Contact stiffness modulation in contact-mode atomic force microscopy

Contact stiffness modulation in contact-mode atomic force microscopy

Ilham Kirrou, Mohamed Belhaq

ⁿ

Q1

Laboratory of Mechanics, University Hassan II-Casablanca, Morocco

a r t i c l e i n f o

Article history:

Received 27 January 2012 Received in revised form 17 April 2013

Accepted 22 April 2013

Keywords:

Atomic force microscopy Contact-mode Non-linear dynamics Stiffness modulation Frequency shift Nanomechanics

a b s t r a c t

The effect of fast contact stiffness modulation on the frequency response in contact-mode atomic force microscopy is studied analytically near primary resonance. Based on the Hertzian contact theory, a lumped single degree of freedom oscillator is considered for modeling the contact-mode dynamics between the tip of the microbeam and the sample. Averaging method and perturbation analysis are performed to obtain the modulation equations of the slow dynamic. The influence of the contact stiffness modulation on the non-linear characteristic of the frequency response is examined. Wefind that the amplitude of the contact stiffness modulation influences significantly the amplitude of the tip oscillation as well as the shift direction of the frequency response indicating that such a modulation can be used to characterize the local elastic properties of the sample. Comparison between the analytical predictions and the numerical simulations is given and application to a real atomic force microscope example is provided.

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1. Introduction

In atomic force microscopy (AFM) [1], a micro-scale cantilever beam with a sharp tip is employed to scan the topography of a specimen surface. Typically, the contact-mode AFM is used in such applications to con

fi

ne the surface force to a Hertzian contact regime between the tip and the moving surface. The performance of this contact-mode AFM in scanning requires the contact-mode regime to be maintained during the scan in order to obtain quanti

fi

ed results in terms of vibrational amplitude and amplitude response. It is known that in macro-scale mechanisms, contact- mode in a harmonically forced Hertzian contact regime has soft- ening characteristic and for a slight increase of the amplitude of the excitation, contact losses occur near resonances causing impacts, thereby a possible deterioration of the device [2]. This loss of contact phenomenon has been observed for an idealized preloaded and non-sliding dry Hertzian contact modeled by a single-degree-of-freedom (sdof) system [3]. Based on numerical simulations, analytical approximation and experimental testing [3,4], it was concluded that the loss of contact is generally initiated by jumps near resonances. In order to control the location of such jumps, three strategies were developed [5,6]. The

fi

rst strategy introduced a fast harmonic excitation (added to the basic harmo- nic forcing) from above, the second strategy used a fast harmonic base displacement, while the third one considered a fast harmonic parametric stiffness. It was concluded that fast harmonic base

displacement shifts the resonance curve left, whereas fast para- metric stiffness shifts the resonance curve right, suggesting methods for controlling contact losses in systems evolving in Hertzian contact regime.

Although the effect of fast excitation has been analyzed analytically and numerically in various engineering applica- tions [5

–

8] showing various non-trivial effects, it has received limited attention in micro-electromechanical devices [9,10] and in AFM [11].

In this paper, we report on the effect of fast contact stiffness modulation on the frequency response to primary resonance in contact-mode AFM considering a lumped sdof oscillator modeling the contact-mode dynamics between the tip and the sample.

In the case where the stiffness modulation is absence, Turner [12] investigated the non-linear vibrations of a linear beam with cantilever Hertzian contact boundary conditions assuming that the beam remains in contact with the moving surface at all times. He used the method of multiple scales (MMS) [13] to approximate the response of the probe-tip sample system to primary resonance excitation. The softening behavior of the response was obtained for the

fi

rst four modes, and it was concluded that the response of the

fi

rst mode is more willing to loose contact near the resonance before a signi

fi

cant change of parameters. Following Hajj et al.

[14], Abdel-Rahman and Nayfeh [15] used the MMS to estimate the non-linear coef

fi

cients of the contact stiffness using the subhar- monic resonance of the contact-mode AFM. Contact-mode AFM can be used in various applications including, for instance, the determination of the viscoelastic properties of materials [16], identi

fi

cation of the interaction modes [17] and evaluation of adhesion energy [18].

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

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International Journal of Non-Linear Mechanics

0020-7462/$ - see front matter&2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.ijnonlinmec.2013.04.013

nCorresponding author.

Q2 Tel.:+212 2522230674.

Q3

E-mail address:mbelhaq@yahoo.fr (M. Belhaq).

Owing to the fact that the

fi

rst mode is predisposed to lose contact promptly with a slight change of parameters [12], atten- tion will be restricted to the analysis of the response to the

fi

rst mode of the microbeam. Thus, we consider a sdof system model- ing the cantilever dynamics of contact-mode AFM based on the Hertzian contact theory. Such a system is often adopted to model the response of AFM cantilever neglecting the higher-order

fl

exural modes.

The next section presents the sdof model under a Hertzian contact condition in which the contact stiffness is assumed to be modulated with high-frequency (HF) excitation. Then, the method of direct partition of motion (DPM) is applied to obtain the main equation describing the slow dynamic of the tip-sample system.

In Section 3, the MMS is applied on the slow dynamic to obtain the corresponding slow

fl

ow to primary resonance. This section includes results of various parameters effect on the frequency response and on jumps phenomena. An application to a real AFM case is also provided in this section. Section 4 concludes the work.

2. Model and slow dynamic equation

A representative model of contact-mode AFM operation with contact stiffness modulation is proposed. It consists of a lumped sdof model, as shown in Fig. 1 [19], described by the equation of motion

m€ x þ c

1

x

_

þ kx ¼

−ð

k

0

þ k

1

cos

Ω²

t Þð z

0−

x Þ

³⁼²

þ mg þ F cos

Ω¹

t ð 1 Þ where x denotes the effective displacement of the cantilever tip, m is the lumped cantilever mass, c

1

( ¼ c

0

þ c

_n

) is the effective damping constant, k is the free cantilever stiffness, k

0

is the constant given by the Hertz theory [20] which is associated with the radius, AFM tip and substrate moduli and Poisson

'

s ratios, k

1

,

Ω2

are the amplitude and the frequency of the contact stiffness modulation, respectively, z

0

is the surface offset, g is the accelera- tion gravity, and F,

Ω1

are, respectively, the amplitude and the frequency of the excitation of the sample vibration, as considered in atomic acoustic microscopy [21,22]. The displacement x is de

fi

ned by considering the static problem as x ¼ x

s

þ X, and the quantity

Δ

¼ z

0−

x

s

as the static Hertz deformation, where x

s

is the static position and X is the displacement from the static position.

From application view point, the contact stiffness modulation can be introduced either by the excited cantilever using ultrasonic transducer attached to the cantilever and monitoring the ampli- tude of the z-piezo [23,24] or by monitoring the amplitude in the

transducer bounded underneath the sample as in the atomic force acoustic microscopy [21,25].

Notice that the contact-mode AFM model under consideration, Eq. (1), takes into account only the Hertzian contact non-linearity.

The non-linear attractive AFM mode opposing the behavior of the Hertzian contact non-linearity is neglected, so that analytical prediction is valid only for a limited amplitude and frequency range of the modulation.

Introducing the variable changes [12]: u ¼ X

=Δ

,

τ

¼

ω0

t,

ω²0

¼ k

=

m, c ¼ c

1=

m

ω⁰

,

β

¼ 3k

0Δ¹⁼²=

2k,

β1

¼

β=

4,

β2

¼

β=

24, f ¼ F

=

m

ω²0Δ

,

ω

¼

Ω1=ω0

and

Ω

¼

Ω2=ω0

, the dimensionless equation of motion takes the form

€

u þ c_ u þ u

−²3β

þ

²3βð

1 þ r cos

ΩτÞð

1

−

u Þ

³⁼²

¼ f cos

ωτ

ð 2 Þ where ðÞ ¼

_

d

=

d

τ

and r ¼ k

1=

k

0

are the dimensionless amplitudes of the contact stiffness modulation given by the ratio between the modulated and the unmodulated contact stiffness coef

fi

cients which is assumed to be smaller than 1 ð k

1o

k

0

Þ or of order 1 (k

1≈

k

0

). The coef

fi

cient

β

de

fi

nes the stiffness of the unmodulated contact relative to the stiffness of the tip.

Expanding the non-linear restoring force in Taylor series in the vicinity of the static load and keeping only terms up to order three in u, Eq. (2) reads

€

u þ

ϖ²

u þ c u

_

þ

β1

u

²

þ

β2

u

³

þ r ð

³2β−β

u þ

β1

u

²

þ

β2

u

³

Þ cos

Ωτ

¼ f cos

ωτ

ð 3 Þ where

ϖ

¼

ffiffiffiffiffiffiffiffiffi

1

−β

p

is the natural frequency of the system. Eq. (3) contains a slow dynamic due to the external excitation of the sample and a fast dynamic produced by the frequency of the contact stiffness modulation

Ω

. Assume that the natural frequency,

ϖ

, may be in resonance with the external excitation,

ω

, but not in resonance with

Ω

(supposed larger than

ϖ

). Further, in order to keep

ϖ

small comparing to

Ω

, values of

β

have to be chosen as close as possible to 1 with the condition

βo

1 to be satis

fi

ed.

Taking these remarks into consideration, the effect of the contact stiffness modulation on the slow dynamic can be investigated using the method of DPM [26,27]. This method consists in introducing two different time scales, a fast time T

0

¼

Ωτ

and a slow time T

1

¼

τ

, and splitting up u ðτÞ into a slow part z ð T

1

Þ and a fast part

ψð

T

0;

T

1

Þ as

u ðτÞ ¼ z ð T

1

Þ þ

ψ

ð T

0;

T

1

Þ ð 4 Þ where z contains a slow dynamic which describes the main motions at time-scale of the tip natural vibrations and

ψ

stands for an overlay of the fast motions at time scale of the parametric excitation. Performing the method of DPM, we obtain the main equation governing the slow dynamic of the motion

€

z þ

ω²1

z þ c_ z þ

ρ1

z

²

þ

ρ2

z

³

þ H ¼ f cos

ωτ

ð 5 Þ where the parameters

ω²1

,

ρ1

,

ρ2

and H are given, respectively, by

ω²1

¼

ϖ²

þ 2

β²

r

²

3

Ω² −

5

β³

r

²

36

Ω⁴− β⁴

r

⁴

48

Ω⁶

ð 6 Þ

ρ1

¼

β

4

−β²

r

²

3

Ω²

þ

β³

r

²

12

Ω⁴

þ 7

β⁴

r

⁴

192

Ω⁶

ð 7 Þ

ρ2

¼

β

24

−β²

r

²

48

Ω²

þ

β³

r

²

36

Ω⁴−

35

β⁴

r

⁴

1152

Ω⁶

ð 8 Þ

H ¼

−β²

r

²

3

Ω²

þ

β³

r

²

18

Ω⁴

þ

β⁴

r

⁴

216

Ω⁶

ð 9 Þ

showing how the natural frequency and the non-linear compo- nents of the slow dynamic are related to the contact stiffness modulation parameters r,

Ω

and coef

fi

cient

β

. Details on the derivation of Eq. (5) are given in Appendix.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67

68

69

70

71

72

73

74

75

76

77

78

79

80

81

82

83

84

85

86

87

88

89

90

91

92

93

94

95

96

97

98

99

100

101

102

103

104

105

106

107

108

109

110

111

112

113

114

115

116

117

118

119

120

121

122

123

124

125

126

127

128

129

130

131

132

Fig. 1. A schematic of sdof model of a tip-sample AFM.

3. Frequency response analysis

In this section we shall analyze the amplitude

–

frequency response of the slow dynamic (5) in the absence and presence of contact stiffness modulation. We apply the MMS to obtain the slow

fl

ow system and we examine the effect of various system parameters on the frequency response near the primary resonance.

3.1. Case without contact stiffness modulation

In order that the cubic non-linearity balances the effect of damping and forcing, we scale parameters in Eq. (5) as c ¼

ϵ²

c,

ρ2

¼

ϵ²ρ2

and f ¼

ϵ²

f (the other parameters being of order

ϵ

) so that they appear together in the modulation equations. Thus, Eq. (5) reads

€

z þ

ω²1

z ¼

−ϵðρ1

z

²

þ H Þ−ϵ

²

ð c_ z þ

ρ2

z

³−

f cos

ωτÞ

ð 10 Þ A two-scale expansion of the solution to Eq. (10) is sought in the form

z ð T

0;

T

1;

T

2

Þ ¼ z

0

ð T

0;

T

1;

T

2

Þ þ

ϵ

z

1

ð T

0;

T

1;

T

2

Þ þ

ϵ²

z

2

ð T

0;

T

1;

T

2

Þ þ 0 ðϵ

³

Þ ð 11 Þ where T

0

¼

τ

, T

1

¼

ϵτ

and T

2

¼

ϵ²τ

. In terms of the variables T

_i

(i ¼ 0

;

1

;

2), the time derivatives become d

=

d

τ

¼ D

0

þ

ϵ

D

1

þ

ϵ²

D

2

þ O ðϵ

³

Þ and d

²=

d

τ²

¼ D

²₀

þ 2

ϵ

D

01

þ

ϵ²

D

²₁

þ 2

ϵ²

D

02

þ O ðϵ

³

Þ , where D

_i

¼

∂=∂

T

_i

. Substituting (11) into (10) and equating the terms with the same order of

ϵ

, yields

D

²₀

z

0

þ

ω²1

z

0

¼ 0 ð 12 Þ

D

²₀

z

1

þ

ω²1

z

1

¼

−

2D

0

D

1

z

0−ρ1

z

²₀−

H ð 13 Þ

D

²₀

z

2

þ

ω²1

z

2

¼

−

2D

0

D

1

z

1−ð

D

²₁

þ 2D

0

D

2

Þ z

0

−

cD

0

z

0−

2

ρ1

z

0

z

1−ρ2

z

³₀

þ f cos

ωτ

ð 14 Þ The solution of Eq. (12) can be written in the form

z

0

ð T

0;

T

1

Þ ¼ A ð T

1

Þ e

^iω1T0

þ cc ð 15 Þ where A ð T

1

Þ is a complex amplitude and cc stands for the complex conjugate of the preceding term. Also, the resonance condition requires that the frequency of excitation is assumed to remain near the natural frequency according to

ω

¼

ω1

þ

ϵs

ð 16 Þ

in which

s

is a detuning parameter representing the deviation from natural frequency. Substituting (15), (16) into (13), (14) and removing secular terms, we obtain

ρn

A

²

A þ

ρl

A þ i ð− 2

ω1

D

2

A

−

c

ω1

A Þ þ f

2 e

^isT2

¼ 0 ð 17 Þ where

ρn

¼ 10

ρ²1

3

ω²1

−

3

ρ2; ρl

¼ 2

ρ1

H

ω²1

ð 18 Þ

are, respectively, the effective non-linearity and the effective linearity induced by the contact stiffness vibration.

To better understand the dynamic of the oscillating tip, the variation of these two quantities,

ρn

,

ρl

, as functions of the amplitude r will be examined below. Eq. (17) can be solved for the complex amplitude by introducing its polar form as

A ¼

¹2

ae

^iθ

ð 19 Þ

Hence, substituting (19) into (17) and separating real and imagin- ary parts, we obtain the modulation equations of amplitude and

phase as da dt ¼ f

2

ω1

sin

φ−

c 2 a d

φ

dt ¼ f 2

ω1

cos

φ

þ 5

ρ²1

12

ω³1

−

3

ρ2

8

ω²1

!

a

³

þ

s

þ

ρ1

H

ω³1

!

a

8>

>>

><

>>

>>

:

ð 20 Þ

in which

φ

¼

s

T

2−θ

. Equilibria of this slow

fl

ow, corresponding to periodic solutions of Eq. (5), are determined by setting da

=

dt ¼ d

φ=

dt ¼ 0. This leads to the amplitude

–

frequency response equation

AJ

³

þ BJ

²

þ CJ þ D ¼ 0 ð 21 Þ

where A ¼ ð

³4ρ2−ð

5

ρ²1=

6

ω²1

ÞÞ

²

, B ¼ 2c

ω1

ð− 2

ω1s−ð

2

ρ1

H

=ω²1

ÞÞ , C ¼ ð c

ω¹

Þ

²

þ ð− 2

ω¹s−ð

2

ρ1

H

=ω²1

ÞÞ

²

, D ¼

−

f

²

and J ¼ a

²

. In the rest of the paper we

fi

x the parameter c ¼ 0.02. Next, the effect of excitation amplitude, f, and contact stiffness,

β

, is analyzed. Fig. 2 shows the variation of the amplitude

–

frequency response, as given by Eq. (21), for different values of the amplitude f. The solid lines denote stable solutions and the dashed lines denote unstable ones.

Results obtained by direct numerical simulation of Eq. (5) (circles) using Runge

–

Kutta method are also plotted for validation. It can be seen from this

fi

gure an increase of the response amplitude and softening behavior when f is increased. In terms of the quantities de

fi

ned in the Hertzian contact case, values of r are larger than 1 mean loss of contact [12]. Fig. 3 depicts the variation of the frequency response for different values of the contact stiffness

β

. One observes that an increase of

β

leads also to an increase in softening behavior. This phenomenon (reported in [12] for a micro cantilever) shows that the contact stiffness in

fl

uences the non- linear characteristic of the system and then the analysis of the 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

−0.2 −0.1 0 0.1 0.2

0 0.2 0.4 0.6 0.8 1

σ

a

f=0.01

f=0.004

Fig. 2.Frequency response forr¼0,β¼0:8 and for different values off. Analytical prediction (solid lines for stable and dashed lines for unstable) and numerical simulation (circles).

−0.2 −0.1 0 0.1 0.2

0 0.2 0.4 0.6 0.8 1

σ

a

β=0.8 β=0.7 β=0.9

Fig. 3.Frequency response forr¼0,f¼0.008 and for different values ofβ.

non-linear behavior of the system can provide important informa- tion on the tip-sample interaction.

3.2. Case with contact stiffness modulation

Now we consider the case where the contact stiffness is modulated and we

fi

x the parameter

β

¼ 0

:

8. Periodic solutions of the slow dynamic, Eq. (5), correspond to the roots of Eq. (21).

This Eq. (21) gives one or three real solutions depending on the sign of its discriminant

Δ

¼ Q

²

þ ð 4P

³=

27 Þ in which P ¼ ð C

=

A Þ

−ð

B

²=

3A

²

Þ and Q ¼ ð 2B

³=

27A

³

Þ−ð BC

=

3A

²

Þ þ ð D

=

C Þ . The bifurcation curves separating the existence domain of periodic solutions given by the condition

Δ

¼ 0 are plotted in Figs. 4 and 5. Fig. 4a shows the bifurcation boundaries in the parameter plane (f

;s

) for two different values of the modulation amplitude r (r ¼ 0 for solid lines and r ¼ 0.4 for dashed lines, respectively). In the regions between the boundaries (

Δ4

0), three solutions exist, two stable and one unstable, while only one stable solution exists outside the bound- aries (

Δo

0). The stability analysis has been done using the Jacobian of the slow

fl

ow system. It can be seen from this

fi

gure that as the amplitude r is increased, the domain of bistability decreases. Fig. 4b presents the bifurcation curves in the parameter plane (r

;s

) showing the zone inside the boundaries where three solutions exist (two stable and one unstable). Outside the bound- aries only one stable solution exists. One observes that beyond a certain value of the amplitude r, the domain of bistability disappears.

Fig. 5 shows the bifurcation curves in the parametric excitation plane (r

;Ω

) for

s

¼ 0 (Fig. 5a) and for

s

¼

−

0

:

08 (Fig. 5b). Between

the boundaries curves of Fig. 6a, two stable and one unstable solutions exist, while one stable solution lies outside. In Fig. 5b, the three solutions exist above the line and the stable solution exists below it.

Next the effect of the amplitude, r, and frequency,

Ω

, of the contact stiffness modulation is examined. Fig. 6 shows the effect of

Ω

on the frequency response indicating that increasing

Ω

softens the contact stiffness between the tip and the sample resulting in a decrease of the bistability domain.

In Fig. 7 we show the frequency response for various values of r.

By inspecting this

fi

gure, one observes that increasing r from 0.4 to 0.7, the amplitude response shifts toward higher frequencies while changing from softening to linear behavior. As the amplitude r continue to increase, the linear frequency response keeps shifting 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

Fig. 4.Bifurcation curves of periodic solutions of(5)in (a) the planeðf;sÞand (b) the planeðr;sÞforf¼0.008 (in subfigure (b)) andΩ¼1.

Fig. 5.Bifurcation curves of periodic solutions in the planeðr;ΩÞforf¼0.008 andΩ¼1.

−0.1 −0.05 0 0.05 0.1

0 0.2 0.4 0.6 0.8 1

σ

a

Ω=1 Ω=2 Ω=1.5

Fig. 6.Frequency response forf¼0.008 andr¼0.5.

right until reaching a maximum position for a certain critical value of r, and then shifts back toward lower frequencies (see the curve for r ¼ 1). This result reveals that increasing the amplitude of the contact stiffness modulation r causes a substantial decreasing in the resonance peak, eliminates the jumps and produces a change in the shift direction of the frequency response. This change in the shift direction results in the variation of the effective contact stiffness. Namely, one observes that for a given value of detuning

s

the amplitude of the tip vibration may increase or decrease providing estimation of the local elastic properties of the surface.

To clarify this change phenomenon in the shift direction, we plot in Fig. 8 the detuning of the pick response as a function of the amplitude r. It can be clearly seen that as r increases from 0 to 1, the values of the pick response increase toward higher frequen- cies, reach a maximum for a certain critical value of r, and then decrease toward lower frequencies (which is coherent with the result shown in Fig. 7). The analytical prediction (solid line) and results obtained by numerical simulations (circles) using Runge

–

Kutta method are compared showing a good match.

To better understand this interesting phenomenon of the change in the frequency shift direction when increasing r, we plot in Fig. 9 the variation of the effective non-linearity,

ρn

, and the effective linearity,

ρl

, given by (18), as functions of r. This

fi

gure depicts two critical values for r; the

fi

rst one, r

l

¼ 0

:

665, corre- sponds to the condition

ρn

þ

ρl

¼ 0 ð 22 Þ

at which the frequency response meets a linear behavior, and the second one, r

s

¼ 0

:

770, given by the condition

ρn

¼ 0 ð 23 Þ

corresponding to the location where the frequency shift changes its direction. Fig. 9 reveals that increasing r from 0 to r

l

, the frequency response shifts right while the softening characteristic

decreases until meeting a linear behavior at r

l

. Increasing r bet- ween r

l

and r

s

, the frequency response keeps shifting right (see Fig. 7 for r ¼ 0.7) until reaching its maximum position at r ¼ r

s

. By increasing r beyond r

s

, the frequency response undergoes a shift back to the left (see Fig. 7 for r ¼ 1).

Fig. 10 shows in the parameter plane ð r

;ΩÞ

the boundary given by the condition (22) separating the regions where the frequency response is softening or has a linear behavior. The curves shown in the small boxes inset Fig. 10 are obtained for values of r and

Ω

as given in the legend.

In Fig. 11 we show the curve given by the condition (23) in the plane ð r

;ΩÞ

. When r increases along the line labeled L

1

, the 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

−0.1 −0.05 0 0.05 0.1 0.15

0 0.2 0.4 0.6 0.8 1

σ

a

r=0.4 r=0.7

r =1

Fig. 7.Frequency response forf¼0.008,Ω¼1 and different values ofr(picked fromFig. 11on the lineL2(dots)).

0 0.2 0.4 0.6 0.8 1

−0.15

−0.1

−0.05 0 0.05

r

Detuning (σ) of the peak response

Fig. 8.Detuning of the pick response versusr. Analytical approximation (solid line) and numerical simulation (circles) forf¼0.008 andΩ¼1.

0 0.2 0.4 0.6 1

−0.1

−0.05 0 0.05 0.1 0.15

r ρi

r_l r_s ρ_n

ρ_l

Fig. 9. The variation of the effective non-linearity and the effective linearity as functions ofrforf¼0.008 andΩ¼1.

Fig. 10. Curve separating softening and linear domains of the response in the plan ðr;ΩÞforf¼0.008. In box (a)Ω¼0:6;r¼1, (b)Ω¼1;r¼1 and (c)Ω¼1:5;r¼0:6.

Fig. 11. Curve corresponding to the change in the frequency shift direction in the planðΩ;rÞforf¼0.008.

frequency response shifts toward higher frequencies while the resonance amplitude decreases (as shown in Fig. 13 for values of r picked on the line L

1

(squares)). In this region no change in the frequency shift direction is observed. When r increases on the line labeled L

2

, the frequency response

fi

rst shifts to the right until reaching a maximum position for a value of r lying on the curve, and then shifts back to the left while increasing r above the curve (as shown in Fig. 7 for values of r picked on the line L

2

(dots)).

In other words, three regions can be distinguished in this

fi

gure.

When increasing r, region I corresponds to softening behavior and a shift to the right, region II corresponds to linear response shifting right, whereas in region III, the response has a linear behavior but shifting left.

To have a complete view of the precedent observations, curves of Figs. 10 and 11 are plotted together in Fig. 12 showing different situations of the frequency response in different regions.

The arrows above the small

fi

gures (in boxes) indicate the direction of the shift. In Fig. 13 we plot the frequency response for

Ω

¼ 0

:

6 and for values of r picked on the line L

1

in Fig. 11 (squares) showing a shift toward higher frequencies accompanied by a substantial reduction of the resonance amplitude.

Finally, Fig. 14 shows the change of the shift direction phenom- enon for values of r picked on the line L

₂

(Fig. 11) but for a different value of

β

¼ 0

:

95.

3.3. Application to a real AFM example

The mathematical model studied in the previous sections is compared with a real AFM example. Those comparisons are made using the parameters typical to those found in AFM [28]. The

elastic modulus and density for

〈

100

〉

silicon, E ¼ 169 GPa and

ρ

¼ 2330 kg

=

m

³

, respectively, were used. The cantilever has width a ¼ 51

μ

m, thickness b ¼ 1

:

5

μ

m, length L ¼ 262

μ

m, the lumped mass m ¼ 1

:

13 10

⁻¹¹

kg and the free stiffness k ¼ 0.404 N/m.

Also, the following parameters, corresponding to a single crystal silicon tip interacting with a chromium surface, were used in the numerical results: R ¼ 20 nm,

Δ

¼ 0

:

26 10

⁻⁶μ

m, E

t

¼ 130 GPa,

νt

¼ 0

:

181, E

s

¼ 204 GPa and

νs

¼ 0

:

26, where R is the tip radius, E

t

, E

s

are the elastic modulus of the tip and surface, respectively, and

ν^t

,

ν^s

are Poisson

'

s ratio of the tip and surface, respectively.

For comparison, Fig. 15 shows in the amplitude and frequency modulation parameter plane ð r

;ΩÞ

the analytical curve given by (22) (solid line) and the curve obtained from the real AFM example (dashed line). Inside the boundary the frequency response is linear and outside the boundary it has a softening behavior.

The chosen value of free stiffness k ¼ 0.404 N/m is rather small in terms of what we usually use for quantitative measurements [28].

However, using the values given above provides the same value for the cantilever stiffness. Probably, it entails that soft cantilevers exhibit much easier non-linear contact resonances.

4. Summary

The effect of contact stiffness modulation on the frequency response of a contact-mode AFM was studied. A lumped sdof system modeling the cantilever dynamics of contact-mode AFM was considered and emphasis was placed on the case when the AFM is driven near the primary resonance. The contact force is produced by the Hertzian contact regime and the external harmonic excitation is induced by the sample vibration. The technique of DPM 1

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66

67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

Fig. 12.Curves ofFigs. 10and11as given by Eqs.(22)and(23).

−0.1 −0.05 0 0.05 0.1 0.15 0.2

0 0.2 0.4 0.6 0.8 1

σ

a

r =0.4

r =0.7

r =0.9

Fig. 13.Frequency response forf¼0.008,Ω¼0:6 and for different values ofrpicked fromFig. 11on the lineL1(squares).

−0.2 −0.1 0 0.1 0.2

0 0.2 0.4 0.6 0.8 1

σ

a r =0

r =0.7

r =1 r =0.4

Fig. 14.Frequency response forf¼0.008,Ω¼1 andβ¼0:95.

Fig. 15.Curve separating softening and linear domains of the response in the plan ðr;ΩÞ. Analytical prediction (solid line, picked fromFig. 10) and result for real AFM example (dashed line).

as well as the MMS was used to determine the non-linear frequency response of the slow dynamic. It was shown that increasing the amplitude of the contact stiffness modulation, the frequency response shifts toward higher-frequencies, becomes linear for a critical value of the amplitude and then shifts back toward lower frequencies while the peak of the amplitude decreases substantially.

This reveals that by monitoring the amplitude of the contact stiffness modulation, the frequency response can be shifted toward higher or lower values of the frequencies leading the amplitude of vibration to increase and decrease for given operating frequencies.

This change of the amplitude oscillation is correlated with the local elastic properties of the surface and thus with the imaging of the sample.

We conclude from this work that for small values of damping and external forcing, modulation of the contact stiffness at large frequency may increase or decrease the Hertzian contact force dynamically as well as the contact area. As a result, the contact- mode AFM can be monitored near the resonance such that a good performance of the AFM operation can be achieved in term of scanning or measuring local elastic proprieties of the specimen.

Appendix A

In the averaging procedure, the fast part

ψ

and its derivatives are assumed to be 2

π

periodic functions of fast time T

0

with zero mean value with respect to this time, so that

〈

u ð t Þ〉 ¼ z ð T

1

Þ where

〈〉≡

1

=

2

πR2π

0

ðÞ dT

0

de

fi

nes time-averaging operator over one period of the fast excitation with the slow time T

1 fi

xed. Introducing D

^j_i≡∂^j=∂^j

T

i

yields d

=

dt ¼

Ω

D

0

þ D

1

, d

²=

dt

²

¼

Ω²

D

²₀

þ 2

Ω

D

0

D

1

þ D

²₁

and substituting Eq. (4) into Eq. (3) gives

€

z þ

ψ€

þ c ð_ z þ

ψ_

Þ þ

ϖ²

ð z þ

ψ

Þ þβ

1

ð z þ

ψ

Þ

²

þ

β2

ð z þ

ψ

Þ

³

þ 3

2 r

β

cos

Ωτ

−

r

βð

z þ

ψÞ

cos

Ωτ

þ r

β1

ð z þ

ψ

Þ

²

cos

Ωτ

þ r

β2

ð z þ

ψÞ³

cos

Ωτ

¼ f cos

ωτ

ð 24 Þ Averaging (24) leads to

€

z þ c_ z þ

ϖ²

z þ

β1

z

²

þ

β1〈ψ²〉

þ

β2

z

³

þ 3

β2

z

〈ψ²〉

þ

β2〈ψ³〉

þ r ½−β〈ψ

〉

þ 2

β1

z

〈ψ〉

þ

β1〈ψ²〉

þ 3

β2

z

²〈ψ〉

þ 3

β2

z

〈ψ²〉

þβ

2〈ψ³〉

cos

Ωτ

¼ f cos

ωτ

ð 25 Þ Subtracting (25) from (24) yields

€

ψ

þ

ψ

þ 2

β1

z

ψ

þ

β1ψ²−β1〈ψ²〉

þ 3

β2

z

²ψ

þ 3

β2

z

ψ²−

3

β2

z

〈ψ²〉

þβ

2ψ³−β2〈ψ³〉

þ r ½−βψ þ

β〈ψ〉

þ 2

β1

z

ψ−

2

β1

z

〈ψ〉

þ

β1ψ²−β1〈ψ²〉

þ 3

β2

z

²ψ−

3

β2

z

²〈ψ〉

þ 3

β2

z

ψ²−

3

β2

z

〈ψ²〉

þβ

2ψ³−β2〈ψ³〉

cos

Ωτ

¼

−

r ½

³2β−β

z þ

β1

z

²

þ

β2

z

³

cos

Ωτ

ð 26 Þ Using the inertial approximation [26], i.e. all terms in the left- hand side of Eq. (26), except the

fi

rst, are ignored, the fast dynamic

ψ

is written as

ψ

¼ r

Ω²

ð

²3β−β

z þ

β1

z

²

þ

β2

z

³

Þ cos

Ωτ

ð 27 Þ This simpli

fi

cation when solving Eq. (26) consists in

fi

nding

ψ

in the form of a sum of a small number of harmonics of the fast time T

0

taking into account that

ψ

is small compared to z and it is possible to consider only the linear dominant terms. For more details on this approximation, the reader can refer to [26, Chapter 2]. Inserting

ψ

from Eq. (27) into Eq. (25), using that

〈

cos

²

T

0〉

¼ 1

=

2, and neglecting terms of orders greater than three in z, give the main equation governing the slow dynamic of the motion (5).

Fig. 16 shows a comparison between the full motion u ðτÞ , (3), and the slow dynamic z ðτÞ , (5), for the given parameters c ¼ 0.05, f ¼ 0.008,

β

¼ 0

:

8,

Ω

¼ 3,

s

¼ 0

:

01 and r ¼ 0.2. The agreement between the full motion and the slow dynamic validates the averaging procedure.

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67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132

2000 2004 2008 2012 2016 2020

−0.4

−0.2 0 0.2 0.4

τ z(τ) u(τ)

Fig. 16.Comparison between the full motionu(t),ð3Þ, and the slow dynamicz(t),(5).

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