Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 53, 1, pp. 235-241, Warsaw 2015 DOI: 10.15632/jtam-pl.53.1.235 THEORETICAL STUDY OF THE EFFECT OF PROBE SHAPE ON ADHESION FORCE BETWEEN PROBE AND SUBSTRATE IN ATOMIC FORCE MICROSCOPE EXPERIMENT Li Yang, Junhui Hu, Lingjiang Kong College of Physics Science and Technology, Guangxi Normal University, Guilin, China e-mail: hujh@mailbox.gxnu.edu.cn. The quantitative description of adhesion force dependence on the probe shapes are of im- portance in many scientific and industrial fields. In order to elucidate how the adhesion force varied with the probe shape in atomic force microscopemanipulation experiment, we performed a theoretical study of the influences of the probe shape (the sphere and parabolic probe) on the adhesion force at different humidity. We found that the combined action of the triple point and the Kelvin radius guiding the trend of the adhesion force, and these two fundamental parameters are closely related to the probe shape.Whilst, the theoretical results demonstrate that the adhesion force is in good agreement with the experiment data if the van derWaals force is taken into account. Keywords: probe shape, adhesion force, van der Waals force, capillary force, relatively hu- midity, liquid bridge 1. Introduction TheAtomicForceMicroscope (AFM)hasagreat impact onvariousareas suchasnanometrology (Eastman and Zhu, 1996), materials science (Thundat et al., 1993a), surface science and biology (Binnig and Quate, 1986). It is a versatile tool for studying nanomaterials properties such as friction and adhesion forces (Werf et al., 1994). The quality of image obtained from an AFM is greatly dependent both on the probe shape and physical property, especially geometry and dimension of the probe end (Thomas et al., 1995). The chemical and physical parameters of the probe significantly affectAFMmeasurements and can reduce resolutionThundat et al., 1993ab). Choosing the correct probe is a crucial part of working on biological samples. Currently, there are many publications devoted to the problem of probe geometries, and differentmodels have been established to investigate the adhesion between objects with different shapes and a solid plane (Butt andKappl, 2009; Chen and Soh, 2008; Tabrizi et al., 2006, 2008). Most of the previous works focused on the capillary force. Tabrizi et al. (2008) numerically calculated the capillary force for a sphere, cone and a flat cylinder in contact with a planar surface. Their model and experiments showed that changes in tip geometry on the sub-10-nm length scale can completely change the adhesion force versus humidity curves. Chen and Soh (2008) studied the influence of the indenter shapes (conical, spherical and truncated conical with a spherical end) on the magnitude of the capillary force in micro-electro-mechanical systems. Different dependences of the capillary force on the indenter shapes and geometric parameters were observed. Butt and Kappl (2009) calculated the capillary force with three different probe shapes, the sphere, coneandcylinder inperfectlywetted surfaces.Theynoticed that thecapillary force and the liquid bridge rupture distance can change fundamentally with different probe shapes. It has already been known that the adhesion force (including van derWaals force, capillary force and electrostatic force) (Quyang et al., 2001) significantly changes with the different probe 236 L. Yang et al. shape in theAFMexperiment, butdetails of howthe tip shape influences the adhesion forcehave not been clearly understood. Therefore, a perspective for exhaustive analysis of key parameters is necessary. In the present paper, we simulated the adhesion force for the sphere probe and the parabolic probe on a substrate in an atomic force microscope experiment. Both the capillary force and the van derWaals force contribute to the adhesion force have been analysed in detail.We found that the triple point and the Kelvin radius that determined by the probe profile are the key parameters that play a decisive role in the adhesion force. 2. Theory In Atomic Force Microscope (AFM) micro-handling manipulation, the liquid bridge formed by the water condensation serves as a channel between the probe and the substrate in the AFM experiment (Piner et al., 1999). The capillary forces are written by (Fisher, 1926) Fcap =2πγxp sin(β+θp)+πx 2 p∆p (2.1) The first term corresponds to the surface tension force and the second denotes the capillary (Laplace) pressure force. Here γ is the surface tension coefficient, xp is the coordinate of the solid-liquid-vapor contact line (triple point), θp is the contact angle of the probe surface, β is the half-filling angle andD is the particle-surface distance. Under thermodynamic equilibrium, the relationship between theRelative Humidity (RH), the pressure difference∆p and theKelvin radius rk obeys the Kelvin equation (Yang et al., 2010). In nanoscale, van der Waals forces become important at the distances below 10-15 nm and may at these distances start to dominate are interactions of different origin that have been observed at large separations. The van derWaals force Fvdw in a single medium is given by Fvdw =− H 6 [R−D D2 + 3R+D (D+2R)2 ] (2.2) where R is probe radius and H is the Hamaker constant. To avoid damaging of the surface, the separation D for most solid contacts must be kept at least several tens of angstroms. For a general probe shape, it is expected that the vanderWaals force between the probe and substrate when the liquid bridge in the gap is given by (Xiao and Qian, 1999) Fvdw =F water vdw { 1− 1 [ 1+ y(x) D ]2 } +Fairvdw { 1− 1 [ 1+ y(x) D ]2 } (2.3) whereFwatervdw andF air vdw are van derWaals forces withwater and air as themedium, respectively. They can be calculated by Eq. (2.2) with differentH,R andD. rk,D and β are related by rk = x−rps[1− cos(β+θp)] x rps −2+sin(β+θp) rps = D+y(x) cos(β+θp)+cosθs (2.4) where rps is the liquid bridge meniscus radius and θs is the contact angle of the substrate. For a spherical probe y(x)= √ R2−x2 β=arcsin x R (2.5) For parabolic probe, we adopt the power function relation y(x)= kxn β=arctany′(x) (2.6) Theoretical study of the effect of probe shape on adhesion force... 237 The adhesion force is the sum of the capillary and van derWaals force (Lazzer et al., 1999). However, inmost previousworks (Jones et al., 2002; Junno et al., 1995), the van derWaals force is usually approximated or omitted in the calculation. In our simulation, we derive the analytical expression of the relation between the probe shape rk and β, Eqs. (2.4)-(2.6), and the adhesion force, which is the sum of the capillary and van derWaals force, is precisely calculated by using the probe geometry y(x). 3. Results and discussion Inorder to illustrate thenecessity of thevanderWaals force,weperformedacomparisonbetween the theoretical simulations and the experimental data for the SiO2 particle on a TiO2 surface, in which the adhesion force is calculated with andwithout the van derWaals force, respectively. When simulating without the van der Waals force, the pull-off force as a function of the RH is plotted inFig. 1. The theoretical capillary force given byEq. (2.1) using the parameters given by Paajanen et al. (2006). The simulation results showed that the capillary force is in a fairly good agreement with the experimental data at RH ­ 40%, but deteriorates when RH < 40%. The large discrepancy between the theoretical results and the experiment data may be attributed to the fact that the adhesion force is dominated by the van der Waals force at a relatively low humidity. Fig. 1. The experimental data (Paajanen et al., 2006) and the theoretical pull-off force for a SiO2 particle on TiO2 surface versus the RH;R=40nm, θp =0 ◦, θs=70◦,D=0.40nm When the van der Waals force is included in simulation, the comparison of the theoretical resultswith theexperimentdata is shown inFig. 2. It is indicated that theadhesion forcebetween probe and substrate increases with the RH. In general, the simulation considering the van der Waals force provides a better description for the experimental data than that of considering the capillary force only. Although the van der Waals force is less dominant in the adhesion force, which is about several nano-Newtons to tens of nano-Newtons, it is should be taken into consideration. Therefore, the van derWaals force is contained in our following stimulation. To clearly understand the influence of the tip shape on the adhesion force, we mainly inve- stigate the adhesion force for the spherical probe and the parabolic probe on a substrate in the AFMexperiment. The comparison between the experimental data and the stimulation results of the adhesion force for the parabolic probe as a function of RH is shown in Fig. 3. The adhesion force between the Si3N4 probe and the SiO2 surface is employed to stimulate to explain the dependence of the adhesion force on the parabolic probe (k = 1.5 · 10−4nm−2), which is the optimum function formof the probe shape for this experimentdefinedaftermany evaluations. In a general AFMexperiment, the generic sketch of the functional relationship between the pull-off force and the Relative Humidity (RH) is inverted U-shaped (Xiao and Qian, 1999; Arai et al., 1996). The force increases steadily at low humidity, rises to a peak at intermediate humidity 238 L. Yang et al. Fig. 2. Adhesion force for the SiO2 partice on the TiO2 surface as a function of RH;R=40nm, θp =0 ◦, θs =70 ◦,D=0.40nm (Paajanen et al., 2006),H =9.46 ·10−20J in air andH =0.69 ·10−20J in water (Xiao andQian, 1999) Fig. 3. The adhesion force between the Si3N4 probe and the SiO2 surface as a function of RH; R=100nm, θp =60 ◦, θs =0 ◦,D=0.8nm (Xiao andQian, 1999) and falls in high humidity. The results predicted in our model show that the adhesion force grows with the increasing RH, which is in accordance with the experiment data shown by Xiao andQian (1999). The good results we obtained might be attributed to the inclusion of the Van der Waals force in low humidity. Some discrepancies appear at high humidity, possibly due to influence of the surface roughness which could reduce the Van der Waals force (Chen and Lin, 2008; Li et al., 2006; Sedin and Rowlen, 2000). There are two different trends of the adhesion force in Figs. 2 and 3with different probe sha- pes. Themain acting force that guided the evolution of the adhesion force should be determined and carefully considered. Figure 4 illustrates the spread of the capillary force and van der Waals force with RH. It is shown that the capillary force grows with an increase in the RH whereas the van der Waals force displays the contrary tendency. The reason is that the formation of the liquid bridge gradually reduces the van der Waals interaction between the particle and the surface (Lennart, 1997). Thus, either of these forcesmay become dominant depending on the humidity. At RH¬ 40%, the capillary force and van derWaals force are in the same ordermagnitude. At RH > 40%, the bridge formed the gap gradually filling with water which reduces the van der Waals force from 5.1nN at a RH of 5% to 0.32nN at 87% RH. The capillary force interaction is very sensitive to the RH, which is much larger than the van der Waals force under high RH conditions (RH ­ 40%). But the van der Waals force should not be omitted from the force equilibrium if the RH is lower than about 40%. Our results are consistent with the previous experiment measurement (Christenson, 1988; Jang et al., 2006). Theoretical study of the effect of probe shape on adhesion force... 239 Fig. 4. The adhesion force together with the capillary force and van derWaals force varying with the RH; (a) spherical probe, (b) parabolic probe From the previous investigation it can be found that the van derWaals force exhibits nonli- near trendswithhumidity.But it doesnot play a key role in the adhesion force variation, thuswe put the emphasis on the capillary force. According to Eq. (2.1), the numerical magnitude of the capillary force is the mutual compensation of these two terms, the capillary (Laplace) pressure force and surface tension force (Aveyard et al., 1999). After numerical simulations, we found that the surface force linearly depends on humidity, while the capillary pressure force expresses fluctuating characters, which are similar to the adhesion force. The core of the adhesion force is the state parameter relation in the capillary pressure force. Since the capillary (Laplace) force term predominated in the capillary force, whereas the xp and rk, are the important factors in the capillary force according toEq. (2.1), so two parameters are chosen to be analyzed in detail. The nonlinear variations of xp (triple point of the liquid bridge)and rk (Kelvin radius)withhumidity for twodifferentprobeshapesare shown inFig. 5. It canbeseen thatxp increaseswithhumidity,whereasrk has theopposite trend.Whenthepositive contribution to the capillary force from increasing xp is larger than the negative effect from decreasing rk, the capillary force rises steadily, and vice versa. These two opposite contributions compete to determine the trend of the capillary force with humidity increment. Fig. 5. Effects of xp and rk in the capillary pressure force; (a) spherical probe, (b) parabolic probe At low humidity (RH < 20%), the liquid bridge between the probe and substrate is very small and thin. Thus the value of xp is less than rk, the total capillary force trend is decreasing. At the intermediate humidity (40%70%), the decrement of rk surpasses the increment of xp, so the total capillary force starts to decrease. On the whole, the capillary force firstly decreases in low humidity, increases in intermediate humidity, and finally decreases at high humidity. Since the 240 L. Yang et al. capillary force is the major determining force, we can see similar nonmonotonous behavior for the adhesion force in Fig. 4. With the development of nano-science, materials science and surface science, various micro- manipulation strategies have been proposed, such as electrostatic, vacuum, inertial handling. The use of the adhesion force is considered to be an effective, controllable and securemethod in micro-manipulation. It is evident that the pickupmanipulation benefits from the increase of the adhesion force, and the release manipulation benefits from the decrease of the adhesion force. Hence a wider adhesion force range induces better controllability during the AFM manipula- tion. Comparing to the spherical probe, the parabolic probe can give a larger interval between the smallest and largest values of the adhesion force, which is in a suitable range for practical operation and would be utilized to improve the handling efficiency. 4. 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