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Powder Technology 191 (2009) 143–148
Contents lists available at ScienceDirect
Powder Technology
j ourna l homepage: www.e lsev ie r.com/ locate /powtec
Mechanical damping using adhesive micro or nano powders
B.L. Severson a, L.M. Keer b, J.M. Ottino a, R.Q. Snurr a,⁎a Department of Chemical and Biological Engineering, Northwestern University, 2145 Sheridan Road, Evanston IL 60208, USAb Department of Civil and Environmental Engineering, Northwestern University, 2145 Sheridan Road, Evanston IL 60208, USA
⁎ Corresponding author.E-mail address: [email protected] (R.Q. Snurr
0032-5910/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.powtec.2008.09.019
a b s t r a c t
a r t i c l e i n f oArticle history:
A particle dynamics simula Received 9 November 2007Received in revised form 26 August 2008Accepted 29 September 2008Available online 15 October 2008Keywords:Particle dynamicsAdhesionParticle dampingJKR
tion model is developed and used to design a mechanical damping device thatuses micro or nano powders as the damping medium. A damping device based on a powder has theadvantage over the traditional use of a viscous fluid of temperature insensitive operation. Adhesion forcessignificantly affect the behavior of small particles. The damping mechanism targeted stems from energydissipated due to the breaking of many adhesive particle contacts. Adhesive contacts can dissipate energydue to hysteresis in the force between particle loading and unloading. In this work adhesion hysteresis isadded to the JKR model of adhesive contact and then used in particle dynamics simulations. The effects ofadhesive properties and packing densities are studied in two device geometries (shear and piston) usingsimulation.
© 2008 Elsevier B.V. All rights reserved.
1. Introduction
It is often desirable to damp mechanical motions or vibrations toprotect mechanical structures from fatigue. Devices engineered fordamping commonly exploit the viscous drag of a liquid to convert theundesirable kinetic energy into heat. However, under extremeconditions—such as wide temperature ranges—liquids are not suitableas a damping medium. For example, most damping fluids cannotwithstand operating temperatures above 500 °C. In addition, fluidviscosity can vary widely with temperature, causing poor perfor-mance [1]. On the other hand, the performance of granularmaterials islargely unaffected by temperature [1,2].
Granular materials are well suited for use as a damping mediumsince they naturally dissipate energy through a number ofmechanismswhen they flow, are jostled or are impacted. Because of these naturalcharacteristics of granular materials, they have recently been targetedto dampmechanical motion in a number of applications. For example,particles can be injected into cavities of beams and plates [3], orattached to help dissipate vibrations in structures. The performance ofthese dampers appears to be effective in damping vibrations of a widerange of frequencies [4] of either harmonic or random excitations [5].Particles are known to dissipate energy through friction as they slidealong each other or during an impact because of their viscoelasticproperties or plastic deformation [6]. The particles employed in theseapplications typically have diameters on the millimeter length scale.
).
l rights reserved.
What we explore here is the use of micron- and nanometer-sizedparticles for use as the damping medium.
Dry micro and nano particles in the form of a powder can dissipateenergy due to impacts and friction just like millimeter-sized particles.However, in addition to these mechanisms, we propose that adhesionbetween micro and nano particles can also be targeted as a noveldampingmechanism. Particles naturally become adhesive as their sizedecreases. The omnipresent van der Waals forces begin to overcomegravitational and inertial forces on the micro scale and can dominatethe nano scale. When particles are adhesive, energy can be dissipatedthrough adhesion hysteresis when particle contacts form and then arebroken. We target this mechanism of energy dissipation to create anovel type of damper based on the dissipation of energy throughadhesion hysteresis. Introducing adhesion via smaller particles givesanother dimension of control that can be tuned in the design of amechanical damping device.
In order to evaluate the possibility of creating a damper based onan adhesive powder as the damping fluid, particle dynamicssimulations are used to connect the adhesive properties of individualparticles to the behavior of the bulk powder. In these simulations,every particle position, velocity and force is recorded throughout thesimulation. This wealth of detail makes particle dynamics a powerfultool to study granular behavior. The accuracy, of course, depends uponthe assumptions made in the model, so development of realisticmodels is critical.
We improve upon the current particle dynamics models foradhesive contact by increasing the level of realism of the energydissipation due to adhesive effects. We then show ways to use themodel to aid in the design of mechanical damping devices filled withadhesive particles. Various studies have used particle dynamics
144 B.L. Severson et al. / Powder Technology 191 (2009) 143–148
simulations to study cohesive granular systems. Many of these studieshowever only model the adhesion between particles in generic wayssuch as using a constant force or square well [7–10]. A few studieshave applied adhesion to the particle dynamics model in a moresophisticatedway by using the Hamaker expression [11,12]. Themodelwhich we describe also uses the Hamaker expression for particles thatare close to contact. However, our method models the adhesion andenergy dissipation of particles during contact in a different way.
Our particle dynamics simulations generally follow the method ofCundall and Strack [13], but simulate particles on a much smallerlength scale. The small micrometer particles simulated here require avery short time-step of 2 ns. The sheared particles generally reach asteady state after 0.01 s. A typical simulation run is 0.05 s.
2. Force models for particle dynamics simulations
2.1. Normal force with adhesion
Van der Waals adhesion can significantly affect the behavior ofsmall particles when they are in close proximity and throughoutcontact. Different theoretical expressions are used depending uponwhether the particles are touching. For particles which are nottouching, the attractive force between two particles whose surfacesare a distance z apart is calculated using Hamaker's expression [14,15]
FHam = −AR⁎
12z2ð1Þ
where A is the Hamaker constant which depends on the chemicalproperties of the solid, and R⁎ is the effective radius defined as
1R⁎
=1R1
+1R2
ð2Þ
where R1 and R2 are the radii of the two particles. Hamaker's equationhas been applied to particle dynamics simulations before to study theeffect of adhesion on packing [16].
Once particles touch, the theory of Johnson, Kendall and Roberts(JKR theory) provides a solution to the contact problem of adhesiveelastic spheres in a quasi-static/thermodynamicmanner. In theories ofelastic contact, the effective properties of two bodies in contact arecommonly defined as
1E⁎
=1−�21E1
+1−�22E2
ð3Þ
1G⁎
=2−�1G1
+2−�2G2
ð4Þ
where E is the Young's modulus, G is the shear modulus and ν is thePoisson ratio of the elastic solids in contact.
JKR theory considers three contributions to the energy of thesystem, namely the stored elastic energy, the mechanical potentialenergy and the surface energy. In their derivation a relationship isgiven between the relative approach of the particle centroids α andthe applied normal load P [17].
ααf
=3 P
Pc
� �+ 2F2 1 + P
Pc
� �1=232=3 P
Pc+ 2F2 1 + P
Pc
� �1=2� �1=3 ð5Þ
This expression is used in our simulations to calculate the normalforce from the positions and radii of the particles. In it, αf is defined as
αf =34
π2γ2R⁎
E⁎2
� �1=3
=3P2
c
16R⁎E⁎2
� �1=3
ð6Þ
where γ is the interfacial energy of the contact. Pc in Eq. (5) is the JKRpull-off force and is defined as
Pc = 3πγR⁎ ð7Þαf represents the tensile deformation of the particle surfaces due totheir attraction. Adhesive elastic surfaces can deform to touch atdistances where rigid bodies would not. When attractive surfaces arebrought near to each other, they can be observed to “jump intocontact.” In JKR theory, the tensile deformation occurs both duringloading and unloading—particle surfaces first touch (during loading)or separate (during unloading) when z=αf. This artifact of JKR theoryis convenient in these simulations because it avoids the singularity inEq. (1) when z=0.
It should be noted that γ in Eqs. (6) and (7) has many names in theliterature depending on the context. It has been called the thermo-dynamic work of adhesion, the interfacial energy, the surface energyor the energy release rate.
The Hamaker constant in Eq. (1) is related to the surface energyused in JKR theory. Both of these variables are intended to capture theattraction stemming from London–van der Waals forces. To ensure acontinuous transition from attraction before contact to adhesionduring contact, we have used a relationship to connect the Hamakerconstant to the surface energy A=24πz02γ where z0 is taken as0.165 nm [15].
2.2. Tangential forces with adhesion
Adhesion also affects the tangential forces and friction betweenparticles. The tangential force model used here was created byThornton and Yin [18,19]. Their work extends thework of Mindlin [20]and Savkoor and Briggs [21] to model the tangential forces betweenadhesive spheres.
The approach of Thornton and Yin is based on the observation thatthe surfaces in an adhesive contact will need to peel before they canslide when there is an applied tangential force. In their model,different equations are used to calculate the tangential force Tdepending upon whether the contact is stationary (not yet peeled)or if it is sliding.
Before the contact has peeled, the application of a tangential forcewill cause micro slip in the area of contact. Mindlin's ‘no slip’ solutionis used to capture the relationship between the tangential force andthe amount of micro slip.
T = 8G⁎aδ ð8Þ
In Mindlin's solution, a is the radius of the contact area and δ is adisplacement representing the amount of micro slip. In our simula-tions, Eq. (8) is used to calculate the tangential force from thedisplacement, or the distance that surfaces have moved relative toeach other since coming into contact. Savkoor and Briggs [21] havesuggested that the application of a tangential force reduces thepotential energy of the contact by an amount Tδ/2 which leads to thefollowing expression for the radius of the contact area
a3 =3R⁎
4E⁎P + 2Pc + 4PPc + 4P2
c −T2E⁎
4G⁎
� �1=2" #ð9Þ
The tangential force is modeled by Eqs. (8) and (9) during thepeeling stage. Note that the radius of the contact area is used incalculating the tangential force and that the tangential force is neededto calculate the radius of the contact area. This is resolved in practiceby using the tangential force from the previous time-step to calculatethe radius of the contact area and then calculating the new tangentialforce.
Table 1The default particle properties used in the simulations reported in this work
Parameter Value Units
Rave 40 μmE 68.95 GPaG 25.92 GPaν 0.33ρ 2700 kg/m3
Γa 0.32 J/m2
Γr 3.2 J/m2
μ 0.3
145B.L. Severson et al. / Powder Technology 191 (2009) 143–148
The tangential force is calculated from (8) until peeling is complete.The completion of peeling is marked by reaching a critical tangentialforce given by
Tc = 4PPc + P2
c
� G⁎
E⁎
" #1=2ð10Þ
Once Tc is reached, peeling is complete and sliding can occur. Thetangential force during sliding is calculated from one of the followingexpressions depending upon the magnitude of the normal force. If Pb−0.3Pc the tangential force is calculated as
T = μP 0 1−P 0−P3P 0
!3=2
ð11Þ
where μ is the friction coefficient and P′ is defined as
P0 = P + 2Pc + 2 PPc + P2c
� 1=2 ð12Þ
When PN−0.3Pc , the tangential force is calculated from
T = μ P + 2Pcð Þ ð13Þ
When used in conjunction, Eqs. (8)–(13) have been shown tomatch experimental measurements of the tangential force duringadhesive contacts well over a wide range of normal loads [19].
2.3. Addition of adhesion hysteresis
The model explained thus far is based upon the quasi-staticassumption used to derive JKR theory. The loading and unloading of aparticle contact following JKR theory, being a quasi-static thermo-dynamic theory, is a perfectly elastic interaction. It does not capturethe energy losses that occur in dynamic collisions. Modifications toJKR theory are necessary to simulate a collision which dissipatesenergy. Thornton's application of JKR theory provides one method toadd some energy dissipation to the JKR model. In their simulations,particle contact begins when α=0 during loading, but duringunloading, particles remain in contact until α=−αf. The energydissipated during a collision is a constant value given by
Ediss =Z 0
−αf
Pdα ð14Þ
where P(α) is obtained by inverting Eq. (5). The energy dissipatedduring a collision using this strategy is constant regardless of thedetails of the collision. The same amount of energy is dissipated nomatter what the incoming velocity is or how compressed the particlesbecome during the collision.
Fig. 1. Example of a normal collision with adhesion hysteresis where Γr=10Γa andvi =0.22 m/s. Pc and af were calculated using Γr.
Wehave extended Thornton's simulationmodel to capture some ofthese details by including adhesion hysteresis. Experiments showhysteresis, or a difference in the contact force between loading andunloading, in adhesive contacts [22,23]. It is thought that adhesionhysteresis arises from surface rearrangements which occur duringcontact [24,25]. Molecules can form bonds or become entangled withmolecules on the opposite surface, which causes the force to separatesurfaces to be greater than the attractive force which brought themtogether [26]. The difference in contact forces between loading andunloading causes a loss of energy. This mechanism of energydissipation is targeted in this work to design a mechanical dampingdevice.
Adhesion hysteresis and the subsequent energy dissipation areincluded in our simulations by replacing the thermodynamic work ofadhesion in the JKR model (γ in Eq. (7)) with an energy release rate γwhich differs depending on whether the particles are advancing Γa orretracting Γr. Since the values are no longer equal to the thermo-dynamic work of adhesion, wewill refer to them as the energy releaserates of advance and retraction. Considerable differences in the energyrelease rate have been measured experimentally to the point where Γrcan be up to 100Γa [23].
A single collision between particles which have a hystereticadhesion force (Γr=10Γa) can be analyzed by looking at a plot of theload vs. displacement (Fig. 1). Two identical particles with propertieslisted in Table 1 were made to collide at a relative velocity of 0.22 m/sand the normal force was recorded and plotted vs. α for each time-step the particles were in contact. The area between the loading andunloading curve represents the energy dissipated during the collision.This area increases as the initial velocity increases because highervalues of α are reached during the collision. We note that for low-velocity collisions, the particles hit and stick without bouncing off.This occurs when the energy dissipation is greater than the initialkinetic energy.
3. Results
With a model that captures the energy dissipated throughadhesive contacts, we can use particle dynamics simulations toconnect the inter-particle forces to bulk powder damping behavior.Design of a damping device includes matching the expected externalforces (impacts, torques or vibrations) to the dissipation of thedamping medium. Parameters that might be altered to change theperformance of the powder include: particle size, adhesive properties,and packing density (solid fraction). We will investigate the effects ofchanging some of these parameters using our particle dynamicssimulations in two possible damper geometries.
3.1. Couette flow
One of the canonical geometries in fluid mechanics is Couette flow.The shearing of adhesive particles in Couette flow may prove to be auseful geometry to design a mechanical damping device. The three-dimensional shear cell in Fig. 2 is operated at constant volume with
Fig. 4. Shear stress as a function of the energy release rate of unloading (Γr). Note thatthe y-axis is on a log scale.
Fig. 2. Shear cell geometry.
146 B.L. Severson et al. / Powder Technology 191 (2009) 143–148
the rigid walls made of 80 (20×4) particles each. The radius of thewallparticles is 50 μm (slightly larger than the free-flowing particles), andthe top wall is sheared at a constant rate of 0.1 m/s. Periodic boundaryconditions are applied at the open faces of the shear cell. The physicalproperties of the particles are shown in Table 1. These values werechosen to simulate silica—a material commonly used to makenanoparticles.
In the results that follow, the shear stress that the top wallexperiences is calculated at each time-step and averaged over a lengthof time after it has stabilized (once there is no long-time trend eitherup or down in the shear stress). The shear stress is related to theamount of energy dissipated by the flowing particles. Particles withhigher dissipation require a higher shear force to keep the top wallmoving at a constant velocity.
As seen in Fig. 3 the packing density (reported here as the fractionsolids) has a large effect on the shear stress. Higher solid fractionsproduce higher shear stresses. This is a logical relationship resultingfrom the greater number of particle contacts being formed and brokenin the denser configurations. As more particle contacts are created andbroken, the total energy dissipated increases, causing a higher shearstress. Fig. 3 shows that the increase in the shear stress can be severalorders of magnitude.
In our contact force model with adhesion hysteresis, the energyrelease rates of loading and unloading can be changed independently.Both the loading and unloading energy release rates have a large effecton the resulting shear stress. The shear stresses reported in Fig. 4 werecalculated by changing the energy release rate during unloading (Γr)while holding the energy release rate during loading constant (Γa).Increasing Γr in this manner increases the force required to breakparticle contacts and increases the energy dissipated for each brokencontact. This increased dissipation again causes a correspondingincrease in the shear stress.
The behavior of the calculated shear stress when Γa is changedholding Γr constant is somewhat more complex, as shown in Fig. 5.Perhaps surprisingly, the shear stress as a function of Γa passesthrough a maximum. The decrease in shear stress at higher values of
Fig. 3. Shear stress as a function of the packing density. Note that the y-axis is on a logscale.
Γa can be understood using our previous logic. As Γa approaches Γr,collisions become increasingly elastic and less energy is dissipated percollision. This translates into a decreasing shear stress with increasedΓa.
However, the increase in shear stress in Fig. 5 at low values of Γadoesn't follow this explanation. Decreasing Γa while holding Γrconstant can only increase the energy dissipated per normal collision.The increase in shear stress with Γa must stem from a differentdissipationmechanism. This behavior can be explained by consideringthe tangential or friction force. Increasing Γa contributes to higherfrictional forces during the loading portion of the contact, andfrictional forces are a form of energy dissipation. The maximumseen in Fig. 5 thus reflects two competing dissipation mechanisms. AsΓa is increased, energy dissipation from friction increases whiledissipation from adhesion hysteresis decreases. This causes the shearstress, a measure of dissipation, to pass through a maximum.
3.2. Modifying the geometry of the shear cell
We imagine a damping devicewhich dissipates energy through thecontinuous breaking and reforming of adhesive contacts. Agglomer-ates of micro or nano particles will dissipate energy as they are brokenand then naturally reform as flow continues, enabling the design of a
Fig. 5. Shear stress as a function of the energy release rate of loading (Γa). Note that thex-axis is on a log scale.
Fig. 6. Couette shear cell with teeth to break up agglomerates.
147B.L. Severson et al. / Powder Technology 191 (2009) 143–148
continual or cyclic process. Some small modifications to the basicshear cell may increase damping by creating flow patterns that moreeffectively break up particle agglomerates. One way to achieve this isto control the surface roughness or add teeth to the solid surface of theshear cell. Fig. 6 displays how a shear cell with teeth would behindered by particle flow and thus increase the shear force.
Fig. 8. The position and force experienced by the disc in the piston for a sinusoidalvibration at 700 Hz.
3.3. Piston geometry
Another geometry that can be targeted for device design is apiston. The shock absorbers on a car are an example of this geometry,which achieves damping by forcing a fluid to pass through orifices in amoving disc. A similar device could be designed to use a fine powder,instead of a viscous liquid. The simulation volume shown in Fig. 7represents one repeating unit of the full piston. The faces not boundedby the top and bottom walls have periodic boundary conditions. Theblack particles near the middle on either side of the picture are movedup and down in the simulation. We refer to these as the disc. As thedisc moves, free-flowing particles (white) are forced through the gapin the disc. The adhesive contacts of the flowing particles are brokenand reformed, dissipating energy along the way.
One way to study the damping performance of the pistongeometry is to apply a prescribed motion to the disc and calculatethe force experienced by the disc as the particles are forced through it.Fig. 8 shows the position and force for an applied sinusoidal vibrationat 700 Hz.
Knowing the force that the disc experiences as it is displacedenables us to evaluate damping by calculating the energy dissipatedper cycle. A plot of the force vs disc position is shown in Fig. 9. Thecyclic process creates a loop. The area bounded by this loop is theenergy dissipated per cycle, which is a function of both the particleand system properties. One of the easiest parameters to control from
Fig. 7. Enclosed container with moving piston. Particles are forced to flow through thehole in the moving piston.
the perspective of a device designer is the packing density or solidfraction. Adding more particles or taking some out should provide astraightforward way to tune the damping of the device to meet theexpected vibrations. Fig. 10 reports the performance of this pistonassembly as the number of particles in the simulation is changed togive a range of solid fractions from 0.35 to 0.55. It should be noted thatthe maximum solid fraction of identical spheres is 0.74.
Fig. 9. Force per area of disc plotted vs. disc position for 3 cycles. The area bounded bythe loop represents the energy dissipated per cycle.
Fig. 10. Energy dissipated per cycle for increasing packing density. Sinusoidal vibrationsare applied at 700 Hz.
148 B.L. Severson et al. / Powder Technology 191 (2009) 143–148
Changing the packing density is just one example of the manyparticle and system properties which could be explored using particledynamics simulations. We have already mentioned that particle sizeand surface adhesion both affect damping. In addition, systemproperties such as the size of the holes in the disc or disc thicknesscould be explored to tailor the damping of the device to a particularapplication.
4. Conclusion
We have added hysteresis to a particle dynamics model foradhesive particles and have used particle dynamics simulations toinvestigate the dependence of damping performance on the adhesiveproperties and packing densities of small particles. The simulationresults offer evidence that micro or nano particles can be used as anovel damping medium which dissipates energy through adhesionhysteresis. This simulation method and model promise to provide auseful tool to design particle dampers based upon this or otherdamping mechanisms.
Acknowledgments
The authors thank the U.S. National Science Foundation IGERTprogram and the Air Force Office of Scientific Research for funding.
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