [17-22]. As the

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strength of these forces increases with decreasing interparticle distance, their accurate

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estimation requires a reduction of the time step with a consequent increase in CPU time.

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Furthermore, when the particles come into contact the use of a specific contact deformation model may also increase the CPU time. This increase in computational cost has forced many researchers and engineers to make use of more simple contact models, which can reasonably

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mirror the real system.

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Contact forces can be simulated using either a “hard-sphere” or a “soft-sphere” model [6, 23, 24]. The soft sphere model was developed by Cundall and Strack [6]. In this model, forces

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are calculated from the overlap of the particles at a given time step. In contrast, the hard-

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sphere model, which was developed by Campbell and Brennen [23, 24], is based on the

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calculation of the particle velocities by momentum conservation of binary collisions. Hertz [25] developed a solution for the contact of two non-conforming elastic bodies under

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the approximation of small deformation. Johnson et al. [26] proposed a contact theory for two adhesive solids that based on an energy balance predicts the external force required to separate two bodies of given surface energy and size. However, these models were developed for quasi-static conditions and their applicability to impact processes still remains to be proven. These limitations have stimulated the analysis and development of new contact models and the analysis of their influence on bulk particle behaviour [27-35]. The work by Di Renzo and Di Maio [31] provided a thorough analysis of the influence of the linearity of the contact model on the actual contact force and velocity during the collision process. They showed how these two parameters are in relatively good agreement when a normalized collision time was used. However, the actual collision duration was very different 3

ACCEPTED MANUSCRIPT for both linear and non-linear models and in addition the collision time depended on the impact velocity when the non-linear contact models were used [31].

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Thornton and Ning [33] derived an expression for the restitution coefficient for elasto-plastic

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deformation of adhesive particles under normal impact. Their work evidenced the influence

despite that no damping force was used.

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of the attraction between particles, defined as surface energy, on the restitution coefficient, They also carried out an energy balance to

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determine the „critical velocity‟ over which particle rebound was observed. Adhesion can be simulated in different ways based on Johnson et al. or DMT models [21, 22,

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25, 26, 36], using the expression of the van der Waals force as given by Hamaker [21, 22, 37, 38] and by considering a force proportional to the deformation [12, 13, 14, 19, 32, 35]. The

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origin of the van der Waals force is found in the interactions between permanent, induced and permanent-induced electric dipoles [38].

Two different approaches have been used to

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quantify the van der Waals force: a) the additivity approach of Hamaker [37], which is based on the sum of the individual interactions between pairs of dipoles and b) the Lifshitz [39]

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theory, which is based on quantum field theory and treats the interacting bodies as continuous macroscopic materials. However, Israelachvili [38] remarked that both approaches provide the same expressions for the interaction energies although the way in which the Hamaker constant is calculated is different. The van der Waals force diverges as the body surfaces approach, and hence a cut-off distance should be considered, setting the force to a constant value once a certain minimum distance of approach has been reached. This cut-off distance provides a maximum value of the inter-particle force for a given particle size [38] and its value should typically reflect the length scale at which the macroscopic properties of the particles dominate over molecular interactions [38, 40, 41]. However, there is no general

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ACCEPTED MANUSCRIPT agreement in the literature on the proper choice for the cut-off distance, although a range of 0.1-1.0 nm has been commonly selected in particle system simulations [21, 22, 42, 43].

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The most important consequence of the use of a given cut-off distance is likely that the

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potential energy of the particles directly depends on this cut-off distance and thus the critical

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velocity will change. For example, for a spherical particle decreasing the van der Waals cutoff distance from 1 nm to 0.1 nm increases the potential energy by about 90 times. The

(1)

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relative difference in potential energy can be calculated as (detailed in Appendix A):

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where ep and d0 are the potential energy and the van der Waals cut-off distance, respectively.

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This paper firstly presents a brief comparison of the linear elastic spring and the linear elastic

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spring-dashpot models [6, 28, 31, 34, 35] with the objective of setting a benchmark to analyse the influence of the cut-off distance on collision parameters. Then, the analysis of the effect of the strength of the van der Waals force and cut-off distance on the collision of a particle

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against a surface is presented. Specifically, we have focused on the effect of cut-off distance on critical velocity, coefficient of restitution, collision duration and maximum overlap. A wide range of impact velocities, particle size and cut-off distances are compared in this study. It is also worth noting that a similar expression to the van der Waals force is applied to the simulation of the hydrophobic force [44] and thus a cut-off distance is needed [45]. Thus the results presented here will also have direct applicability to the simulation of the hydrophobic force. 2. Methodology 2.1 DEM formulation

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ACCEPTED MANUSCRIPT A computer code has been developed in MATLAB to simulate the collision of a single particle against a flat surface. The collision process is schematically shown in Figure 1. A

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finite difference form of Newton‟s second law was used to determine the particle position,

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velocity and acceleration in each time step. The new position of the particle centre, y(t+∆t),

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after one time step, Δt, along the y-direction (normal to the surface) can be found as: (2)

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Where F(t)is the total force acting on the particle at time t, m is the mass of the particle and

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v(t) is the velocity of the particle.

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Figure 1

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2.2 Forces

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The total force is the sum of all forces acting on the particle at any given time. Only two types of forces have been considered here: the van der Waals, FvdW and the elastic contact

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force, Fc (we are only considering a normal impact and therefore we have not simulated any tangential force).

(3)

The driving force for the particle motion is the van der Waals force (Eq. 4). A simplified expression [36, 51] for short distances was used to calculate the value of the van der Waals force between the spherical particle and the flat surface: (4)

where H is the Hamaker constant, d is the separation distance between the particle and wall surfaces and R is the radius of the particle. This equation represents a fairly good fit for a 6

ACCEPTED MANUSCRIPT sphere-plate non-retarded van der Waals interaction for separation distances of up to 10 nm and particle radius of up to 100 nm [38, 46].

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Given that Eq. 4 diverges when the distance between surfaces approaches zero, a cut-off

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distance, d0, is required to avoid this problem. Therefore, for distances smaller than d0 the van

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der Waals force was considered to be constant and equal to HR/6d02.

The soft sphere model was used to simulate particle-surface collision [6]. In the soft-sphere

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approach, the overlap between the two colliding surfaces is represented as a spring-dashpot system. The spring causes the rebound off the colliding particle and the dashpot mimics the

(5)

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dissipation of kinetic energy due to inelastic collision [28].

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where kn, η and v are the normal contact stiffness, damping coefficient and particle velocity, respectively. The damping coefficient, η, can be expressed as a function of the critical

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damping coefficient, ηc, which is the value of the damping coefficient required to stop the oscillations of two bodies in contact in the absence of external forces, as:

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(6)

where β is the damping ratio and the critical damping is given by: (7) with ω0 being the natural frequency of oscillation of the equivalent spring which can be calculated as:

(8)

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ACCEPTED MANUSCRIPT 2.3 Model parameters The particle properties used during the course of the simulations are summarized in Table 1.

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gap between particle surface and wall was set to 10 nm.

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Particle diameter was fixed as 10 µm in all simulations unless otherwise specified. An initial

Table 1

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The time step (Δt) in the DEM simulations should be selected in such a way that the force evolution during the collision process can be predicted with a high level of accuracy. For the

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typical particle properties used in the simulations (Table 1) the time step was selected to ensure negligible numerical errors and was equal to 1×10-9 s, which is slightly less than

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3. Results and discussion

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approximately 0.01 τc [47].

3.1 Contact model comparison in the absence of adhesion

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In order to set up a benchmark for the analysis of the influence of the cut-off distance on impact characteristics a comparison between predictions of linear elastic spring (LS) and linear elastic spring-dashpot (LSD) models has been carried out.

Figure 2 shows the predicted contact force (elastic plus damping) versus particle-surface overlap for the LS and LSD models. The overlap is given by the difference (y-R) between particle position, y and particle radius, R. Note that the loading and unloading profiles for the LS model are identical since the collision is perfectly elastic (non-dissipative). In addition, the elastic force is always positive indicating that it tries to separate the two particles. In contrast, the loading and unloading curves for the dissipative model, LSD, do not overlap as a 8

ACCEPTED MANUSCRIPT consequence of the existence of damping, which has the same direction as the elastic force during loading and the opposite direction to the elastic force during unloading. The area

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enclosed by the curve is the total energy lost during the collision.

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Figure 2

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Furthermore, a nonzero force at δ = 0 is observed for both the loading and unloading phases in the LSD model, though its value is positive (repulsive) during loading and negative during

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unloading [34]. These two latter effects are a direct consequence of the damping force being

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proportional to the loading velocity and thus it predicts a nonzero force at zero deformation.

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3.2 Effect of van der Waals cut-off distance on the dynamic of the particle

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Figure 3 shows the position of the particle centre versus time for three different values of cutoff distance of van der Waals force. The particle was initially positioned at 10 nm above the

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flat surface (surface to surface distance). An initial velocity equal to 0.01 m/s was given to the particle. A damping ratio of 0.1 was used for the contact force calculations.

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Figure 3

As shown in Figure 3, the collision durations as well as the maximum overlaps are quite different from case to case. The curve shows the oscillations typical of an impact event. However, the most important feature is that as the cut-off distance becomes shorter, the maximum overlap increases as a result of the increase in potential energy due to the van der Waals interaction. When the cut-off distance is shorter than approximately 0.15 nm, the particle is no longer able to rebound and it remains adhered to the surface undergoing damped oscillations. Therefore, it is clear that a critical value of cut-off distance over which the particle rebounds exists. In other words the cut-off distance has an important influence on controlling the rebound of the particle. 9

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3.2.1 Critical velocity: Theoretical solution

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The critical velocity is a key parameter of the impact dynamics and it is directly related to the strength of the adhesion force [48] as any collision with a velocity below the critical velocity

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will result in the particle being adhered at the boundary upon which it is colliding. All previous studies [48-56] in the context of finding the critical velocity, required the

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measurement of kinetic energy loss over a wide range of velocities, particle sizes, and material properties so that the effects of plastic deformation as well as adhesion could be

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taken into account. In this study, a constant van der Waals force calculated at the cut-off distance representing the adhesion force was applied. By solving analytically the equation of

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motion during the collision, it is possible to determine the critical velocity as a function of the collision parameters. Accordingly, the expression for particle acceleration during contact is

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given by:

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(9)

where the first, second and third terms on the right hand side of Eq. 9 represent the damping, elastic and adhesion forces, respectively. Eq. 9 can be rewritten as a function of the damping ratio and the natural frequency of oscillation of the equivalent linear elastic spring as: Where (10)

The solution to Eq. 10 depends on the value of the damping ratio β. For β<1 (underdamped system) an analytical expression describing particle motion during contact considering the

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ACCEPTED MANUSCRIPT initial conditions v(t=0) = Vimp and y(t=0) = R has been obtained in the form (see Appendix B for details):

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(11)

Hence, we can establish a rebound criterion as follows: „The critical velocity corresponds to the minimum impact velocity which allows the position of the particle centre to be equal to

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the particle radius (y=R) during collision‟. Therefore, when y=R the particle separates from the surface. Therefore, the critical velocity, Vcrit, can be obtained by solving the following

(12)

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equation:

while Eq. 12 has been derived for the impact of a spherical body upon a planar surface it may

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readily be extended to include the case of impact between two spherical bodies by using the reduced radius of the two bodies [54].

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3.2.1.1 Approximation for small values of β Eq. 12 can be approximated for small values of the damping ratio, β, considering the McLaurin expansion of the square root and the exponential: (13) (14) Eq. 13 is a second order approximation as the first order term is null. Therefore, we should be able to approximate the square root as 1 while using a first order approximation of the exponential.

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ACCEPTED MANUSCRIPT Therefore, considering that the product ω0τ is of the order of 2π, βω0τ is close to zero and thus the exponential in Eq. 12 is close to 1. In addition, the argument of the trigonometric

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functions can be considered to be approximately equal to ω0τ. We can now write Eq. 12 for

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small values of β as:

(15)

Clearly, Eq. 15 shows an approximately linear relationship between critical velocity and

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damping ratio for small values of the latter parameter. It is important to note that for small

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values of β the collision time, τ, will approximate as 2π/ω0, which is the value of the collision for a linear elastic spring model. The actual linearity of this expression will be demonstrated

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in the following section using DEM.

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3.2.1.2 Effect of cut-off distance, particle diameter and damping parameter

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In Figure 4a, b and c the values of the critical velocity obtained by using Eq. 12 are plotted versus cut-off distance, particle diameter and β, respectively. The particle size was fixed as

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10 µm in diameter in Figs 4a and c (Table 1). In Fig. 4a-c the area below each curve represents a region of parameters where the incident velocities are smaller than the critical values (adhesion region) and therefore particle capture due to the van der Waals force takes place. On the other hand, the area above each curve corresponds to the cases in which the impact velocity is larger than the critical value (rebound region) and therefore, particle rebound is observed. All three figures reveal that a decrease in cut-off distance or an increase in damping ratio, β, results in higher critical velocity.

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ACCEPTED MANUSCRIPT Figure 4 The influence of cut-off distance is better observed in Fig. 4a. It is clear that the curves for

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difference β values tend asymptotically to infinity when the cut-off distance approaches zero.

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This result should be expected as the van der Waals force equation diverges for small values

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of the separation distance and thus the adhesive potential energy will become infinity. Thus, no particle would be able to rebound.

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The effect of particle size on the critical velocity for various values of β and cut-off distance is shown in Figure 4b. The increase in particle size decreases the critical velocity following a

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power law relationship (Fig. 4a is on a log-log scale). Larger particles possess greater kinetic energy and thus for the same impact velocity the adhesion energy is proportionally smaller,

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thus facilitating the rebound of the particle. It is also important to observe that for a constant

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particle size the effect of cut-off distance is stronger than the effect of the damping ratio. For

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instance, let‟s consider the case of β = 0.05 and d0=1 nm as a reference. The reduction of the cut-off distance by one order of magnitude results in an increase of nearly two orders of magnitude in the critical velocity. In contrast, reducing the damping ratio by one order of

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magnitude results in the critical velocity being reduced by approximately the same order of magnitude.

Figure 4c reveals the dependency of critical velocity on β at a given particle diameter and different cut-off distances. The increase of critical velocity with β is clearly a non-linear relationship. The plot suggests that the curves will tend asymptotically to infinity as β approaches 1. This result is expected as a value of β = 1 would produce the full damping of the collision in the absence of adhesion. The results for small values of β show a linear relationship between critical velocity and damping ratio, which is consistent with Eq. 14. 3.2.1.3 Comparison to experimental data

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ACCEPTED MANUSCRIPT The predictions of the critical velocity as a function of particle size obtained by solving Eq. 12 are plotted in Figure 5 alongside other relevant data from the literature for comparison

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[49]. In general, all slopes for these widely different particle and surface combinations were

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significantly steeper than negative unity. A slope significantly less than unity (-0.39) occurred

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only for the results of Esmen [53]. The steepest slopes (-4.79) were reported by Rogers and Reed for impacts between copper and harder materials (steel) [54]. The slope of the present model (-0.5) is within the same range as the slopes of the experimental data (Fig. 5). As can

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be seen (Fig. 5) the critical velocity changes by selecting different cut-off distances.

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However, the slope of the theoretical lines remained unchanged. The difference in the slope of the relationship between critical impact velocity and particle size with respect to the

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different experimental cases may be due to a number of different reasons. These reasons

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include material properties such as plasticity or strain rate dependency of the deformation of the real particles, and thus the use of a linear elastic spring which is a simplified version of

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reality will not be appropriate. Furthermore, the damping model predicts a constant restitution coefficient independent of the impact velocity, which is very well known to be unrealistic.

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Therefore, the study of the influence of particle size on critical velocity will need a further consideration to adapt the stiffness of the particle to the actual size. A non-linear relationship between the particle stiffness and particle diameter represents a better agreement with the experimental data. The coloured dashed line (Fig. 5) shows the corresponding non-linear relationship which is defined as: (16)

where dp* and kn* are the particle diameter, 10 µm, and the particle stiffness, 1000 N/m respectively, that were used as the basic input data throughout all simulations (Table 1). Plotting the simulation results using Eq. 12 combined with Eq. 16 provides the slope equal to 14

ACCEPTED MANUSCRIPT -1.4 which is exactly the same as the results of Wall et al. [49] for ammonium fluorescein particles colliding against a silica target surface. In our model it has been assumed that both

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particle and surface are made of silica.

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Figure 5

3.3 DEM analysis of the influence of cut-off and damping ratio on collision parameters

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3.3.1 Effect of initial particle velocity on impact velocity

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Figure 6 shows the simulation predictions for the impact velocity as a function of the cut-off distance for different initial velocities, V0. In addition, we have plotted the criterion for the critical velocity corresponding to β = 0.3. The shaded area represents the values of the impact

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velocities for which the particle remains attached to the surface. Please note that due to the van der Waals attraction, the particle accelerates towards the surface and thus the actual

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impact velocity is different from the initial velocity given to the particle at its initial position with a separation gap of 10 nm between the particle and surface.

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Figure 6 reveals that for initial velocities above 0.05 m/s, the impact velocity is approximately equal to the initial particle velocity and independent of the cut-off distance, as the initial kinetic energy is much larger than the adhesion energy. In addition, for initial velocities of less than 0.01 m/s the impact velocity is much larger than its initial velocity. This fact suggests that the van der Waals potential energy was much larger than the initial kinetic energy. Therefore, we should expect that the impact velocity is independent of the initial velocity of the particle. Furthermore, as the adhesion energy decays as 1/d0, the impact kinetic energy for the case of small initial velocity compared to the adhesion energy decreases by the same factor (1/d0). Therefore, the actual impact velocity decreases with the square root of the inverse of the cut15

ACCEPTED MANUSCRIPT off distance (1/d00.5). In comparison, the critical velocity (Fig. 6 dashed line) decays as 1/d02 according to Eq. 12. Therefore, both the curves for impact velocity and critical velocity cross

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each other at some stage.

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Figure 6

3.3.2 Maximum overlap during collision

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Figure 7a-c shows the maximum overlap for values of β equal to 0.1, 0.3 and 0.5. For the

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cases in which the particle undergoes damped oscillation during contact, the first trough corresponds to the maximum overlap as shown in Figure 3a.

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Figure 7

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The same trend of the maximum overlap as a function of cut-off distance can be seen in

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Figure 7a-c. However, the actual values of the maximum overlap are different since as the damping force decreases the maximum overlap increases (Figure 7 a-c). We can also observe that the influence of the cut-off distance increases with decreasing impact velocity as the

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initial kinetic energy becomes smaller than the adhesion energy. Specifically, for impact velocities between 0.1 m/s and 0.05 m/s the maximum overlap is relatively insensitive to the cut-off distance for values larger than 0.4 and 0.5 nm, respectively. The maximum overlap of the particle is a direct consequence of the balance between kinetic, elastic, damping and van der Waals potential energy. The larger the cut-off distance, the smaller the contribution of the van der Waals energy compared to the initial kinetic energy. Therefore, the final particle deformation becomes independent of the van der Waals interaction (and thus on the cut-off distance) and only becomes dependent on the initial kinetic energy. These results agree well with the insensitivity of the impact velocity to the cut-off distance observed in Figure 6 for values of initial velocity of 0.05 and 0.1 m/s. However, for values of initial velocity of less 16

ACCEPTED MANUSCRIPT than 0.01 m/s the maximum overlap decreases monotonically with cut-off distance within the range of values of the latter parameter studied here. It is also important to note that even for

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value of this parameter is less than 1% of the particle diameter.

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the largest value of maximum overlap (Fig. 7a, β = 0.1, d0 = 0.1nm and V0 = 0.1 m/s), the

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3.3.3 Collision duration

The collision durations as a function of cut-off distance for damping ratios of 0.1, 0.3 and 0.5

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are shown in Figure 8a-c, respectively. The curves correspond to values of initial impact velocity equal to 0, 0.01, 0.05, 0.10 m/s. Only the cases in which particle rebound is observed

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(Fig. 6) has been plotted in this figure. In addition, we have added for reference the coefficient of restitution predicted for the case of non-adhesion force.

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The collision duration decreases with increasing cut-off distance and the curves seem to converge into a single value for each damping ratio (Fig.8 a-c). The increase in cut-off

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distance decreases the maximum attractive force and thus the particle is able to rebound faster. Therefore, the increase in adhesion force as the cut-off distance approaches zero

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produces an asymptotically increasing value of the collision duration as the adhesion energy becomes infinity.

Figure 8

By comparison of Figs. 8a to c, it is clear that the increase in initial velocity produces a decrease in collision duration. This is a very well-known and understood fact [25]. It is also possible to note that the increase in damping ratio, β produces an increase in the duration of the collision as a consequence of the increase in the amount of kinetic energy that has been dissipated during the impact process. Nevertheless, the relationship between damping, cutoff distance and collision duration seems clearly non-linear.

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ACCEPTED MANUSCRIPT 3.3.4 Coefficient of restitution The coefficient of restitution, en, is the ratio of the impact velocity to the rebound velocity.

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The square of the restitution coefficient is directly proportional to the loss of kinetic energy

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and in our case therefore, it is related to the adhesive energy and energy loss in damping.

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The dependency of the coefficient of restitution on the cut-off distance is displayed in Figure 9 a-c for values of β equal to 0.1, 0.3 and 0.5, respectively. In addition, we have added for

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reference the coefficient of restitution predicted for the case of non-adhesion force. Referring to Figure 6, it can be seen that obviously only those data points corresponding to the rebound

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region are considered for the computation of the coefficient of restitution.

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Figure 9

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The decrease in cut-off distance increases the attraction of the particle towards to the surface and thus the collisions become longer as shown in Fig. 8. Therefore, the amount of energy

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dissipated in the form of damping increases with respect to the case of no-adhesion and the actual restitution coefficient consequently decreases. In contrast, the larger the cut-off

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distance the smaller the attraction is and the curves for the different impact velocities seem to converge into a single value which corresponds to the no-adhesion case (Fig. 8a-c). Similarly, when the initial velocity increases, the kinetic energy component increases with respect to the adhesion energy and thus the restitution coefficient starts to increase and approach to the non-adhesion case. Finally, it is clear by comparing Figures 9a, b and c that the coefficient of restitution strongly depends on the damping ratio. The larger the damping ratio, the more energy is dissipated and the smaller the values of the coefficient of restitutions are.

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ACCEPTED MANUSCRIPT 4. Conclusion The implementation of the van der Waals force in computer simulations requires the

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necessity of setting a minimum cut-off distance in order to avoid the divergence of the force

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expression when two particles come into contact. The maximum adhesion force has been

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considered to be a constant value determined at a cut-off distance. This work presents the effect of this cut-off distance on particle rebound, coefficient of restitution, collision duration

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and maximum overlap during the collision.

We have derived a criterion for the critical velocity in the presence of adhesion by solving the

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motion equation. The solution for the critical velocity as a function of particle size obtained in this study is plotted along other relevant data from the literature for comparison.

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It was found that the critical velocity under which the particle rebounds was clearly affected by the value of the cut-off distance and its influence depended on the value of damping ratio

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and particle size. The critical velocity seems to approach asymptotically to infinity as the damping ratio approaches 1. Similarly, we have shown theoretically and by DEM simulations

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that the critical velocity approaches asymptotically to infinity when the cut-off distance approaches zero. All the results suggest that significant changes in the collision parameters are observable particularly for smaller cut-off distances and particle velocities. The analysis of the impact velocity as a function of the cut-off distance has shown that for initial velocities smaller than 0.01 m/s the particle accelerates towards the target due to the attractive potential energy. The impact velocity was independent of the initial particle velocity for small cut-off distances (and thus large adhesion energies). Furthermore, the final impact velocity decreases more slowly with cut-off distance than the critical velocity showing that both velocities profiles cross each other. Therefore, for a given initial particle velocity the particle will rebound or adhere depending on the cut-off distance.

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ACCEPTED MANUSCRIPT The influence of cut-off distance on the maximum overlap, and collision duration becomes more significant for small values of cut-off distance and particle velocity as a consequence of

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the increase in adhesion energy compared to the initial kinetic energy.

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The coefficient of restitution increased with cut-off distance and reached a maximum value

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independent of the latter parameter. This is a direct consequence of the decrease in van der Waals potential energy with increase in cut-off value and thus the system starts to behave as a

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linear elastic spring dashpot in which the restitution coefficient is independent of the impact velocity. Then, the restitution coefficient becomes only a function of β and independent of

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the impact velocity and material properties.

Therefore, careful consideration of the selection of the cut-off distance needs be taken and the

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the specific problem.

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cut-off distance should be selected depending on the range of collision velocities involved in

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In conclusion, the system examined here is restricted to normal impacts between a particle and flat surface. The extension of these findings to the more complex case of oblique

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collisions and collisions involving many particles remains largely untested.

Acknowledgment Mr Abbasfard would like to thank The University of Newcastle for the granting of the Postgraduate Research Scholarship (UNPRS).

Appendix A The potential energy for a spherical particle in contact with a surface, and thus at zero separation distance, under the influence of the van der Waals force can be calculated as: 20

ACCEPTED MANUSCRIPT (A.1) The integ If we consider the necessity of using a cut-off distance equal to d0 below which the van der

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Waals force becomes constant, we can decompose the integral into two terms:

Where (A.3)

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Therefore, if we integrate we obtain

Where (A.2)

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If we consider now the relative difference between two cut-off distances we obtain:

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(A.4)

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Appendix B

The particle-surface collision is treated using the LSD model thus the expression for particle

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acceleration during contact is given by: (B.1)

For convenience the terms β, ω0 (as Eqs. 9 and 10) and B.1 can be rewritten as:

are introduced so that Eq. Where T (B.2)

This equation is a non-homogeneous linear second-order ODE whose corresponding general Where solution is: (B.3)

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Where

ACCEPTED MANUSCRIPT Where y1(t) and y2(t) are the general solutions of the homogeneous problem and yp(t) is a particular solution of the non-homogeneous problem. For solving the homogeneous problem

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we try a solution of the form y=ert, where r is an unknown constant. Substituting this function

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into the homogeneous ODE we obtain:

F(r)

is

called

the

characteristic

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Hence;

polynomial.

Solving

(B.4)

(B.5) Where the

original

ODE

is

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the general solution can be written:

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reduced to solving an algebraic equation. For an underdamped system the characteristic Where polynomial has complex conjugate roots of forms a+ib and a-ib where a and b are real. So,

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(B.6)

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Using Euler‟s formula we have:

(B.7) Where

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On the other hand, the particular solution can be obtained by assuming yp as a constant parameter and substituting it into Eq. B.2. Then, it is found as:

Where (B.8)

Using the initial conditions at t=0, y(0)=R and dy/dt(0)=Vimp then C1 and C2 are determined and then the complete solution is given as:

Letting

Where (B.9)

, Eq. B.9 is exactly the same as Eq. 11.

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ACCEPTED MANUSCRIPT LIST OF SYMBOLS

d

Separation distance between particle and wall

Particle diameter (m)

d0

van der Waals cut-off distance (m)

ep

Potential energy (N.m)

en

Coefficient of restitution

F

Force (N)

Fc

Contact force (N)

FvdW

van der Waals force (N)

H

Hamaker constant (N.m)

kn

Normal contact stiffness (N/m)

LS

Linear Spring contact model

LSD

Linear Spring-Dashpot contact model

m

Particle mass (kg)

R

Particle radius (m)

t

Time (s)

Vcrit

Critical velocity (m/s)

Vimp

Impact velocity (m/s)

V0

Initial velocity (m/s)

y yp

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v

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dp

Particle velocity (m/s) Position of particle centre (m) Particular solution

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surface

Initial position of particle centre (m)

Greek letters α0

van der Waals force at cut-off distance (N)

β

Damping ratio

Δt

Time step (s)

δ

Particle overlap (m)

η

Damping coefficient (kg/s)

ηc

Critical damping coefficient (kg/s) 23

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Particle density (kg/m3)

τ

Collision duration (s)

τc

Critical time step (s)

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ω0 Natural frequency of oscillation (rad/s)

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References

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Eng. Res. Des. 76 (1998) 775-796.

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plastic spheres, Powder Technol. 99 (1998) 154-162.

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[37] H.C. Hamaker, The London van der Waals attraction between spherical particles, Physica IV 10

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[45] R. Moreno-Atanasio, Influence of the hydrophobic force model on the capture of particles by

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bubbles: A computational study using Discrete Element Method, Adv. Powder Technol. 24 (4) (2013)

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786-795.

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New York, 2003.

[47] S. Ji, H.H. Shen, Effect of contact force models on granular flow dynamics, J. Eng. Mech. 132

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[48] J. Reed, Energy losses due to elastic wave propagation during an elastic impact, J. Phys. D: Appl.

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Phys. 18 (1985) 2329-2337.

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[49] S. Wall, W. John, H.C. Wang, S.L. Goren, Measurements of kinetic energy loss for particles

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impacting surfaces, Aerosol Sci. Technol. 12 (1990) 926-946. [50] H.C. Wang and W. John, In Particles on Surfaces 1: Detection, Adhesion and Removal, Edited by K.L. Mittal. Plenum Press, New York, 1988.

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[51] Y.S. Cheng and H.C. Yeh, Particle bounce in cascade impactors, Environ. Sci. Techno1 13 (1979) 1392-1396.

[52] T. D'Ottavio, and S.L. Goren, Aerosol Capture in Granular Beds in the Impaction Dominated Regime, Aerosol Sci. Technol. 2 (1983) 91-108. [53] N.A. Esmen, P. Ziegler, R. Whitfield, The adhesion of particles upon impaction, J. Aerosol Sci. 9 (1978) 547-556. [54] L.N. Rogers and J. Reed, The adhesion of particles undergoing an elastic-plastic impact with a surface, J. Phys. D Appl. Phys. 17 (1984) 677-689.

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ACCEPTED MANUSCRIPT [55] M. Dong, S. Li, J. Xie, J. Han, Experimental Studies on the Normal Impact of Fly Ash Particles with Planar Surfaces, Energies 6 (2013) 3245-3262.

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[56] S.Y. Fu, X.Q. Feng, B. Lauke, Y.W. Mai, Effects of particle size, particle/matrix interface

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adhesion and particle loading on mechanical properties of particulate–polymer composites,

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Composites: Part B 39 (2008) 933-961.

Table caption

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Table 1. Simulation input parameters

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Figure captions Figure 1. Schematic of the simulated system.

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Figure 2. Contact force versus overlap for LS and LSD models for an impact velocity equal

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to 0.01 m/s. The values of damping ratios are 0 and 0.015 for the LS and LSD models,

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respectively.

Figure 3. Particle centre position versus time for d0 = 0.1, 0.15, 0.2 nm for initial particle

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velocity equal to 0.01 m/s.

Figure 4. Critical velocities predicted using Eq. 12 versus (a) cut-off distance (b) particle diameter (c) damping ratio, β. The area below each curve represents a region where the incident velocities are smaller than the critical values (adhesion region).The area above each curve corresponds to the cases in which the impact velocities are larger than the critical value (rebound region) and therefore, particle rebound is observed.

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ACCEPTED MANUSCRIPT Figure 5. Critical velocity versus particle diameter; experimental to simulation comparison (Eq. 12). Different slope of dashed purple line, -1.4, for simulation is due to using Eq. 16

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letting the stiffness change with particle size.

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Figure 6. Impact velocity versus cut-off distance for initial values of the velocity equal to

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0.10, 0.05, 0.01 and β = 0.3. The shaded area represents the values of impact velocity for which the particle will remain adhered to the surface.

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Figure 7. Maximum overlap versus cut-off distance for V0 = 0.10, 0.05, 0.01, 0.0 and (a) β =

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0.5 (b) = 0.3 and (c) = 0.1.

Figure 8. Collision duration versus cut-off distance for V0 = 0.10, 0.05, 0.01, 0.0 and (a) β =

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0.5, (b) = 0.3 and (c) = 0.1.

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Figure 9. Coefficient of restitution versus cut-off distance for V0 = 0.1, 0.05, 0.01, 0.0 and (a)

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β = 0.5, (b) = 0.3 and (c) = 0.1.

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y

FvdW

d

+

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Figure 1

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Figure 2

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Figure 3

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figure 4

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Figure 5

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Figure 6

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Figure 7

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Figure 9

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Properties

Value 0.0, 0.01, 0.05, 0.10 0.01

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Initial velocity (m/s) Particle-wall gap (µm) Particle diameter (µm)

10

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Particle density (kg/m3)

2600

1.36×10-12

Particle mass (kg) Stiffness (N/m)

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Critical time (s) Time step (s)

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Hamaker constant (J) Critical damping (kg/s)

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Damping Ratio (-)

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1000

1.16×10-7 1.00×10-9

1.00×10-19 7.38×10-5

0.01, 0.05, 0.10, 0.30, 0.50

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Graphical abstract

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Highlights

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Analysis of van der Waals cut-off distance on collision parameters Analytical solution for critical velocity versus cut-off distance Finding criteria for rebound and adhesion of particles colliding a surface Comparison of simulation to experimental data

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   

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