In atomic force microscope (AFM) experiments,adhesion forces (van der Waals,capillary, and electrostatic forces) become dominant in micrograsping,with respect to other ones (weight, inertial forces,external loads and so on). These interactions have strong influence on the microfabrication techniques,which play a crucial role in the operation of the assembly^[1],nanolithography^[2],and aggregation and morphology^[3].

Adhesion force measurements are very sensitive to many factors,such as size,material, structure,shape,and mass. In previous studies,the shape geometry effects have been emphasized to explain the adhesion force between AFM tips and substrate. It is reported that the geometrical changes of AFM probes affected the measurement results in AFM operation^[4],and the a pplica tio n o f mo dels w ith the ina ccur a te^[5] or ill-defined probe shape^[6] may lead to serious deviations between the theoretical results and the experiment measurements. Therefore, systemic studies to the effects of probe shapes become necessary.

In theoretical models,most of the investigations were done by assuming tip shapes as the perfect spherical^{[7, 8]} ,truncated cone^[9] ,cylinder^[10],or conical^[11] geometries for simplification. These simple models presented the monotonous increase of the adhesion force with the humidity increasing,but these trends were inconsistent with experimental non-monotonous trends, which looked like a peak-to-valley corrugation,i.e.,the adhesion force firstly increased and then decreased with humidity^[12] . Tabrizi el al.^[10] even proposed two sphere models to calculate the adhesion force between two sphere surfaces,but did not acquire the consistent trends with experiments. In fact,in various manipulations,chemical,biological,electromagnetic,etc.,the pyramidal geometry tip (V-shaped) is a common option^{[13, 14]} ,but it could become a paraboliclike tip shape^{[15, 16]} due to the abrasive wear during its repeated use in Refs. ^{[17, 18]}.

In this paper,we adopt parabolic-like shapes to approximate V-shaped tips to calculate the adhesion force between AFM tips and substrates. We find that the effects of the probe profile on the adhesion force are directly related to the solid-liquid-vapor contact lines (triple point x_p) of the liquid bridge and Kelvin radius. The nonlinear variation of adhesion force exhibiting experimentally in relative humidity (R_H) from 40% to 70% mainly results from the curvature change of the probe. These results are helpful for analyzing and understanding the probe behaviors in AFM operation. 2 Model 2.1 Spherical approximation model

In this paper,we propose a mathematical model to describe the dominant adhesion force, including the capillary force and the van der Waals force,between the probe and the substrate. For the characteristic length smaller than the capillary length,gravity hardly affects the movement of a liquid^[19] ,and thus,gravity effects of the liquid bridge and the elastic deformation of materials are omitted in our calculation. Also,we do not consider the electrostatic interaction in the situation with explicitly-charged materials. As shown in Fig. 1,the interaction between a micro sphere particle and a surface represents a typical configuration when contacting a micro object with a probe in an AFM experiment.

Fig. 1. Scheme of liquid bridge formed between spherical probe and substrate due to humidity condensation (R is probe radius,θp and θs are contact angles of liquid with probe and substrate, respectively,β is half-filling angle,xp is coordinate of solid-liquid-vapor contact lines (triple point) with probe,and D is probe substrate distance).

Figure options

The adhesion force mainly includes the capillary force and the van der Waals force that can be expressed as follows :

Click Browse formula

Many studies about adhesion forces in the micro manipulations are mainly concerned about the capillary force^{[12, 20, 21]} between the probe and the substrate,which is composed of the capillary pressure force and the surface tension force^[22],

Click Browse formula

Under thermodynamic equilibrium,the pressure difference and the Kelvin radius follow the Kelvin equation^[23]. The relationship between the R_H and the pressure difference Δp across the vapour-liquid interface is obtained from the Laplace-Young and Kelvin equations^[24]:

Click Browse formula

Click Browse formula

The curvature of the liquid bridge is characterized by two radii,the azimuthal radius r₁ and the meridional radius r₂.

The van der Waals force plays an important role in controlling or influencing the behavior of microscopic objects. Based on the additivity law,Hamaker^[25] proposed an expression for the van der Waals force between macroscopic particles in 1937. Feddema et al. ^[26] numerically calculated the van der Waals force between the diameter micron spherical object and a rectangular-shaped tool in micro assembly. He detailed the influence of van der Waals force and the electrostatic force as parts approaching 1-10 µm. The van der Waals force F_vdw between the sphere and a surface,which is expected valid for bodies sized from several nanometers and larger,is given as

Click Browse formula

where H is the Hamker constant. According to the theoretical analysis and experimental results,it is concluded that the attractive van der Waals forces must be considered for the probe substrate distance up to a few nanometers,D > 0.2 nm^{[22, 27, 28]} .

UndeR^Humidity conditions,the capillary bridge is presented on the surface that produces a significant contribution to practical micro-handling. Figure 2 shows the results of the adhesion force as a function of the humidity calculated by Eqs. (1)-(4) and compared with the experiment data presented in Ref. ^[12]. The drastic increase of adsorbed water around contact surfaces is closely related to the capillary condensation,which results in the increases of the adhesion force. We can see that,with the increase of humidity,the van der Waals force between the micro sphere and the surface reduces gradually,but the capillary force increases greatly and the whole adhesion force increases accordingly.

Overall,the spherical model would be a successful model in rough estimations on adhesion measurements^[24]. However,as shown in Fig. 2,the tendency in our calculations with spherical model does not match the measured data in Ref. ^[12]. The discrepancy between the theory and the experiment can be attributed to three following aspects. Firstly,the model is very simple,in which the AFM probe is often described as spherical at their ends in many previous works. It would be noted that a practical probe may not be spherical. Some studies suggest that slight changes in the tip shape can change the adhesion force significantly^{[29, 30]} . Secondly, the assumption is generally not valid when considering the scale involved in AFM. In our calculations,the classical van der Waals theory is used to predict the adhesion force,which assumes that the separation distance is small compared with the tip radius. In fact,the AFM tip is so small that separation distances between the probe and experimental samples are the same order of magnitude (about several nanometers). This model based on the approximated relation (sphere-plane) cannot match the experiment data well ^[11]. Thirdly,the value of the Hamker constant in single media cannot be directly used for the system that is linked by a liquid bridge due to the capillary condensation. Thus,the inaccurate calculation of the van der Waals force would lead to the deviation of the adhesion^[31]. Therefore,the adhesion forces derived for these methods are inaccurate in the scale of AFM and further considerations become urgently requisite.

Fig. 2. Adhesion force as function of relative humidity calculated by spherical model.

Figure options

For a general probe shape y(x),the van der Waals force between a sphere and substrate in a humid atmosphere is given by^[12]:

Click Browse formula

where F^water_vdw and F^air_vdw represent the van der Waals forces between the probe and the substrate in water and air,respectively.

Click Browse formula

Click Browse formula

Click Browse formula

Considering all the aforementioned forces,a dynamic model of the handling system is formulated^[32] . For a Si₃N₄/SiO₂ contact,the contact angles are θ₁=60 ^◦ and θ₂= 0 ^◦. The probe radius is 100 nm,Hair=10.38 × 10 ₋₂₀ J,and Hwater=1.9 × 10 ₋₂₀ J^[33] .

For the pyramidal geometry tip (V-shaped) adopted in various AFM measurements,we adopt the approximation of parabolic-like shapes by the power function y= kxⁿ to calculate the adhesion force between AFM tips and substrates. The constant parameters k and n are determined by fitting the AFM image of experimental tips,and the experimental data are taken from Ref. ^[12],and they are tuned to generate the subtle changes in the parabolic tip profile to mimic the variation on abrasive wear from its repeated use.

Figure 3 shows the calculated adhesion forces as a function of the humidity. Compared with the tendency in Fig. 2,the power function approximation model is much better than the spherical model. For the third-order power function model,with the increasing of the humidity, the adhesion forces monotonously increase with the humidity until R_H reaches 70% and then gradually decreases. It is remarkable that the third-order power function can reproduce the delicate tendency of the adhesion force on the humidity,including the subtle increase when humidity is below 20%.

Fig. 3. Adhesion force as function of relative humidity with third-order power function model with y = kx3,k = 1.5 × 10−4 nm−2,and D=0.8 nm.

Figure options

To investigate the non-linear variation of the adhesion force,Fig. 3 also shows two important components in adhesion,which are the capillary force and the van der Waals force. We can see from Fig. 3 that the capillary force increases with increasing humidity while the van der Waals force has an opposite tendency. The variations of the adhesion force come from the combined effects of these two important components. At low humidity (R^H ≤40%),the adhesion force is dominated by the van der Waals force and the capillary force maintains small values. At intermediate humidity (40%< R^H <70%),the capillary force increases but the van der Waals force decreases and loses its dominate status. When the humidity is up to 70%,the capillary force become dominated and reaches its maximum. After the humidity is larger than 70%, the capillary force also decreases. As it can be seen,the variations of the adhesion force form a reversed-spoon-like trend with humidity varying,and the third-order power function model reproduces this trend well. 3 Discussion 3.1 Effects of probeshapes

Next,we discuss the effects of probe shapes. Figure 4 re-plots three different power function shapes,y=kx²,y=kx³,together with y=kx⁴ models. Compared with two previous models, the high-order-power model (y=kx⁴) also presents a peak-to-valley corrugation with humidity varying,but the valley depth become much more shallow. Thus,the question is what happens to the adhesion forces for different shapes of AFM tips.

Fig. 4.Effect of probe shape on humidity dependence of adhesion force (probe shape: y = k1x2, k1 =5 × 10−3,D=0.3 nm; y = k2x3,k1 =1.5 × 10 −4,D=0.8 nm; y = k3x4,k1=0.6 × 10 −5,D= 1.0 nm,and D is vertical distance between probe and substrate)

Figure options

Fig. 5. Effect of probe shape on humidity dependence of xp (curvature of probe profiles represented in inset)

Figure options

Our analysis indicates the high correlation between adhesion force and probe curvatures. xp is an inter mediate quantity,which is deter mined by the probe curvature,and also closely related to the adhesion force (from (3)-(4)). For R^H >40%,the capillary force is the major determining force because the van der Waals force becomes much weak (see Figs.2-3). Meanwhile,the capillary force comes from the mutual compensation of two terms,i.e.,the capillary pressure force and surface tension force^[12],and the effect from increasing xp is larger than the one from decreasing Δp,which causes the increase of the capillary force. Only for RH >70%,the decrement of Δp surpasses the increment of xp,the total capillary force begins to decrease. When 40%< RH <70%,the intermediate quantity xp plays the crucial role on the adhesion force. Therefore,the probe shapes generate strong influence on adhesion force between tips of AFM and experimental substrates with the changing of the solid-liquid-vapor contact lines (triple point). 4 Conclusions

In this paper,we investigate the factors which influence the adhesion force between AFM probe and substrate on micromanipulation process variation. We notice that the tip shape can strongly affect on the adhesion force between the substrate and the tip,and the blunt tip generally produces abrupt change with broad ranges of the adhesion force. A more broad range of the adhesion force is quite suitable for AFM applications. The change of the curvatures of the probe profile strongly influences the solid-liquid-vapor contact line (triple point) xp results in evolutions of the adhesion force in the corresponding humidity regions. Our analysis indicates that a much smooth and steady tip-head is suitable to apply in experiments of AFM micrograsping ,because this probe tip can provide the bigger difference of adhesion forces at high or low humidity.

In practical experiments,the lateral resolution of AFM is mainly determined by the probe shape and physical property,especially the geometry and dimension of the probe end^{[11, 34]} . On the otheR^Hand,it has been reported that AFM images are found to closely related to the adhesion force^{[35, 36]} . The larger adhesion force generally produces images with lower resolutions, and the wider range of atomic resolution is obtainable over a wide range of adhesion force^[37]. Choosing the correct probe is a crucial part of working on nano-manipulation. A large number of commercially available probes have been used in a variety of applications,and skilled users are increasingly modifying these probes to achieve their specific goals.

Acknowledgements The authors would like to thank Ling-jiang KONG (College of Physics Science and Technology,Guangxi Normal University) and Hai-ping FANG (Shanghai Institute of Applied Physics,Chinese Academy of Sciences) for valuable discussion and comments,and thank Shanghai University Supercomputer (ZQ3000 & 4000) for the support of computing resources.