Effect of Tip Mass on Frequency Response and Sensitivity of AFM Cantilever in Liquid

Amir Farokh Payam1,2, Morteza Fathipour2

1Instituto de Ciencia de Materiales de Madrid (ICMM), CSIC, Madrid, Spain

2School of Electrical & Computer Engineering, University of Tehran, Tehran, Iran.

Email: amir.farokh@csic.es

Abstract

The effect of tip mass on the frequency response and sensitivity of atomic force microscope (AFM)

cantilever in the liquid environment is investigated. For this purpose, using Euler-Bernoulli beam theory

and considering tip mass and hydrodynamic functions in a liquid environment, an expression for the

resonance frequencies of AFM cantilever in liquid is derived. Then, based on this expression, the effect of

the surface contact stiffness on the flexural mode of a rectangular AFM cantilever in fluid is investigated

and compared with the case where the AFM cantilever operates in the air. The results show that in contest

with an air environment, the tip mass has no significant impact on the resonance frequency and sensitivity

of the AFM cantilever in the liquid. Hence, analysis of AFM behaviour in liquid environment by

neglecting the tip mass is logical.

Keywords- Tip Mass, Cantilever, Sensitivity, Frequency Response, Liquid.

1. Introduction

Nowadays, due to the rapid development of Nano-Biotechnology and the need to measure bio materials in

fluid environments, application of AFM in this field has been expanded (Gan, 2009). In comparison with

the air or vacuum environments, AFM cantilever dynamics in liquids remains less understood and

requires further investigations (Raman et al., 2008). Several theoretical models have been proposed for

the AFM cantilevers immersed in liquid which consider the effect of the viscosity (Bassak et al., 2006;

Chon et al., 2000; Chu, 1963; Dorignac et al., 2006; Elmer et al., 1997; Eysden et al., 2006; Green et al.,

2005; Kawak, 1996; Paul et al., 2004, 2006; Raman et al., 2008). However, the effect of tip mass on the

dynamic behaviour of AFM cantilever is not considered. In (Kracofe et al., 2010), the effect of tip mass

on the mode shape, stiffness and optical lever sensitivity calibration is studied. But, the

effect of tip mass on the resonance frequencies and modal sensitivity analysis was

neglected. In most of the mentioned methods, the hydrodynamics of vibrating cantilever is solved in an

unbounded fluid (Sader, 1998). Thus, surface-coupled hydrodynamic effects were absent. Recently,

Ploscariu and Szoskiewicz (Ploscariu et al., 2013) and Tung et al. (Tung et al., 2014), investigated the

dynamics behaviour of contact resonance (CR) system in the fluid environment and they elucidate the fact

that the hydrodynamics of a surface-coupled interaction can be reconstructed. They show that the

polynomial expressions of (Maali et al., 2005) can be extended to model hydrodynamic function in the

case where there is a surface-coupled interaction.

Several studies have been carried out on the analysis of AFM cantilever sensitivity

to the stiffness of the samples (Abbasi et al., 2014; Chang, 2002; Chang et al.,

2008; Kahrobiyan et al., 2010; Lee et al., 2005; Shen et al., 2004; Turner et al.,

2001; Wu et al., 2004). However, all of these studies concentrate on the operation

of AFM in the air environment. An analysis of the modal sensitivity of the AFM

cantilever in liquid has only been carried out in (Payam, 2013a, b). Hence, in this

paper by considering tip mass effect, the characteristic equation of AFM cantilever

in liquid is derived. Then, based on the derived expression for resonance

frequencies, the modal sensitivity of AFM cantilever to surface stiffness variations is

investigated. Finally, using numerical simulation the results are analysed and

compared with the air environment.

2. Modelling of AFM cantilever in liquid by considering tip mass

In this section, using Euler-Bernoulli beam theory and by considering tip mass of the

cantilever, the resonance frequencies of the AFM cantilever is calculated. The

schematic of AFM cantilever with tip mass is shown in Fig.1.

Following equation represents the AFM cantilever system immersed in liquid

environment:

EI ∂∂ x4 [W ( x , t )+a1

∂W ( x , t )∂ t ]+ ρc bh ∂2W (x , t )

∂ t2 =Fh+Fexc

(1)

Where E is the Young’s Modulus, I is the area moment of inertia, a1 is the internal damping coefficient,

b ,h and L are width, height and length of the cantilever, respectively.

W ( x , t )is the time-dependent displacement of the cantilever, Fext ( t ) is the excitation force. Fh is the

hydrodynamic force which can be described by a separate added mass mh and viscose dampingch

(Kiracofe et al., 2010; Maali et al., 2005; Tuck, 1969):

Fh( t )=−mh∂2

∂ t2W ( x , t )−ch

∂∂ t

W ( x , t ) (2)

where mh and ch are defined as (Kiracofe et al., 2010):

mh=π4

ρf b2Re al [ Γ (ω )] (3)

ch=π4

ρf ωb2 Im [Γ (ω )] (4)

where

Γ=Γr+ jΓ i (5)

Γr=a1' +

a2'

√Re, Γ i=

b1'

√Re+

b2'

Re (6)

which a i'and b i

'are regression constants of hydrodynamic function (Maali et al., 2005; Ploscariu et al.,

2013; Tung et al., 2014). The Reynolds numberRe is calculated by:

Re=ρf ωb2

4 η (7)

whereρ f is the density and η is the viscosity of the fluid, respectively.

The boundary conditions of the cantilever beam by considering the effect of tip mass are given by:

W (0 , t )=∂ W (0 , t )∂ x

=0 (8)

EI ∂2W ( L ,t )∂ x2 =−J e

∂3W ( L ,t )∂ x ∂ t2

(9)

EI ∂3 W ( L ,t )∂ x3 =K f W (L , t )−me

∂2 W ( L, t )∂ t2

(10)

where me is the tip mass, Je=me ht2/3 is the mass moment of inertia of the tip height (ht ) andK f is the

normal contact stiffness which is calculated by linearizing the interaction force around equilibrium point

and expressed as effective spring constant.

A general solution of equations (1), (8)- (10) can be expressed as:

W ( x , t )=(C1 sin kn x+C2coskn x+C3 sinh k n x+C4 cosh kn x )eiωt (11)

where C i are constant coefficient, ω is the angular frequency and k n is the flexural wave number.

From equations (8)-(11), the characteristics equation is obtained as:

C(κn , β )=κn3(1−

J e ω2 LEI κn

(sinh κncos κn+cosh κnsin κn)+cosκncosh κn)+

(me L3ωn

2

EI− β )(J e ω2L

EI κn(1−cosκncosh κn )+cossinh κncosκn−sin κncosh κn) (12)

If the effect of tip height is neglected, the characteristics equation is converted to (Payam et al., 2013, 2014):

C (κn , β )=κn3(1+cosκn cosh κn )+(

me L3 ωn2

EI−β )(sinh κn cos κn−sin κn cosh κn) (13)

where κn=kn L and β=

K f

EI /L3is the normal stiffness ratio between the normal contact stiffness and

that of the cantilever .

By considering the calculated equation for the resonance frequencies of AFM cantilever in the liquid (Payam, 2013b):

ωn2[ π

4a1

' ρ f b2+ρ cbh ]+ωn3/2[ π

2a2

' b√η√ ρf ]=EIkn4

(14)

Following equation can be obtained:

κn=(ωn

2 [ π4

a1' ρf b2+ ρc bh]+ωn

3/2 [ π2

a2' b√η√ρ f ])

1/ 4

( L/ EI1 /4 ) (15)

By substituting (15) in (12), the resonance frequencies of AFM cantilever in liquid with the effect of tip

mass can be calculated.

3. Modal sensitivity of flexural vibration of AFM in liquid

The flexural sensitivity of the cantilever to surface stiffness variations can be calculated from the

following equation:

∂ f n

∂ βn=

∂ f n

∂ κn

dκn

dβn (16)

where

∂ f n

∂κn= 1

2 π [ 4 EI κn3 /L4

2 ωn (π a1' ρf b2 /4+ ρc bh )+3 π √ωn a2

' b√η√ ρf /4 ] (17)

and

dκn

dβn=−

∂C /∂ βn

∂C /∂κn (18)

which:

∂C∂ β

=−( Je ω2 LEI κn

(1−cosκn cosh κn)+cossinh κn cosκn−sin κn coshκn) (19)

∂C∂κn

=3κn2(1−

Je ω2 LEI κn

(sinh κncos κn+cosh κnsin κn)+cos κncosh κn)+

κn3(sinhκncos κn−coshκnsin κn+(

J e ω2LEI κn

2 −2J e ωLEI κn

∂ω∂κn

)(sinhκncos κn+cosh κnsin κn)−2J e ω2 LEI κn

cosκncosh κn)+

2me L3ωEI

∂ω∂κn (J e ω2 L

EI κn(1−cos κncosh κn)−sinκncosh κn+cos κnsinh κn)−

(β−me ω2 L3

EI )((2J e LωEI κn

∂ω∂κn

−J e ω2 LEI κn

2 )(1−cos κn coshκn)+J e ω2 LEI κn

(coshκnsin κn−sinh κncosκn)−2sinκnsinh κn)

(20)

Based on the equations (17)-(20) ∂ f n /∂ βn can be calculated.

A dimensionless form of flexural sensitivity in liquid is given by:

σ f=∂ f /∂ βn

12 πL2 √ EI

ρc A

(21)

4. Numerical Analysis

By considering the parameters of AFM from Table 1, and

a1' =1 . 0553 , a2

' =3.7997 , b1' =3 .8018 , b2

' =2 . 273642 for hydrodynamic function (Maali et al., 2005), the

effect of tip mass and interaction force on the resonance frequencies are studied. Tables 2-3, show the

effect of tip mass on the normalized frequencies of AFM cantilever in liquid environment for the cases of

free vibration and interaction in the repulsive regime (β=1 ). It is noted that in the liquid environment,

the interaction in the attractive regime can be neglected. The frequencies are normalized with respect to

the first mode of free vibrational frequency of the cantilever (ω1=56901(rad /sec) ) when the tip mass is

neglected. Also, the different tip masses are normalized with respect to the nominal value of tip mass

given in Table 1. As it can be seen, especially in comparison with the air environment (Payam et al.,

2009, 2013, 2014), the effect of tip mass in the liquid environment is negligible. Table 4 shows the effect

of tip height on the resonance frequency of cantilever. The frequencies are normalized with respect to the

first mode of free vibrational frequency when the tip height is 10 µm and the tip mass is 2 .25×10−12kg

(nominal value). From the results, it is clear that the height of the tip is negligible in the liquid

environment which is in coincidence with the experimental results of (Kiracofe et al., 2010).

Figures 2-5 show the effect of tip mass on the sensitivity of first four modes. As it can be seen, in general,

the tip mass has not significant effect on the sensitivity and its effect can be neglected in the liquid

environment, while in the air environment, for higher modes and soft materials, the effect of tip mass is

significant (Payam et al., 2014). From figures 4 and 5 it is depicted that for the large tip mass and soft

samples, the sensitivities of higher modes in the air are less than in the liquid. This means that in contrast

of liquid which tip mass has no significant impact on the sensitivity, for soft materials in air it has

significant impact which may reduce the sensitivity of cantilever in comparison with liquid. In these

figures me is considered as nominal tip mass of the cantilever which is given in Table 1. Moreover, the

effect of tip height on the sensitivities of AFM cantilever is shown in Fig.6. From the results, it is clear

that the tip height has not effect on the sensitivity. Hence, in the modeling and analysis of AFM cantilever

immersed in the liquid environment it can be neglected. This numerical result is in full agreement with

experiments of (Kiracofe et al., 2010) that depicts the tip height is negligible in liquid.

5. Conclusion

Using Euler-Bernoulli beam theory and considering tip mass, the characteristic equation of a

microcantilever in liquid is derived. Then, the effect of tip mass and interaction force on the resonance

frequencies and sensitivity of microcantilever to surface stiffness variations is analyzed. The results

demonstrate that in contrast to an air environment, the effect of tip mass on cantilever dynamics in liquid

is negligible.

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Fig.1. Schematic of AFM cantilever with tip mass.

Fig. 2. Normalized flexural sensitivity of mode 1 in air and liquid.

Fig. 3. Normalized flexural sensitivity of mode 2 in air and liquid.

Fig. 4. Normalized flexural sensitivity of mode 3 in air and liquid.

Fig. 5. Normalized flexural sensitivity of mode 4 in air and liquid.

Fig. 6. Normalized flexural sensitivities as a function of tip height in liquid.

Table 1. Parameters of the AFM cantilever

Elastic modulus E (GPa) 170

Density of the cantilever ρc (kg/m3) 2320

Density of the Fluid ρ f (kg/m3)1000

Length L (µm) 225Width b (µm) 40

Thickness h (µm) 1.8Tip Height ht(µm) 10

Tip mass me (kg) 2 .25×10−12

Viscosity η ¿10−3(Pa.s) 1

Internal damping of the cantiever a1(s) 2×10−10

Table 2. Normalized resonance frequencies of the micro-cantilever as a function of normalized tip mass in the absent of interaction.

modes\normalized mass

0 .2 .4 .6 .8 1 2 4

1 1 0.9982 0.9965 0.9947 0.993 0.9912 0.9824 0.96662 7.2459 7.2301 7.2143 7.1985 7.1826 7.1668 7.0913 6.95593 21.1842 21.1314 21.0804 21.0295 20.9803 20.9328 20.7061 20.31424 42.3068 42.1942 42.0853 41.9798 41.8779 41.7777 41.319 40.5633

Table 3. Normalized resonance frequencies of the micro-cantilever as a function of normalized tip mass in the repulsive regime (

β=1¿ .

modes\normalized mass

0 .2 .4 .6 .8 1 2 4

1 1.167 1.1652 1.1634 1.1617 1.1582 1.1564 1.1476 1.12832 7.2776 7.2617 7.2442 7.2283 7.2125 7.1967 7.1194 6.98233 21.1964 21.1437 21.091 21.0418 20.990

820.9434 20.7149 20.323

4 42.312 42.2013 42.0923 41.9851 41.8832

41.783 41.3226 40.5669

Table 4. Normalized resonance frequencies of the micro-cantilever as a function of tip height.

modes\Height (µm) 0 5 101 1 1 12 7.2304 7.2304 7.23043 21.129

121.1255 21.1184

4 42.189 42.1783 42.1482