Effective friction in adhesive mixtures intended for inhalation: Simulation of oblique impact of adhesive units

Oblique impact between adhesive units was studied using the Discrete Element Method (DEM) as a means to assess the effective friction coefficient and its dependence on particle shape, surface coverage ratio (SCR; 0.5, 0.75 and 1) and impact angle ( 15 ◦ − 75 ◦ ) in a low handling velocity regime (0.6 – 1 . 6m∕s ). Adhesive units were created in silico from a spherical carrier particle ( 100μm diameter) and monodisperse fine (drug) particles ( 3μm ) of a spherical, triangular bipyramidal or tetrahedral shape. The particle interaction was modelled using the Hertz–Mindlin contact model with JKR adhesion (surface energy 0 . 03J∕m 2 ). A total of over 300 simulations were performed and the effective friction coefficient was extracted from the normal and tangential forces experienced by the carrier particles. The adhered fines resulted in a significantly reduced friction coefficient in comparison to the bare carriers but no additional reduction was observed with increasing SCR, suggesting a saturation already at an SCR of 0


Keywords: Oblique impact Adhesive mixtures Inhalation Modelling
A B S T R A C T Oblique impact between adhesive units was studied using the Discrete Element Method (DEM) as a means to assess the effective friction coefficient and its dependence on particle shape, surface coverage ratio (SCR; 0.5, 0.75 and 1) and impact angle (15 • − 75 • ) in a low handling velocity regime (0.6 -1.6 m∕s).Adhesive units were created in silico from a spherical carrier particle (100 μm diameter) and monodisperse fine (drug) particles (3 μm) of a spherical, triangular bipyramidal or tetrahedral shape.The particle interaction was modelled using the Hertz-Mindlin contact model with JKR adhesion (surface energy 0.03 J∕m 2 ).A total of over 300 simulations were performed and the effective friction coefficient was extracted from the normal and tangential forces experienced by the carrier particles.The adhered fines resulted in a significantly reduced friction coefficient in comparison to the bare carriers but no additional reduction was observed with increasing SCR, suggesting a saturation already at an SCR of 0.5.The effective friction coefficient was independent of the impact velocity

Introduction
The inhalation pathway for delivery of drugs/Active Pharmaceutical Ingredients (APIs) to the lungs has been extensively studied during the last few decades.Over time, Dry Powder Inhalers (DPIs) have become a standard device for delivery of APIs with local and systemic effects [1].Most DPIs use a special type of mixture known as an adhesive mixture [2,3] that consists of micronised API (fine particles) of size < 5 μm and comparatively larger sized carrier particles (∼ 100 μm).Owing to their high surface-to-volume ratio, the fine particles have a significant effect of van der Waals forces, leading to high cohesiveness and agglomeration, resulting in low flowability [2,4].Adhesive units, consisting of fine particles on the surface of carrier particles, improve flowability, maximize homogeneity and improve dispersibility of the API during the aersolisation process [4].Understanding the effective behaviour of these units during formation and handling would aid optimization of drug delivery to the lungs and pave the way for high drug loads [5,6] using DPIs.
Adhesive mixtures are complex and difficult to understand.The interplay between interdependent parameters is critical for the understanding of the formulation, handling and dispersion performance of DPIs [7].Even the most simplified systems consisting of lactose carriers and lactose fine particles seem to depend on an array of factors such as the cohesive-adhesive balance [8] and the size, shape and morphology of the fines and carriers [9][10][11], making correlation between the factors difficult to disentangle.Numerical modelling and simulation enable systematic parametric variations that avoid confounding of different factors, providing a clearer picture.
Numerical studies have been used to better understand the formation mechanisms of adhesive mixtures [12][13][14], their micromechanics during handling [15][16][17] and the dynamics of particles during aersolisation [18].There have been studies conducted focusing on agglomeration and deagglomeration of aggregates in DPIs [19,20] and the influence of surface energy thereon [21].Computational Fluid Dynamics (CFD) and the Discrete Element Method (DEM) are the two most commonly used tools to study dynamics of particles in air streams.Recent studies have addressed the effects of carrier morphology [14] and fine-particle shape [22] as well as surface energy during the manufacturing and handling phase of adhesive mixtures.
Previous studies of the interaction between adhesive units have to a large extent focused on normal (head on) impact.The overall mechanical response may in this case be summarized in terms of a (normal) coefficient of restitution.The situation is more complex for non-normal, i.e. oblique, impacts.Experimental studies and theoretical analysis of (large) non-cohesive spherical particles indicate that two regimes can be identified; a sticking regime for low impact angles (near head-on impacts) and a sliding regime for larger angles (near grazing impacts) [23][24][25].The response in the sticking regime is typically characterized by a tangential coefficient of restitution and that in the sliding regime by a friction coefficient.This situation corresponds to oblique impact between bare carriers, in which case two parameters, the tangential coefficient of restitution and the friction coefficient, are typically at play.
Oblique impact of adhesive elastic particles was studied by Thornton et al. [26] who explained the departure of from the results obtained with the Hertz-Mindlin model.However, studies trying to explain the effective behaviour of adhesive units undergoing oblique impact are scarce.Adhered fines are expected to dramatically influence the effective behaviour of the particles, at least when present in sufficient amounts, since the interaction then is mediated via the fines whereas the carrier particles typically do not touch.
The current study is the first attempt to assess the effective (frictional) behaviour of adhesive units undergoing oblique impact in a handling velocity regime [27][28][29].To this end, the DEM was used to study the effect of particle shape, impact angle, impact velocity and number density of fines (represented as a surface coverage ratio) on the effective tangential force between adhesive units.The shape of the particle was derived from our previous study [16] which focused on spherical, triangular bipyramidal and tetrahedral fine particles.The current study thus complements our previous work [15][16][17], in which normal (head-on) impact between adhesive units was investigated, and provides information needed for a more complete understanding of the effective mechanical behaviour of adhesive units.For brevity the carrier particles are referred to as carriers and fine (drug/API) particles are referred to as fines.

The Discrete Element Method (DEM)
The interaction between the adhesive units was modelled using the DEM.The constituent particles are treated as discrete entities in the DEM and interact predominantly by short-range forces.The contact forces are calculated based on the effective overlap between the particles at the point of contact.The translational motion is governed by Newton's laws represented as, where  is the target particle and  is a neighbouring particle,   is the mass and   is the position of particle ,  is acceleration due to gravity,  nc  and  tc  are normal and tangential forces acting on particle  due to contact with particle  and  f luid  is the force on particle  resulting from fluid interaction.As in our previous study [22], a low velocity regime is considered to create an effective multiscale model to study the stability of adhesive mixtures during handling.The fluid forces in Eq. ( 1) are neglected owing to the low drag forces resulting from small velocities.The impact of gravity, though implemented, is negligible for individual particles owing to the small mass and time scale of study.The equation of motion for the rotational degrees of freedom could be expressed as, where  is the inertia tensor,  is the applied torque and  is the angular velocity about the centre of mass of the particle.In this work, cohesive contact laws similar to the ones stated by Thornton et al. [30,31] are used.The effective properties between the particles could be defined as follows: • The effective radius ( * ), and effective mass ( * ) of the two particles are represented as the harmonic mean of the respective mass and radii as stated in Eqs. ( 3) and ( 4) where   is the radius and   the mass for particle  and analogously for particle .
• The effective Young's Modulus  * and shear modulus  * of two particles  and  in contact are defined as [32], In the above expressions,   is Young's modulus,   is the shear modulus and   Poisson's ratio for particle  (and analogously for particle ).

The contact model
The adhesive/cohesive interaction between the particles could be governed by the DMT (Derjaguin-Muller-Toporov) [33] model or the JKR (Johnson-Kendall-Roberts) model.The Tabor parameter [34] is a dimensionless parameter which provides guidelines for selecting the appropriate contact model for adhesive/cohesive contacts given by, where  is the equilibrium distance between two spherical particles and  is the surface energy, representing the strength of adhesion/cohesion between the two particles.The Tabor parameter was calculated for interactions between pairs of particles and was found to be greater than 1, which corresponds to the JKR model [35].Specifically, the Hertz-Mindlin contact model with JKR was adopted in this work [35][36][37][38].The Hertz-Mindlin model defines the normal and tangential forces acting on the particles and relies of the overlap between the interacting particles and JKR adds adhesion/cohesion.
The normal elastic force ( ne ) between two particles is represented as, where  c = 3  * is the pull-off force between the two particles,  is the contact radius at a given time and is the finite contact radius due to the attractive force in the absence of external load [39,40].The contact radius is obtained from the normal overlap  n via an expression of the form where Likewise, the tangential elastic force ( te ) is defined as [37], where  t is the tangential stiffness and  t is the tangential overlap.The detailed calculation of overlap and tangential stiffness is described in our previous work [15].Normal ( nd ) and tangential ( td ) damping forces are defined as [38], where  is a dimensionless factor dependent of the coefficient of restitution,  n ( t ) is normal (tangential) stiffness and  n ( t ) is the normal (tangential) relative velocity between the particles.Hence the coefficient of restitution is used to parameterize the magnitude of the damping forces.Although alternative contact models predict a (weak) dependence of the coefficient of restitution on impact velocity [41], experimental data indicate that this parameter can be treated as a constant for a relatively large range of conditions [24], suggesting that the model used is sufficient in the present context.Standard Coulomb-type friction was used, such that the maximal tangential force is given by where μ s is the coefficient of static friction.Likewise, standard rolling friction was used, with a torque  r given by where μ r is the coefficient of rolling friction and  is the distance from the particle centre to the contact point.

Primary particles
The carrier particles were modelled as elastic spheres with contact damping.The diameter of the carrier particles was 100 μm with properties mimicking those of lactose (Table 1).Fine particles were modelled as multisphere composite particles of three different shapes; spherical, triangular bipyramidal and tetrahedral, in line with our previous work [22] and also illustrated in Fig. 1.The maximal diameter of the fine particles was fixed at 3 μm.The composite shapes with multispheres were designed so as to ensure conservation of volume across shapes [16].The multispheres represent various degrees of complexity of particle shape which controls their interlocking and rolling ability.Thus, the triangular bipyramidal particles represent a modest (in relative terms) and the tetrahedral particles a significant deviation from a spherical shape.The surface energy that controls the strength of adhesion/cohesion between the carrier-carrier, carrier-fines and fines-fines was fixed at 0.03 J∕m 2 [12,42,43].

Adhesive units
The adhesive units were created in-silico by pseudo-randomizing the fine particles on the surface of the carrier particles [15].This was attained by creating a uniformly spaced envelope of the desired particle number density.Each of the fine particles was given a random velocity and the system was evolved until smaller agglomerates of fine particles were formed.Thereafter the fine particles were given a velocity directed towards the centre of the carrier particle.The system was evolved until an envelope of weakly attached fine particles on the surface of the carrier particle was formed.The number density of fine particles on the surface of the carrier particle was quantified in terms of the surface coverage ratio (SCR) represented as,  where  is the number of fine particles on the surface of the carrier particle,  f ines is the radius of the fine particles and  carrier is the radius of the carrier particle.The definition is consistent with our previous work and the work conducted by Rudén et al. [5,6].Specifically, adhesive units with three different SCRs (0.5, 0.7 and 1) and three different shapes of fine particles were created in-silico as shown in Fig. 1.Different values of SCR represent different traditional drug loads [46] and potential new high dosage formulations [5].

Simulations
Oblique binary collisions between adhesive units were simulated using the setup illustrated in Fig. 2. The initial velocities of the adhesive units were directed along the positive and negative  axis and their centres were displaced along the  axis (displacement   ) to obtain oblique impacts in the  plane with different (nominal) impact angles  (Fig. 2).The nominal impact angle relates the magnitude of the displacement of the particle centres along the  axis to the nominal particle distance  =   +   at the instant of impact, Hence head on collisions between the two particles correspond to an  of 0 • and grazing impacts correspond to 90 • .Binary collisions between adhesive units with varying angles (15 • , 30 • , 45 • , 60 • and 75 • ) and velocity (0.6, 0.8, 1.0, 1.2, 1.4, and 1.6 m/s) [27,28] were studied.The velocity range (0.6-1.6 m/s) was chosen to represent handling conditions and is in line with our previous works [15,16,22].The procedure was repeated for different SCRs and shapes of fine particles.A total of over 300 independent simulations were performed.
DEM modelling and simulation were performed using the Altair ® EDEM TM 2021 bulk material simulation software package.A Verlet integration scheme with a time step of 1 ns was used to maintain numerical consistency.The initial setup was created using Matlab and Visual Basic and detailed analysis and post processing were performed using MATLAB ® and the graphs, linear regression and statistical analysis were performed using Graphpad Prism.Total force, position and velocity of particles were extracted for analysis.

Analysis
The total force  tot experienced by one of the carrier particles (carrier  in Fig. 2) was recorded from the simulation.A unit normal vector n along  =   −   (where   and   represent the location of particle  and ) and a unit tangent vector t in the  plane were introduced (Fig. 2).These enabled the total force to be decomposed into normal ( n ) and tangential ( t ) components, The effective coefficient of friction (μ) could be represented by, and was extracted as the slope of the graph of  t vs.  n .This procedure was considered satisfactory in the present context, because the adhered fines precluded direct interaction between the carriers and, as a result, sticking was not observed.

Statistical analysis
To establish an understanding of the data trends and the overall behaviour of μ with changing impact angle (), SCR and shape of fine particles; two-way ANNOVA followed by Tukey's multiple comparison test was performed with the level of significance fixed at 0.05.While calculating the coefficient of friction (μ), relative velocities above 1 m∕s were considered (as explained further in Section 3.3).The standard deviation calculated from the three velocities exceeding 1 m∕s is represented as error bars in the plots.

Results and discussion
Micromechanical studies of adhesive units during oblique impact provide insights into the effective frictional behaviour between the units and aid the development of a holistic multiscale model to improve the understanding of adhesive unit dynamics both during manufacturing and handling.It also throws light on the complex behaviour of adhesive units and their departure from a simplistic elastic collision with adhesion as studied by Thornton et al. [47].The implications could be extended to further studying the interaction with a hopper wall and the dynamics inside inhalers, capsules and blisters as a few examples.
In order to quantify and predict the dynamics of the adhesive units during oblique collision, the normal and tangential forces between them were determined and the coefficient of friction was calculated and compared.

Evolution of contact forces with time
Fig. 3 shows the variation of normal and tangential forces with respect to time as measured during collision between adhesive units.The representative systems shown in Fig. 3a, b and c represents adhesive units with an SCR of 0.5, an impact angle  of 45 • and a relative impact velocity of 1.2 m∕s with (a) spherical, (b) triangular bipyramidal and (c) tetrahedral fine particles.
It could be observed from Fig. 3a that the collision between adhesive units with spherical fine particles results in multiple peaks, henceforth termed events, in the normal and tangential force in the time range of 0.2 to 0.4 μs.The smaller peaks are characterized as minor events and are likely due to dynamic interlocking of the fine particles between the carrier particles at the point of initial contact.The minor events are analogous to a stick-slip behaviour.The event with longest time span and highest value of the normal force is most significant and hence characterized as the major event.An SCR of 0.5 is sufficient to preclude direct interaction between carriers, and hence this event represents rolling of fines on the surface of the carriers analogously to a ball bearing.
Fig. 3b represents the variation in force for adhesive units with triangular bipyramidal fine particles.Multiple minor events and one major event could be observed between 0.5 and 0.7 μs.The minor event is characterized by the initial interaction of the fine particles resulting in a stick-slip type behaviour.The major event (0.6-0.7 μs) is representative of rolling of the fine particles between the interacting carrier particles.
Lastly, Fig. 3c represents the force due to collision of adhesive units with tetrahedron fine particles.There are no significant minor events but a single major event is observed with a tangential force that is significantly higher than that for the spherical and triangular bipyramidal fines.This results is a direct consequence of the particle shape; tetrahedral particles are expected to interlock due to the presence of ridges and valleys to a larger extent than the rounder triangular bipyramidal particles and the spherical particles.
In summary, there is a significant effect of the shape of the fine particles on the magnitude of the tangential force.The tangential forces are lowest for the spherical fines, intermediate for the triangular bipyramidal fines and highest for the tetrahedral fines.This is expected, since composite particles tend to approach a spherical shape when formed from a larger number of spheres, as for the triangular bipyramidal fines compared to the tetrahedral fines.The observations stated above were consistent across the entire data set for the given shape of fines on the adhesive unit.The major event was captured and analysed further to understand the parametric dependence of the coefficient of friction on shape, SCR and collision velocity.

Calculation of coefficient of friction
The effective coefficient of friction (μ) is defined as the slope of the curve between the normal and tangential force.As explained in  Section 3.1, the analysis was restricted to data corresponding to the major events, characterized by the largest normal force and generally also the longest duration.In Fig. 4, the tangential force is plotted vs. the normal force for the major events in Fig. 3.A linear regression line was fitted to each of the data sets, with a regression coefficient ( 2 ) close to 0.99, and the slope of the curve was recorded.It could be observed that adhesive units with spherical fines have the smallest coefficient of friction followed by triangular bipyramidal and tetrahedral as anticipated from the results presented in Fig. 3 and elaborated upon in the following sections.

Effect of velocity
The coefficient of friction was calculated for the major events for the interactions between adhesive units during oblique impact.Variations with the impact angle, SCR and shape of fine particles were also noted.It was observed that the adhesive units tended to stick to each other for low relative impact velocity (≤ 1 m∕s) as illustrated in Fig. 5.In these cases, other forces than friction dominate the response and it is not meaningful to extract any friction coefficients.The particles stick because the initial kinetic energy cannot compensate for the energy loss caused by breakage of adhesive bonds, interlocking and rearrangement of particles.It was observed from the data set that a relative impact velocity greater than 1 m∕s ensured that the adhesive units moved apart after the collision.The change in coefficient of friction was minimal for higher relative impact velocities as could be seen from the representative data set displayed in Fig. 6.In contrast, normal restitution coefficients for head-on impact between adhesive units with spherical fines generally increase with increasing velocity in this range, irrespective of the SCR [17].Such a behaviour is also observed for triangular bipyramidal and tetrahedral fines at low SCRs (up to about 0.5), until larger interlinked chains (contact chains/force chains) of fines as formed [16].Ultimately, these results have their roots in the contact law governing the behaviour of binary particle interaction: For normal interactions, contact damping and cohesion result in a velocitydependent energy loss.However, for tangential interactions, the energy loss caused by contact damping and friction is velocity independent.
To further process the data, the mean and standard deviation of the friction coefficients obtained at 1.2, 1.4 and 1.6 m∕s were calculated.

Impact of shape and impact angle
Fig. 7 shows the effect of the impact angle () on the coefficient of friction.Each data point represents the mean value obtained for higher velocity as elucidated in Section 3.3 and the error bars represent the corresponding standard deviations.In order to better represent and interpret the data, Tukey multiple comparison tests were performed as discussed below.

Spherical fines
Fig. 7a shows how the friction coefficient varies with impact angle for spherical fine particles.No significant difference was observed across SCR for the same impact angle (Table 2).A pattern emerges when comparing the changes across impact angles for the same SCR (Table 3).The adhesive units with an  of 15 • , 30 • and 45 • have similar friction coefficients whereas the friction coefficients for 60 • and 75 • are significantly higher.The data set can thus be separated into two clusters, one characterized by relatively low friction coefficients obtained for head on-like impacts ( ≤ 45 • ) and the other characterized by larger friction coefficients obtained for grazing-like impact ( ≥ 60 • ) with larger interaction between the fine particles.

Table 2
Significant differences at the 5% level between friction coefficients obtained for different surface coverage ratios (SCRs).The entries represent fine-particle shapes for which significant differences were obtained: spherical (S), triangular bipyramidal (B) and tetrahedral (T).Significant differences at the 5% level between friction coefficients obtained for different impact angles.The entries represent fine-particle shapes for which significant differences were obtained: spherical (S), triangular bipyramidal (B) and tetrahedral (T).
Angle (°) SCR 0. These results can be contrasted to the one obtained for (considerably larger) bare spheres.For these, the effective friction coefficient exhibits a nearly linear increase with impact angle for small angles [23,25] in the sticking regime, followed by an angle-independent value for larger angles in the sliding regime.The much lower effective friction coefficients observed for adhesive units in the current study are most likely due to the fact that interaction is mediated via the fine (spheres) and that, as a consequence, sticking does not occur.
On comparing adhesive units with different SCRs and different impact angles, a similar behaviour as explained in the previous paragraph is found.Significant differences are observed between the grazing angle and head on angle on comparing among SCR values for 0.5, 0.7 and 1.A distinct effect is observed for extreme angles such as an  of 75 • .One possible explanation for this behaviour is that the region of interaction increases with increasing impact angle, and that the rearrangement of more fines are involved for larger SCRs, leading to an increased friction coefficient.

Triangular bipyramidal fines
Fig. 7b shows the variation of coefficient of friction with impact angle for adhesive units with triangular bipyramidal fine particles.On comparing the dependence of μ on SCR for the same value of  there were significant changes observed which was corroborated through ANNOVA and Tukey's test (Table 2).The angles close to grazing angle such as 60 • and 75 • result in a higher coefficient of friction compared to the smaller angles (Table 3).The larger angles increase the contact region and therefore involve more fine particles, resulting in stronger interlocking and thus higher resistance to the motion.
On comparing adhesive units as a function of SCR and , a pattern similar to that for adhesive units with spherical fine particles was observed with higher friction coefficient.The grazing-like impacts (60 • and 75 • ) result in significantly different friction coefficient and the 45 • impact angle behaves either as a grazing-like or an head on-like angle.The triangular bipyramidal fine particles offer a higher resistance than spherical particles as their tendency to roll reduces which results in a higher coefficient of friction.

Tetrahedral fines
Fig. 7c shows the variation of coefficient of friction with changing impact angle for different SCRs for adhesive units containing tetrahedral fine particles.On comparing the  across each SCR it could be   observed that for an SCR of 0.5, the friction coefficients obtained for impact angles of 15 • and 30 • are lower compared to those obtained for higher angles (Table 3).In contrast, for SCRs of 0.7 and 1, a difference is observed only for comparison with an impact angle of 75 • .
Significant differences between coefficients of friction obtained for different SCRs (0.5 vs. 0.75 and 0.75 vs. 1) are obtained for some impact angles  (Table 2).There is some tendency for the differences in μ to increase with an increase in the impact angle.On cross comparing , a similar trend as that within the SCR is observed.The variation in friction coefficient is not as distinct as that for adhesive unit with spherical fines.One possible explanation for this is that the tetrahedral fines tend to interlock more compared to that of triangular bipyramidal and especially spherical fine particles for all impact angles.It could also be noted from Fig. 3 that adhesive units with tetrahedral fines only have a major event as opposed to those with the spherical and triangular bipyramidal particles, for which a number of minor events were also observed.

Cross comparison
Fig. 8 shows μ vs.  for all SCRs and fine particle shapes.Adhesive units with spherical fines have a lower coefficient of friction compared to the triangular bipyramidal and tetrahedral fines for  ≤ 45 • (head on-like collisions).The reason is that tetrahedral and triangular bipyramidal fines have ridges and valley and require a higher torque to initiate the rolling motion as compared to spherical particles and in increased tendency to interlock.On comparing across the different shapes of fine particles it could be observed that the maximum value for coefficient of friction is attained for tetrahedral followed by triangular bipyramidal and spherical fine particles.This is also observed in Fig. 3 where adhesive units with tetrahedral fines have a single event compared to units with spherical and triangular bipyramidal fines which have multiple smaller events followed by a larger event before the particles start to roll.For higher values of SCR and  ≥ 60 •

Table 4
Significant differences at the 5% level between friction coefficients obtained for different fine-particle shapes.The entries represent the surface coverage ratios for which significant differences were obtained.(grazing-like angles) μ tends to merge with one another.Spherical fine particles tend to form smaller aggregates which effectively behaves as composite particles, which makes the effect of shape minimal between spherical and triangular bipyramidal particles.As could be observed from Table 4, the tetrahedral particles have higher μ because of their nonspherical shape.

Self agglomerates
It was observed that near grazing impact between adhesive units with an SCR of 1 resulted in a peeling of the fine particle envelope from the surface of carrier.The peeling predominantly took place between the two interacting carriers and the particles that were peeled off tended to cohere.The peeling effect could be observed distinctly for tetrahedron followed by triangular bipyramidal and spherical fine particles as illustrated in Fig. 9.The particles peeled off the adhesive units formed smaller self agglomerates.Such self agglomerates could potentially coalesce to form larger self agglomerates of the type observed by Rudén et al. [6] during the formulation of adhesive mixtures with high drug loads.With increasing drug loads, corresponding to higher number density of fines and higher SCRs, secondary and tertiary layers are formed.This results in the formation of larger interlinked contact or force chains [22] so that the particles in the adhered layer start to act cooperatively, explaining the increased tendency of the fine particle envelope to peel off.

Conclusion
Oblique impacts between adhesive units were studied as a means to extract their effective friction coefficient.The effect of fine-particle shape, impact angle, surface coverage ratio and collision velocity was assessed.The fine particles ranged from spheres via triangular bipyramids (formed from 5 spheres) with a relatively small deviation from a spherical shape (in relative terms) to tetrahedra (formed from 4 spheres) with a significant deviation.Simulations were performed for an impact angle between 15 • and 75 • and three different surface coverage ratios (0.5, 0.7 and 1) for a fixed surface energy of (0.03 J∕m 2 ) in a handling velocity regime (0.6 -1.6 m∕s).

Fig. 1 .
Fig. 1.Illustration of multisphere composite fine particles of (a) spherical, (b) triangular bipyramidal and (c) tetrahedral shape on the surface of a carrier particle (for a surface coverage ratio of 0.5).

Fig. 2 .
Fig. 2. (a) Definition of the impact angle  and the unit normal and tangent vectors n and t for oblique impact between two adhesive units A and B. (b) Snapshot from the simulation (for a surface coverage ratio of 0.5 and an impact angle of 75 • ).

Fig. 3 .
Fig. 3. Variation of normal and tangential forces with time for different fine-particle shapes (for systems with a surface coverage ratio of 0.5, an impact angle of 45 • and a relative impact velocity of 1.2 m∕s).

Fig. 4 .
Fig. 4. Tangential vs. normal force on the carrier during the major collision event for different fine-particle shapes (for systems with a surface coverage ratio of 0.5, an impact angle of 45 • and a relative impact velocity of 1.2 m∕s).The slope of the curves represents the coefficient of friction.

Fig. 5 .
Fig. 5. Sticking observed for oblique impact of adhesive units with spherical fines at low relative velocity; (a) 0.6 (b) 0.8 and (c) 1 m∕s.The surface coverage ratio was 0.5 and the impact angle 45 • .

Fig. 6 .
Fig. 6.Representative data set for coefficient of friction vs. velocity for different fineparticle shapes [spherical (S), triangular bipyramidal (B) and tetrahedral (T)], surface coverage ratios (SCRs) and impact angles showing minimal dependence on collision velocity.

S
.Sarangi and G. Frenning

Fig. 7 .
Fig. 7. Coefficient of friction vs. impact angle () for adhesive units with different fine-particle shapes and surface coverage ratios (SCRs).