W. DING, A. J. HOWARD, M.D. MURTHY PERI and C. CETINKAYA*

Transcription

1 Philosophical Magazine, Vol. 87, No. 36, 21 December 2007, Rolling resistance moment of microspheres on surfaces: contact measurements W. DING, A. J. HOWARD, M.D. MURTHY PERI and C. CETINKAYA* Dept. of Mechanical and Aeronautical Engineering, Nanomechanics Nanomaterials Lab, Wallace H. Coulter School of Engineering, Clarkson University, Potsdam, NY , USA (Received 10 February 2007; accepted in revised form 25 September 2007) Using contact measurements, experimental evidence was obtained for the existence of the rolling resistance moment. The critical rolling distance prior to detachment is reported. Previously it has been argued that the critical rolling distance should be related to the lattice size and/or the molecular length of the particle and surface materials. However, there has been no theoretical prediction for the critical value and, currently, the reasons for its existence are not fully understood. For polystyrene latex (PSL) particles, measurements presented in the current study on silicon suggest much higher values for the critical rolling distance than previous anticipated levels. The current approach can also be employed to measure the work of adhesion between a spherical particle and a flat surface without the prior knowledge of the particle diameter since the rolling moment stiffness is directly proportional to the work of adhesion with no dependence on the diameter of the particle. Experimental results are compared with the available data and good agreement between the theoretical predictions and the experimental values is found. 1. Introduction The rolling resistance moment is a restoration moment that opposes the rotational motion of a particle adhered on a surface. Particle removal and strength/stability of particle networks depend on rolling resistance more than axial adhesion since the rolling forces are a few orders of magnitude lower than the axial detachment forces. The rolling resistance moment of an adhesion bond between a particle and flat substrate controls the motion of particles on surfaces and the strength/stability of networks of adhered round objects in a diverse spectrum of applications (e.g. particles, powders, blood cells and nanotubes) on the micro/nanoscale, as in sintering and compaction. Also, it is essential in detachment and removal of micro/nanoparticles from surfaces. Many adhesion models and theories have been proposed and discussed to understand the axial (out-of-plane) stiffness and strengths of particle surface bonds. *Corresponding author. cetin@clarkson.edu Philosophical Magazine ISSN print/issn online ß 2007 Taylor & Francis DOI: /

2 5686 W. Ding et al. Johnson and Greenwood [1] proposed a unifying framework for existing theories, i.e. Hertz, JKR (Johnson Kendall Roberts), DMT (Derjaguin Muller Toporov), MD (Maugis Dugdale), and established the transition between these theories and their applicability zones for ranges of external loads and an elasticity parameter. In one-dimensional (axial) adhesion models and theories, the pressure field in the contact area is assumed symmetric. Studying particle rolling, however, requires a two-dimensional adhesion model and analysis where the stress at the bond interface is asymmetric during pre-rolling and rolling. Dominik and Tielens [2] derived an expression for this rolling moment by considering an external force applied at the centre of the spherical particle. This external force creating a moment with respect to the bond zone causes an asymmetric pressure distribution in the contact area, resulting in the pressure variations in contact area at the leading edge and at the trailing edge of the contact with the surface. The resulting asymmetric pressure field in the contact area creates a restoring moment in small rotational angles. Under external excitation, this restoring force along with the rotational inertia of the particle could result in free oscillatory vibrations of the particle with respect to its contact [3, 4]. The rolling friction of millimetre-scale balls on rubber surfaces was reported for the first time in 1975 [5 7]. In spite of the fact that, using atomic force microscope (AFM) techniques, extensive research has been dedicated for measuring the adhesion force between a micron-scale particle and a substrate by detaching particles, the rolling motion and resistance of a particle on a substrate has rarely been experimentally explored with AFM. The main disadvantage of axial detachment-based AFM techniques in such experiments is that the particle has to be fixed/glued to the tip of a probe; therefore it is essentially a destructive technique for the particle. The rolling resistance moment is particularly critical for various applications from biology to semiconductor manufacture since it controls the stability of a network of adhered round objects (e.g. particles, cells and nanotubes) on the micro/nanoscale. Adhesion and frictional forces between spherical micron-sized particles on the basis of rolling resistance moment was studied by Heim and Blum [8] by means of AFM. The rolling resistant moment can be estimated based on an adhesion model [2]. The rolling resistance moment and, hence, the adhesion force between a microparticle and a substrate were determined based on acoustic excitation and interferometric motion sensing [3, 4]. Particle adhesion and force studies have also been performed with the aid of micromanipulators [9 11]. For example, Saito et al. [9] analyzed the interaction forces between the microsphere, the substrate and the manipulation probe, and proposed a method to pickup and manipulate a microsphere. They also suggested the existence of a maximum rolling resistance, i.e. an external moment has to be greater than a certain threshold to roll a particle. Sitti [11] performed a pushing study on 500 nm Au-coated latex particles on a silicon substrate with an AFM probe. The sliding, rolling and rotational motions of the particles were observed, and the particle substrate frictional properties were estimated. No rolling resistance data was provided, although its effect in the control loop was reported. However, the pushing process cannot be visually observed in situ in real time because the pushing and imaging processes use the same AFM cantilever probe. It is often argued that the (critical) detachment distance

3 Rolling resistance moment of microspheres 5687 (the shift in contact area) prior to the onset of detachment should be related to the lattice size and/or the molecular length of the particle and surface materials. However, there is no theoretical prediction for this critical value. 2. Axial and rocking motions of a microsphere The Johnson Kendall Roberts (JKR) theory is a one-dimensional (axial) adhesion model for a spherical particle attached to a flat substrate [12]. The JKR model states that the external force axial displacement relationship consists of two components (Hertzian contact and adhesion, respectively): sffiffiffiffiffi F A ¼ 1 r3=2 3=2 2 r 3=4 3=4, ð1þ where is the axial displacement of the centre of the particle with respect to the surface of the substrate due to the applied external force (F A ) and ¼ r=k, where K is the stiffness coefficient of the adhesion bond given by 4E 1 E 2 K ¼ 3ðE 2 ð1 2 1 ÞþE 1ð1 2 2 ÞÞ, ð2þ where E 1 and E 2 are the Young s modulus and 1 and 2 are the Poisson s ratios of the substrate and the particle, respectively and ¼ 3W A r, where W A is the work of adhesion. The stiffness expression [equation (1)] is a nonlinear relationship between the applied external force (F) and the axial displacement (). The stiffness expression is linearized at the stable equilibrium point at ¼ðð2Þ 3=2 =rþ to determine the axial natural frequency of the particle substrate system in the neighbourhood of its stable equilibrium: K ¼ df d ¼ ¼ 3r1=3 2 5=3 : ð3þ 2=3 The axial stiffness for a 21.4 mm polystyrene latex (PSL) spherical particle on a silicon substrate is calculated as K* ¼ N m 1. In addition to its axial motion, it is known that the particle can have a rotational degree of freedom due to radial modes of motion of the base and can make rotational vibrations on a flat surface (figure 1a) with respect to the centre of the contact area. Unlike the axial motion, in order to model this rotational motion of the particle, a two-degree-of-freedom adhesion theory needs to be utilized. Under a lateral pushing force, the non-uniform stress distribution in the contact area during the rotational motion creates a restoring moment (also referred to as resistance moment) to rolling motion (figure 1b) [2], which is proportional to the angle of rotation and leads to angular free vibrations of the particle. The pressure distribution, p(x, y), for a spherical particle on a flat substrate according to the JKR adhesion model has to be cylindrically symmetric when no external rotational moment is exerted on the spherical particle. The moment of resistance in case of symmetric pressure distribution is, therefore,

4 5688 W. Ding et al. (a) (b) A (c) Figure 1. (a) Rolling is initiated if the lateral force, F, is increased above a certain level (not to scale); (b) an image of a PSL particle on a silicon substrate subjected to an external pushing force via an AFM cantilever beam; (c) an overall SEM view of the particles tested and the AFM cantilever beams. M y ¼ RR xpðx, yþdx dy ¼ 0. However, if an external shear force or horizontal displacement field is applied to the substrate, the moment associated with rotational (rocking) motion of the particle results in shifting of contact area. In other words, the contact area is no longer centered around the point that is located at the centre of the original contact circle and the pressure distribution becomes asymmetric. The pressure distribution, p(x, y), and the moment associated with the rocking motion can be calculated from the following assumptions [2]: (i) the true shifted contact area is approximated by decomposing the contact area, into two circles of different radii a þ and a, where is the shift in the contact area due to rocking motion, (ii) the half circle x50 is one half of a symmetrical contact with contact radius a þ and pressure distribution, p(r, a þ, z ); (iii) the half circle x40 has a smaller contact radius a and a corresponding pressure distribution, p(r, a, z ). The total moment due to both the half circles is calculated. The resulting distribution

5 Rolling resistance moment of microspheres 5689 is discontinuous at x ¼ 0. From the derivations reported by Dominik and Tielens [2], an approximation for M y is given as: M y 4P c ^a 3=2 : This expression indicates that the moment associated with the pressure asymmetry is proportional to the pull-off force, P c, and the shift of the centre of contact area. For a spherical particle in contact with a flat surface, P c ¼ð3=2ÞW A r, where W A is the work of adhesion (equation (2)), and if the normal forces stay within the range P c 5P5P c, the factor ^a 3=2 varies in the range and assuming ^a ¼ 1, the rolling resistance moment for a particle on a flat substrate is further simplified from equation (4): M y 6W A r. Clearly, the rocking natural frequency is dependent on the radius of the particle, work of the adhesion of the particle substrate system and the density of the particle. It is noteworthy that the elastic properties of the particle and substrate material appear to play no role in rocking motion, whereas the natural frequency of the axial motion is a function of the elastic properties of the particle substrate system. Based on the reported value of the work of adhesion for these dry-deposited PSL particles on silicon, W A ¼ Jm 2 [3, 4], the expected force displacement slope (rotational stiffness) should be N m 1, whereas the axial stiffness for a 21.4 mm PSL particle on silicon substrate is calculated as K* ¼ N m 1. Thus, the stiffness of the rolling bond is much softer than that of the axial detachment, which is why it takes less effort to move and/or detach a particle on a surface by rolling it. ð4þ 3. Experimental procedure PSL spherical particles with an average diameter of 21.4 mm (Duke Scientific) were dry-deposited on a single-crystal silicon substrate (Polishing Corporation of America). In the contact rolling experiments, an in-plane contact force (F) on a PSL spherical particle on the silicon substrate was applied by an AFM cantilever beam (MikroMasch, Inc., CSC 38) with a length of 350 mm such that a rolling moment with respect to the bonding area was generated with a moment arm of the radius of the PSL particle (figure 1). Prior to the onset of rolling motion, the particle substrate bond deforms and the particle rotates with respect to the centre of the bond (A) (figure 1a). All the contact experiments were conducted in the vacuum chamber of a scanning electron microscope (SEM) (JEOL 7400) with the aid of a custom-made manipulator with nanometre precision (figure 2). The nanomanipulator is composed of two opposing positioning stages consisting of an integrated X Y linear motion stage and a Z-linear motion stage (OptoSigma, Inc., /1155) on the opposing plane. The stages are driven by piezoelectric actuators (New Focus, Inc., MRA 8351) that provide linear motion with a motion resolution of approximately 30 nm. A piezoelectric bender (Noliac A/S, Denmark, CMBP 05) that provides fine positioning at sub-nanometre resolution is mounted on the top of the X Y stage. For the pushing trials, the base chip of the AFM cantilever

6 5690 W. Ding et al. Figure 2. The instrumentation diagram of the experimental set-up featuring the custommade SEM nanomanipulator and the supporting motion control and imaging systems. was attached at the free end of the piezoelectric bender, and the silicon substrate with the PSL particles deposited was mounted on the opposing Z-stage. The nanomanipulator allows the precise positioning and motion of the tip of the cantilever beam with respect to the PSL particle on the silicon substrate as the cantilever beam and the particle are viewed by the SEM. In the experiments reported in this study, the AFM cantilever was first brought into the close proximity of a selected PSL particle on the substrate through a series of translational motion of the stages of the nanomanipulator. Then a dc voltage incremented in discrete steps was applied to the piezoelectric bender to actuate the AFM cantilever until the cantilever came into contact with the PSL particle at point C (figure 1a). Since its motion became constrained by the adhesion bond and associated rolling resistance in the particle-substrate interface, the AFM cantilever began deflecting as a lateral force was externally exerted on the PSL particle moving from C to C 0. The increment of the pushing force corresponding to each actuation step was in the range of 3 5 nn. This incremental force level is sufficiently low not to slide the particle without first experiencing bond deformation (with no rolling). By acquiring a series of SEM images, the entire pushing process was recorded for each of the eight PSL particles tested. The AFM cantilever serves as the force sensing element and the applied force (F) is calculated from the relative cantilever deflection using the linear bending stiffness of the cantilever beam. Yet, direct measurement of the cantilever deflection from the recorded SEM image is non-trivial due to the lack of a reference point. A method based on the calibration of the piezoelectric bender response was employed to extract the cantilever deflection from the series of SEM images [13]. The stiffness constant of the AFM cantilever beam needed for determining the force F was calibrated/measured in situ in the SEM vacuum chamber prior to the contact experiment with a resonance method developed by Sader et al. [14]. The stiffness value for the specific AFM cantilever beam used in this study was determined as N m 1. The displacement of the particle in the

7 Rolling resistance moment of microspheres 5691 x-direction (x) was obtained from the processing of recorded images by tracking the pixel positions of the contact point between the particle and AFM cantilever surface. A previously developed data/image analysis procedure for a sequence of SEM images was employed for extracting the force displacement (F x) curves for the particles tested [13]. 4. Experimental results The experimental procedure developed for the current rolling resistance study was applied to eight PSL particles dry-deposited on a silicon substrate (figure 1c). A sequence of SEM images for each particle was obtained. As an incremental force was exerted to the particle, the AFM cantilever deflection and particle displacement at each pushing step were obtained through the image analysis procedure described above. Then the force (F) displacement (x) relationships were extracted (figure 3a). It is clear that the displacement of the particle increases with the increasing pushing force as the particle moves. Passing a certain force level for each particle (with an exception of particle 7), it was observed that the external force drops rapidly as the particle position in the direction x increases substantially. This observation suggests that the particle substrate adhesion bond was broken and/or significantly weakened. The maximum forces before this sudden drop (possibly due to sliding), as depicted for the seven particles in figure 3a, range from 32 to 128 nn. Note that the sudden increase of the particle displacement occurred immediately after the increase of the pushing force, so rapid that no SEM image could be acquired. The maximum force presented in figure 3a should thus be the resolution of the applied force (3 5 nn) lower than the true values. For particle 7, the pushing force was gradually released after reaching a maximum value of around 29 nn. Therefore, this particle did not experience the sudden displacement increase. From the force displacement relationship in figure 3, the work of adhesion between the particle and substrate can be extracted. According to Dominik and Tielens [2], the rolling resistance moment of a particle on a flat substrate can be approximated as: D 2 M ¼ 6W A ð5þ 2 where W A is the work of adhesion, D is the diameter of the spherical particle and is the angle of rotation of the particle with respect to the particle substrate bond (A) (figure 1a). For the PSL silicon substrate system used in the reported experiments, the work of adhesion (W A ) is taken as Jm 2. For the current contact testing experimental configuration, assuming small contact areas and no sliding between the particle and the substrate in the initial phases of the pushing, the rolling moment induced by the pushing force (F) with respect to the contact area on the substrate can be approximated as M FðD=2Þ and the angle of rotation () of the particle can be approximated from the measured displacement (x) as

8 5692 W. Ding et al. (a) (b) Figure 3. (a) The force displacement curves of the pushing trials for the eight particles used in the experiments and (b) the initial part of the force displacement curves indicating the transition from pre-rolling deformation to rolling motions for particles 2, 5, 6 and 7. The end of pre-rolling points for particles 2, 5, 6, and 7 are marked with circles. No pre-rolling deformation is observed for particles 1, 3, 4 and 8. tan x=ðd=2þ, assuming the rotation is small and that the displacement at the particle centre is the same as the displacement at particle cantilever contact point (C). The work of adhesion between the particle and substrate can be characterized from the slope of the force displacement curve. From these two

9 Rolling resistance moment of microspheres 5693 expressions, the slope of the force displacement curve (k) can be approximated in a displacement range corresponding to the pre-rolling phase of motion as k ¼ F x ¼ 2M=D D=2 ¼ 4M D 2 : ð6þ From equations (5) and (6), the work of adhesion can be expressed as W A ¼ k 6 : ð7þ It is noteworthy that equation (7) contains no particle diameter, D, dependence; therefore, the knowledge of the size of the particle is not required for measuring the work of adhesion using the force displacement curves generated in current study. The experimental force displacement data depicted in figure 3a indicates that the slope of the force displacement curve varies for each trial, especially between two groups of particles: (1, 3, 4 and 8) and (2, 5, 6, and 7). The slope values of pre-rolling part and the rolling part of the force displacement curves from the pushing trials on eight different particles are listed in table 1. The initial parts of the eight curves are depicted in figure 3b. Based on their force displacement curve slopes, the eight pushing trials can be classified into two categories. For particles 1, 3, 4 and 8, the slopes roughly remain constant throughout the pushing process, and the values of the slopes are in the range of N m 1. For particles 2, 5, 6 and 7, the slopes, however, decrease after the first few loading steps. The corresponding initial slope values (figure 3b) are N m 1, larger than the slopes of the other four particles, indicating stiffer bonds. After the first few loading steps, their slope values drop to N m 1, which is closer to those of the first group (particles 1, 3, 4 and 8). Based on the reported work of adhesion value, Jm 2 [3, 4], the expected force displacement slope should be N m 1. We believe that in pushing trials for particles 2, 5, 6 and 7, rolling resistances exist from the adhesion bonds at the initial few loading steps. Therefore, the initial particle displacement is due to the pre-rolling motion of the particle. The sudden slope change indicates that the adhesion bond yields (or breaks the leading edge of the bond closes as the trial end opens) and loses its ability to resist rolling. For particles 1, 3, 4 and 8, the rolling resistance might be too small to be detected with our current set-up, and thus these particles appear to start rolling instantly upon the exertion of force with no rolling resistance. Whereas all other particles were undisturbed before being pushed, particle 3 was accidentally pushed before the recorded trials. Thus, the pushing trial for particle 3 was performed after the complete destruction of the initial bonds between the particle and the substrate. This initial bond breaking might explain the low value of the particle 3 slope (table 1). Based on the initial slopes of trials 2, 5, 6 and 7, the work of adhesion values between the PSL particle and silicon substrate are calculated with equation (6) and are reported in table 1. The work of adhesion obtained experimentally varies from Jm 2 to Jm 2, reasonably close to the reported theoretical value, W A ¼ Jm 2 for a PSL particle on a silicon surface. From the

10 5694 W. Ding et al. Table 1. A summary of the experimental data for the bond stiffnesses in the pre-rolling and rolling phases of the motion, the critical angle and distance of rotation and the measured work-of-adhesion of the particles studied in the experiments. For particles 6 and 7, the nominal sphere diameter 21.4 mm is used. Particle no. Particle diameter D (mm) Pre-rolling stiffness k (N m 1 ) Rolling stiffness ko (N m 1 ) Critical distance (onset of rolling) * (nm) Critical angle of rotation * (10 3 rad) Work of adhesion (experimental) WA (10 3 Jm 2 ) n/a 0.13 n/a n/a no pre-rolling (0.40 ) n/a 0.09 n/a n/a no pre-rolling n/a 0.17 n/a n/a no pre-rolling (0.44 ) 24 6 no data * (0.39 ) 17 7 no data * (0.40 ) n/a 0.14 n/a n/a no pre-rolling Theoretical prediction no prediction no prediction no prediction 23.5

11 Rolling resistance moment of microspheres 5695 recorded SEM images, the diameter of the PSL particles were measured and listed in table 1. The corresponding critical angles of rolling are also estimated for trials 2, 5, 6 and 7 using the approximation tan x=ðd=2þ. As summarized in figure 3b, the maximum displacements before rolling are in the range of nm, and the corresponding critical angles of rolling range from * ¼ rad to * ¼ rad (table 1). The contact radius of the 21.4 mm PSL particle on a silicon substrate is approximately 231 nm and the force of adhesion is 1.18 mn. It is found that, for the particles tested, the critical rolling distance is approximately 16 20% of the contact radius. To our knowledge, the reasons for the existence of this critical angle * (and corresponding translation motion x* of the particle) are not fully understood. It is argued that it could be related to the atomic length and/or characteristic lattice distance of the materials involved (on the order of nm) [2, 8]. However, the measurement reported in the study indicates the critical distances x* are on the order of 100 nm (table 1) and much larger than the atomic length scale suggested in the reported studies. 5. Conclusions and remarks A contact experimental study for investigating the rolling moment resistance of a microsphere on a flat surface and the associated critical rotational angle is detailed. Results of the presented work provide further experimental evidence to the existence of rolling resistance moment and, for the first time, experimental data on the critical angle for microspheres prior to the onset of rolling is reported. In the literature, it has been argued that this critical rolling angle that separates pre-rolling from rolling regimes should be related to the lattice size and/or the molecular length of the particle and surface materials. However, there has been no theoretical prediction for this critical value. The measurement provided by this study for the critical detachment distance is two orders of magnitude higher than previous estimates. As reported in this study, the current approach can also be employed to measure the work of adhesion between a spherical particle and a flat surface without the prior knowledge of the particle diameter since the rolling moment stiffness is directly proportional to the work of adhesion with no dependence on the diameter of the particle. This observation eliminates the need for the measurement of the diameter of the particle and simplifies the AFM cantilever beam contact measurements of the work of adhesion, especially compared to detachment tests with glued particles. Using the experimental set-up, the work of adhesion and critical rolling angle measurements for eight PSL microspheres with the diameters ranging from 22.5 mm to 26.8 mm on a silicon substrate are determined. The measured values for the work of adhesion are in the range of mj m 2. For the particles tested, it is found that the critical rolling distance is approximately 16 20% of the contact radius. Except for one particle, adhesion data is in line with previous measurements. Good agreement between the reported data in the literature and the experimental values obtained in current study is found.

12 5696 Rolling resistance moment of microspheres Rolling resistance moment is critical in various areas from biological systems to semiconductor manufacture since it controls the strength of adhesion bonds, the stability of networks of adhered round objects on micro/nanoscale, as well as the detachment and removal of micro/nano-particles from surfaces. Acknowledgements Partial funding for this work from Intel Corp. for the EUV photomask clean programme and a seed award from the Coulter School of Engineering at Clarkson University are acknowledged. References [1] K.L. Johnson and J.A. Greenwood, J. Colloid Interface Sci (1997). [2] C. Dominik and A.G.G.M. Tielens, Phil. Mag. A (1995). [3] M.D.M. Peri and C. Cetinkaya, Phil. Mag (2005). [4] M.D.M. Peri and C. Cetinkaya, J. Colloid Interface Sci (2005). [5] K. Kendall, Wear (1975). [6] A.D. Roberts and A.G. Thomas, Wear (1975). [7] K. Kendall, Molecular Adhesion and its Applications (Kluwer Academic, New York, 2001), p [8] L. Heim and J. Blum, Phys. Rev. Lett (1999). [9] S. Saito, H.T. Miyazaki, T. Sato, et al., J. Appl. Phys (2002). [10] M. Sitti and H. Hashimoto, IEEE ASME Trans. Mechatron (2000). [11] M. Sitti, IEEE ASME Trans. Mechatron (2004). [12] K.L. Johnson, K. Kendall and A.D. Roberts, Proc. R. Soc. A (1971). [13] W. Ding, L. Calabri, K.M. Kohlhaas, et al., Expl Mech (2007). [14] J.E. Sader, I. Larson, P. Mulvaney, et al., Rev. Scient. Instrum (1995).