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The role of damping in phase imaging in tapping mode atomic force microscopy
Surface Science 429 (1999) 178–185
www.elsevier.nl/locate/susc
The role of damping in phase imaging in tapping mode atomic force microscopy Lugen Wang * Edison Joining Technology Center, Welding Engineering, The Ohio State University, Columbus, OH 43210, USA Received 25 August 1998; accepted for publication 23 February 1999
Abstract The influences of damping on phase imaging in tapping mode atomic force microscopy have been investigated numerically and experimentally. It shows that the adhesion force (attractive) and elastic force (repulsive) divide the phase response into two distinct regimes. Jumps and hysteresis may happen between these two regimes. The magnitude of jumps and hysteresis decreases in the presence of material damping. The sensitivity of phase to surface stiffness and adhesion is significantly dependent on the material damping. © 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Atomic force microscopy (AFM ); Energy dissipation; Phase imaging
1. Introduction The phase image makes the tapping mode atomic force microscopy (AFM ) go beyond topography to image material surface properties such as elasticity and viscoelasticity on a nanometer scale [1–3]. It has been demonstrated that phase imaging can provide more information and contrast in many situations than amplitude imaging or height imaging. The experimental results given by Magonov et al. [4] showed the phase response can reflect material elasticity in small setpoint amplitude ratio. The influence of elasticity on phase response has been numerically investigated by Winkler et al. [5]. In their model, the complicated tip–sample interactions are simplified as a linear spring without an attractive force. They showed that the phase increases as the Young’s modulus * Fax: +1-614-292-6842. E-mail address: [email protected] (L. Wang)
increases. However, the experimental results show much more complicated phenomena. The complicated features in phase response can be explained when one considers the attractive, repulsive force and material damping together. The importance of damping in phase response has already been pointed out by several authors [6–9]. Kuhle et al. [8] measured the damped frequency response. In this paper, by using a model including attractive force (adhesion), repulsive force (elasticity) and damping, the influence of the driving frequency, setpoint amplitude ratio and material properties on phase response will be discussed.
2. Model of the system A simple model of tapping mode AFM is a single degree-of-freedom nonlinear oscillator in which the cantilever is modeled by a spring with spring constant k , a point mass m and quality c
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L. Wang / Surface Science 429 (1999) 178–185
factor Q [9]. A bimorph which oscillates with 0 frequency v and amplitude u at one end of the d 0 cantilever forces the tip to vibrate, thus making intermittent contact with the surface. The tip offset z as shown in Fig. 1 can be controlled by setpoint 0 amplitude ratio (vibration amplitude normalized by free vibration amplitude). The vibration system is described by the differential equation d2z dt2
+2a
dz dt
+[z+F(z)/k ]=u cos(vt) c 0
(1)
where v =앀k /m, t=v t is the nondimensional c c c time and v=v /v is the normalized driving fred c quency. a is the damping constant which includes free vibration damping a and material damping 0 a . F(z) represents the tip–sample interaction. s The static tip–sample interactions have been described by macroscopic continuum theories such as Hertzian, Johnson–Kendall–Roberts (JKR) and Derjaguin–Muller–Toporov (DMT ) contact models [10–13]. Recently, Maugis [12] proposed a more general model based on the Dugdale theory. In atomic force microscopy, the contact radius reaches nanometer; the tip–sample interactions become sensitive to the material properties, experimental conditions and tip shape. The experimental results given by Burnham et al. [10] show
Fig. 1. A model of tapping mode AFM. The cantilever and tip are modeled as a vibration system with spring constant k and c mass m. z is the tip–sample separation. u cos(vt) is the driv0 0 ing signal.
the complicated relation between the force and tip–sample separation. However, in a typical force curve, the force is attractive at small tip–sample separation and approaches Hertzian contact force at large deformation. Fig. 2 shows the relations between the force and tip–sample separation described by Hertzian, JKR, DMT, and Maugis theories. The formulations for these theories can be found in Ref. [13]. The influence of these models on phase response will be discussed in Section 3. In most of our calculation, we will use the simple DMT model to represent the tip–sample interaction in the contact region. Burnham et al. [10] also used an equation similar to the DMT model to fit their typical experimental force curve. The tip–sample interaction in the DMT model may be written as F=K앀Rd3/2−2pwR where w is the surface energy and 3
A
B
1−n2 1−n2 s + c . K 4 E E s c n , n are the Poisson ratio of the sample and s c cantilever respectively and E , E are the Young’s s c modulus of the sample and cantilever respectively. R represents the tip radius and d=z−z is the tip– 0 sample separation. If one considers other attractive forces at noncontact region (d<0), the relation 1
=
Fig. 2. The relation between forces and tip–sample separation for difference models. The long range forces are indicated by the dashed line.
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L. Wang / Surface Science 429 (1999) 178–185
between force and tip–sample separation may be written as F=f (d )
d<0
F=K앀Rd3/2−f
d≥0
0
de dt
.
(2)
(3)
As shown in Eq. (2), the Hertzian contact force is dependent on the power of deformation (d3/2), therefore we assume the viscous force also has a similar relation with deformation and the damping term may be written as a =a (d/R)n s 1
d≥0
u 0 (1+v)앀a2 +(v−v )2 e e v −v tan(Q)= e a e where
(5)
A=
where f is the adhesion force 2pwR, K앀Rd3/2 0 is the repulsive Hertzian contact force. f(d ) represents the long range attractive forces. Burnham et al. [13] wrote this term as f(d)=−f /(1−d/d )2, where d is a constant. This 0 0 0 selection makes the interaction forces F in contact (d>0) and noncontact (d<0) regions join at d=0. When the tip impacts the material surface, energy dissipation occurs due to viscoelasticity and adhesion, especially for polymers and biologic materials. This energy dissipation may be represented by a damping constant a [2,6 ]. In the s Voiget model, the relation of stress s and strain e in the presence of viscoelasticity is given by s=Ee+g
totic approximation [14], the amplitude A and phase Q can be found analytically:
(4)
where a and n are constants and we select n=0.5. 1 A similar relation between damping and deformation has also been used by Tamayo and Garcia [6 ] The analytical solution of Eq. (1) cannot be found. The Runge–Kutta method has been widely used to obtain the numerical results. However, because the interaction force is only nonzero in very short duration in one vibration cycle, i.e. d>d , in tapping mode. In the first order approxi0 mation, the solution may be written as z=A +A cos(vt+Q)=A +A cos(h). 0 0 This first order approximation is similar to the harmonic approximation forwarded by Whangbo et al. [2]. Substituting this solution in Eq. (1) and using the Krylov–Bogolubov–Mitropolsky asymp-
(6)
P
h0 a sin2(h) dh s 0 1 h0 F [A cos(h)] cos(h) dh. v2 =1+ e pk A 0 c h is determined from the contact position 0 z +d 0. cos h = 0 0 A
2 a =a+ e p
P
(7)
(8)
One can consider a as the effective damping and e v as the effective resonance frequency. More than e one solutions can be obtained from Eqs. (5) and (6). The solution is stable if it satisfies a
A
B
dv2 da e >[(v−2)+v2 ] 1−v− e . e e dA dA
Comparison between this approximation solution with the numerical results will be discussed in Section 4.
3. Frequency response under different damping In this section, the influence of damping on frequency response is investigated. The experimental results are measured on a commercial scanning probe microscopy (NanoScopeIII). The etched silicon probe consists of an integrated crystal silicon cantilever and tip. The dimensions of the rectangular cantilever are 125×40×1.6 mm3. Measurements are performed on a polyethylene film with thickness 0.1 mm. All measurements are performed in ambient environment at room temperature. The frequency responses are obtained by
L. Wang / Surface Science 429 (1999) 178–185
tuning the tip, i.e. slowly scanning the driving frequency from high to low or low to high. Fig. 3a and b shows a typical frequency response measured on polyethylene. The quality factor Q , driving amplitude u and free resonance 0 0 frequency v are obtained from the free vibration 0 frequency response. The cantilever stiffness k and c the tip radius R are selected as the typical values provided by the manufacturer (k =40 N/m and c R=30 nm). The material properties K, w and a 1
Fig. 3. A typical frequency response in tapping mode AFM ((a) amplitude, (b) phase). The open circles show the experimental result measured on polyethylene. The lines show the theoretical results. Driving amplitude u =0.62 nm, free vibration damping 0 a =0.002, free resonance frequency 148.24 kHz and cantilever 0 spring constant 40 N/m. The arrows show the frequency scanning directions and jumps. The dotted lines represent unstable solutions.
181
are obtained by minimizing the difference between experimental and theoretical results using the least square algorithm. We obtained K=1.35 GPa, w= 0.37 N/m and a =0.014. If one assumes a Poisson 1 ratio of 0.35, the determined Young’s modulus of the specimen (E ) is 0.89 GPa. This value is less s than the Young’s modulus (1.5 GPa) of this sample in the handbook. The determined attractive force f is 69.7 nN. It agrees with direct force measure0 ments [10,15] that show the maximum attractive force is very large in ambient measurements (>50 nN ). The typical frequency response shown by the solid line includes three branches, i.e. free vibration (D∞A and CB∞), attractive force dominated branches (DA∞ and CC∞) and repulsive force dominated branch (A∞B). These branches are linked by unstable solution shown by the dash lines. The vibration responses (amplitude and phase) jump among these branches as driving frequency sweeps. As the driving frequency sweeps from low frequency to high (D∞ to B∞), the vibration amplitude increases as the driving frequency approaches the free resonance frequency (R). The tip reaches the sample surface at point A and the amplitude jumps from A to A∞. The amplitude follows A∞B as the driving frequency increases until it drops suddenly to free vibration at point B. As the driving frequency sweeps from high to low (B∞ to D∞), the amplitude and phase slowly increase along CC◊ when the tip reaches the sample surface at point C. At point C∞, the amplitude jumps from C∞ to C◊. Similar to branch C◊B at the higher frequency side, there is a branch at the lower frequency side A∞D and the amplitude jumps to free vibration at D instead of A∞. As described by Eq. (2), the interaction force is attractive (adhesion force) when the tip begins to touch the surface. The tip reaches the sample surface when the driving frequency approaches the free resonance frequency on both sides (point A and C ). Because attractive force shifts the resonance to lower frequency according to Eq. (8), the vibration amplitude becomes larger than that of the free vibration at lower frequency side (DA∞) and smaller at higher frequency side (CC∞) in the attractive force dominated regimes. The interaction becomes repulsive when deformation increases (A∞B). This repulsive
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force shifts the resonance to higher frequency and leads to larger amplitude at higher frequency side (C◊B). Fig. 4a and b shows the results under different interactions. The experimental result shows the frequency response at high damping condition. The two curves representing without elastic force (K=0) or adhesion force ( f =0) clearly show the 0 effects of these forces: (1) the attractive force leads to decreasing the resonance frequency and bending
Fig. 4. Frequency response under three interaction conditions ((a) amplitude, (b) phase). The three interaction conditions are without adhesion force (w=0), without elastic force (K=0) and without sample damping (a =0). The open circles represent the s experimental result. The parameters for the vibration system are driving amplitude u =0.152 nm, free vibration damping 0 a =0.0006, free resonance frequency 148.51 kHz and cantilever 0 spring constant 40 N/m.
the resonance curve to lower frequency and (2) the repulsive force leads to increasing the resonance frequency and bending the resonance curve to higher frequency. Without damping, the transitions among the attractive force dominated vibration, repulsive force dominated vibration and free vibration have very large hysteresis; the amplitude jumps from one vibration state to another vibration state and the phase changes slowly with driving frequency and jumps between negative and positive values. The damping effect decreases the unstable ranges and smooths the transition between the two states dominated by adhesion or elastic force. According to Eq. (7), the amplitude increase leads to effective damp a increasing. But e the effective damping increase will lead to amplitude decrease as described by Eq. (5). Therefore the damping can smooth the amplitude variation and significantly decrease the unstable range. The sample also becomes effectively ‘stiff ’ due to the damping and the deformation becomes small as shown by the solid line which corresponds to high damping. Therefore the interaction force is limited within the attractive force dominated range at high damping, and damping increase leads to phase decrease. Fig. 5 shows the frequency of the jump positions versus the damping constant. The frequency has been normalized by the free resonance
Fig. 5. The frequency of the jump positions (A, B, C∞, D) shown in Fig. 3 versus material damping. The frequency has been normalized by the free resonance frequency. The jumps at high frequency side (B, C∞) disappear at high damping.
L. Wang / Surface Science 429 (1999) 178–185
frequency. When damping is small, there are four bifurcation positions (A, B, C∞, D) as shown in Fig. 3a. The frequencies of A and D are less than the free resonance frequency and the frequency of B is much larger than the free resonance frequency. As the damping increases, the frequency of B decreases quickly and the frequency of D increases. A and C∞ are not much influenced by damping. C∞ and B disappear as the frequency of B reaches free resonance frequency (B=1). Fig. 5 clearly shows that the responses are more significantly influenced by the damping in the repulsive dominated regime (B) than in the attractive force dominated regimes (C∞ and D). This phenomenon has been experimentally observed by Cleveland et al. [16 ] and can be explained by Eqs. (2) and (7). In order to reach repulsive force regime, the deformation d has to be larger as shown in Eq. (2). This large deformation leads to larger contact phase h and effective 0 damping a . e Fig. 6 shows the frequency responses (phase) calculated by Hertzian, DMT, JKR, and Maugis models. The corresponding interaction forces described by these models are shown in Fig. 2. At slight tapping regimes (AD and CC∞), the phase is determined by the long range force. At regime (A∞B), the phase calculated by the DMT model is less than that calculated by the JKR model because the DMT model predicts larger attractive force. The phase calculated by the Maugis model is
Fig. 6. The influence of different interaction models on frequency response.
183
between the values calculated by the DMT and JKR models.
4. Setpoint amplitude ratio response under difference damping Another important adjustable parameter in tapping mode AFM is the setpoint amplitude ratio that is defined as the ratio of tapping vibration amplitude and free vibration amplitude. In tapping mode AFM, the vibration amplitude is maintained constant by feedback. Then the movement of the Z scanner represents the height image and the phase difference between driving signal and output signal shows the phase image. Fig. 7 shows the relation between phase and setpoint amplitude ratio under different damping. The solid lines are calculated by sixth order Runge–Kutta method and the dot line is calculated by the approximated analytical solution as described above. The analytical solution has good agreement with the numerical method when setpoint amplitude ratio is larger than 0.4. In smaller setpoint amplitude ratio, the contact phase h becomes large because the vibra0
Fig. 7. Phase versus setpoint amplitude ratio under different interaction conditions. Driving frequency is equal to the free resonance frequency. The relation of phase and setpoint amplitude ratio in the absence of damping (a =0) and adhesion force s (w=0) is represented by F . F shows the relation in the + − absence of damping and elastic force (K=0). The other curves show the influence of material damping. The unit for force f 0 is nN and for modulus K is GPa.
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L. Wang / Surface Science 429 (1999) 178–185
tion amplitude decreases and the vibration system becomes a strong nonlinear vibration system. Therefore the solution cannot be expressed as a simple harmonic function. As shown in Fig. 7, without damping the phase setpoint amplitude ratio relation simply becomes two curves F and F which are independent of + − the magnitude of the interactions. F corresponds + to the repulsive force control and F corresponds − to the attractive force control. As the setpoint decreases, because the initial interaction is attractive, the phase first follows the F branch. Then − at point A the phase jumps to F branch as the + setpoint amplitude ratio decreases. The phase increases as setpoint amplitude ratio decreases after the phase jumps to the repulsive force control branch F . The phase reaches maximum (C ) and + then decreases as setpoint amplitude ratio decreases and may jump again to attractive control branch (BB∞). At the end, the tapping mode vibration changes to force modulation vibration and the phase approaches 90°. According to Eq. (5), if the sample damping is zero (a =a ), the effective e 0 resonance v can be determined by the amplitude e which is adjusted by the setpoint amplitude ratio. Then the phase Q can also be obtained by Eq. (6). Therefore the magnitude of attractive and repulsive force only affects the transition position (A) without sample damping. In the presence of damping, the amplitude variation affects not only the effective resonance frequency v but also the effece tive damping a . Therefore the phase cannot be e absolutely determined by the vibration amplitude and it becomes dependent on the magnitude of repulsive or attractive forces, i.e. adhesion and elasticity. As shown in Fig. 7, the phase is significantly dependent on the damping. As damping increases, the phase decreases and shifts from repulsive force control F to attractive force con+ trol F . As discussed in the frequency response − section, the damping increase leads to an effective ‘stiff ’ surface and small deformation. As the deformation decreases, the average interaction force in one vibration cycle is shifted from positive to negative. It leads to the phase decreasing and negative regimes increasing. At high damping, the phase becomes negative in all setpoint amplitude
ratio. These phenomena agree with the experimental observations shown by Magonov et al. [4].
5. The influence of damping on mechanical properties mapping It has been shown that the phase response is more sensitive to the sample mechanical properties than amplitude or height response. Winkler et al. [5] showed the phase and modulus relation under different damping without the attractive force. As described above, the repulsive, attractive force and damping influence each other and make the responses in tapping mode AFM complicated. Figs. 8 and 9 show the influences of damping on elasticity and adhesion mapping. As shown by the solid line in these two figures, without damping, the phase only jumps between about 45° and –45°. These two states correspond to repulsive or attractive force control and the values are determined by the setpoint amplitude ratio as described in Section 4. As the modulus K increases, the phase stays on the attractive force control branch until the modulus K reaches a critical value (A). The repulsive and attractive force control regimes have
Fig. 8. The phase versus Young’s modulus for three different damping values (a =0, 0.005, 0.1). The adhesion force is con1 stant ( f =40 nN ). The setpoint amplitude ratio is 0.7 and the 0 driving frequency is equal to the free resonance frequency. The solid line shows that the phase jumps between two values without damping. In the presence of damping, the phase increases as the Young’s modulus increases.
L. Wang / Surface Science 429 (1999) 178–185
185
in tapping mode AFM have been analyzed by an analytical solution. All features can be divided into two regimes which are dominated by attractive or repulsive forces. The transition between these two regimes is significantly dependent on the damping. The damping increase leads to the attractive force control regime increasing and the magnitude of hysteresis decreasing. The sensitivity of phase imaging to surface properties is significantly dependent on the damping. To map the material surface stiffness and adhesion using phase imaging, the vibration conditions (driving frequency, amplitude and setpoint amplitude ratio) should be optimized to obtain high sensitivity. Fig. 9. The phase versus adhesion force modulus for three different damping values (a =0, 0.005, 0.1). The Young’s mod1 ulus is 5 GPa. Other conditions are the same as those in Fig. 7. The solid line shows that the phase jumps between two values without damping. In the presence of damping, the phase decreases as adhesion force decreases.
an overlapping range AB. Within this range, the phase can be on either of these branches. In the presence of damping, the transition between these states becomes smoother and the overlap regime decreases. As shown in Fig. 8, the phase increases as the modulus increases. However, unstable regimes exist for small damping. The same phenomenon happens as the adhesion force changes. The phase decreases as attractive force increases. As shown in these figures, the best sensitivity is around the transition range where the phase changes around 0°. Therefore, for given sample, we can adjust the vibration parameter (setpoint amplitude ratio, drive amplitude and frequency) to make the work point around the transition ranges. The phase is insensitive to modulus or attractive force in the attractive force control regimes.
6. Conclusion By using a model which includes the attractive, repulsive force and damping, the complex features
References [1] Q. Zhong, D. Innis, K. Kjoller, V.B. Elings, Surf. Sci. 290 (1993) L688. [2] M.H. Whangbo, G. Bar, R. Brandsch, Surf. Sci. 411 (1998) 794. [3] R. Ho¨per, T. Gesang, W. Possart, O.D. Hennemann, S. Boseck, Ultramicroscopy 60 (1995) 17. [4] S.N. Magonov, V. Elings, M.H. Whangbo, Surf. Sci. 375 (1997) L385. [5] R.G. Winkler, J.P. Sppatz, S. Sheiko, M. Moller, P. Reineker, O. Marti, Phys. Rev. B 54 (1996) 8908. [6 ] J. Tamayo, R. Garcia, Langmuir 12 (1996) 4430. [7] C.J. Putman, K.V. der Werf, B.G. deGrooth, N.V. Hulst, J. Greve, Appl. Phys. Lett. 64 (1994) 2454. [8] A. Kuhle, A.H. Sorensen, J. Bohr, J. Appl. Phys. 81 (1997) 6562. [9] D. Sarid, T.G. Ruskell, R.K. Workman, D. Chen, J. Vac. Sci. Technol. B 14 (1996) 864. [10] N.A. Burnham, R.J. Colton, H.M. Pollock, Nanotechnology 4 (1993) 64. [11] K.J. Johnson, Contact Mechanics, Cambridge University Press, New York, 1985. [12] D. Maugis, J. Colloid Interface Sci. 150 (1992) 243. [13] N.A. Burham, O.P. Behrend, G. Gremand, P.J. Gallo, D. Gourdon, E. Dupas, A.J. Kulik, H.M. Pollock, G.D. Briggs, Nanotechnology 8 (1997) 67. [14] N. Minorsky, Nonlinear Oscillations, Van Nostrand, Princeton, NJ, 1962. [15] A.L. Weisenhorm, P.K. Hansma, T.R. Albrecht, C.F. Quate, Appl. Phys. Lett. 54 (26) (1989) 2651. [16 ] J.P. Cleveland, B. Anczykowski, A.E. Schmid, V.B. Elings, Appl. Phys. Lett. 72 (1998) 2613.
www.elsevier.nl/locate/susc
The role of damping in phase imaging in tapping mode atomic force microscopy Lugen Wang * Edison Joining Technology Center, Welding Engineering, The Ohio State University, Columbus, OH 43210, USA Received 25 August 1998; accepted for publication 23 February 1999
Abstract The influences of damping on phase imaging in tapping mode atomic force microscopy have been investigated numerically and experimentally. It shows that the adhesion force (attractive) and elastic force (repulsive) divide the phase response into two distinct regimes. Jumps and hysteresis may happen between these two regimes. The magnitude of jumps and hysteresis decreases in the presence of material damping. The sensitivity of phase to surface stiffness and adhesion is significantly dependent on the material damping. © 1999 Published by Elsevier Science B.V. All rights reserved. Keywords: Atomic force microscopy (AFM ); Energy dissipation; Phase imaging
1. Introduction The phase image makes the tapping mode atomic force microscopy (AFM ) go beyond topography to image material surface properties such as elasticity and viscoelasticity on a nanometer scale [1–3]. It has been demonstrated that phase imaging can provide more information and contrast in many situations than amplitude imaging or height imaging. The experimental results given by Magonov et al. [4] showed the phase response can reflect material elasticity in small setpoint amplitude ratio. The influence of elasticity on phase response has been numerically investigated by Winkler et al. [5]. In their model, the complicated tip–sample interactions are simplified as a linear spring without an attractive force. They showed that the phase increases as the Young’s modulus * Fax: +1-614-292-6842. E-mail address: [email protected] (L. Wang)
increases. However, the experimental results show much more complicated phenomena. The complicated features in phase response can be explained when one considers the attractive, repulsive force and material damping together. The importance of damping in phase response has already been pointed out by several authors [6–9]. Kuhle et al. [8] measured the damped frequency response. In this paper, by using a model including attractive force (adhesion), repulsive force (elasticity) and damping, the influence of the driving frequency, setpoint amplitude ratio and material properties on phase response will be discussed.
2. Model of the system A simple model of tapping mode AFM is a single degree-of-freedom nonlinear oscillator in which the cantilever is modeled by a spring with spring constant k , a point mass m and quality c
0039-6028/99/$ – see front matter © 1999 Published by Elsevier Science B.V. All rights reserved. PII: S0 0 39 - 6 0 28 ( 99 ) 0 03 6 8 -4
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L. Wang / Surface Science 429 (1999) 178–185
factor Q [9]. A bimorph which oscillates with 0 frequency v and amplitude u at one end of the d 0 cantilever forces the tip to vibrate, thus making intermittent contact with the surface. The tip offset z as shown in Fig. 1 can be controlled by setpoint 0 amplitude ratio (vibration amplitude normalized by free vibration amplitude). The vibration system is described by the differential equation d2z dt2
+2a
dz dt
+[z+F(z)/k ]=u cos(vt) c 0
(1)
where v =앀k /m, t=v t is the nondimensional c c c time and v=v /v is the normalized driving fred c quency. a is the damping constant which includes free vibration damping a and material damping 0 a . F(z) represents the tip–sample interaction. s The static tip–sample interactions have been described by macroscopic continuum theories such as Hertzian, Johnson–Kendall–Roberts (JKR) and Derjaguin–Muller–Toporov (DMT ) contact models [10–13]. Recently, Maugis [12] proposed a more general model based on the Dugdale theory. In atomic force microscopy, the contact radius reaches nanometer; the tip–sample interactions become sensitive to the material properties, experimental conditions and tip shape. The experimental results given by Burnham et al. [10] show
Fig. 1. A model of tapping mode AFM. The cantilever and tip are modeled as a vibration system with spring constant k and c mass m. z is the tip–sample separation. u cos(vt) is the driv0 0 ing signal.
the complicated relation between the force and tip–sample separation. However, in a typical force curve, the force is attractive at small tip–sample separation and approaches Hertzian contact force at large deformation. Fig. 2 shows the relations between the force and tip–sample separation described by Hertzian, JKR, DMT, and Maugis theories. The formulations for these theories can be found in Ref. [13]. The influence of these models on phase response will be discussed in Section 3. In most of our calculation, we will use the simple DMT model to represent the tip–sample interaction in the contact region. Burnham et al. [10] also used an equation similar to the DMT model to fit their typical experimental force curve. The tip–sample interaction in the DMT model may be written as F=K앀Rd3/2−2pwR where w is the surface energy and 3
A
B
1−n2 1−n2 s + c . K 4 E E s c n , n are the Poisson ratio of the sample and s c cantilever respectively and E , E are the Young’s s c modulus of the sample and cantilever respectively. R represents the tip radius and d=z−z is the tip– 0 sample separation. If one considers other attractive forces at noncontact region (d<0), the relation 1
=
Fig. 2. The relation between forces and tip–sample separation for difference models. The long range forces are indicated by the dashed line.
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L. Wang / Surface Science 429 (1999) 178–185
between force and tip–sample separation may be written as F=f (d )
d<0
F=K앀Rd3/2−f
d≥0
0
de dt
.
(2)
(3)
As shown in Eq. (2), the Hertzian contact force is dependent on the power of deformation (d3/2), therefore we assume the viscous force also has a similar relation with deformation and the damping term may be written as a =a (d/R)n s 1
d≥0
u 0 (1+v)앀a2 +(v−v )2 e e v −v tan(Q)= e a e where
(5)
A=
where f is the adhesion force 2pwR, K앀Rd3/2 0 is the repulsive Hertzian contact force. f(d ) represents the long range attractive forces. Burnham et al. [13] wrote this term as f(d)=−f /(1−d/d )2, where d is a constant. This 0 0 0 selection makes the interaction forces F in contact (d>0) and noncontact (d<0) regions join at d=0. When the tip impacts the material surface, energy dissipation occurs due to viscoelasticity and adhesion, especially for polymers and biologic materials. This energy dissipation may be represented by a damping constant a [2,6 ]. In the s Voiget model, the relation of stress s and strain e in the presence of viscoelasticity is given by s=Ee+g
totic approximation [14], the amplitude A and phase Q can be found analytically:
(4)
where a and n are constants and we select n=0.5. 1 A similar relation between damping and deformation has also been used by Tamayo and Garcia [6 ] The analytical solution of Eq. (1) cannot be found. The Runge–Kutta method has been widely used to obtain the numerical results. However, because the interaction force is only nonzero in very short duration in one vibration cycle, i.e. d>d , in tapping mode. In the first order approxi0 mation, the solution may be written as z=A +A cos(vt+Q)=A +A cos(h). 0 0 This first order approximation is similar to the harmonic approximation forwarded by Whangbo et al. [2]. Substituting this solution in Eq. (1) and using the Krylov–Bogolubov–Mitropolsky asymp-
(6)
P
h0 a sin2(h) dh s 0 1 h0 F [A cos(h)] cos(h) dh. v2 =1+ e pk A 0 c h is determined from the contact position 0 z +d 0. cos h = 0 0 A
2 a =a+ e p
P
(7)
(8)
One can consider a as the effective damping and e v as the effective resonance frequency. More than e one solutions can be obtained from Eqs. (5) and (6). The solution is stable if it satisfies a
A
B
dv2 da e >[(v−2)+v2 ] 1−v− e . e e dA dA
Comparison between this approximation solution with the numerical results will be discussed in Section 4.
3. Frequency response under different damping In this section, the influence of damping on frequency response is investigated. The experimental results are measured on a commercial scanning probe microscopy (NanoScopeIII). The etched silicon probe consists of an integrated crystal silicon cantilever and tip. The dimensions of the rectangular cantilever are 125×40×1.6 mm3. Measurements are performed on a polyethylene film with thickness 0.1 mm. All measurements are performed in ambient environment at room temperature. The frequency responses are obtained by
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tuning the tip, i.e. slowly scanning the driving frequency from high to low or low to high. Fig. 3a and b shows a typical frequency response measured on polyethylene. The quality factor Q , driving amplitude u and free resonance 0 0 frequency v are obtained from the free vibration 0 frequency response. The cantilever stiffness k and c the tip radius R are selected as the typical values provided by the manufacturer (k =40 N/m and c R=30 nm). The material properties K, w and a 1
Fig. 3. A typical frequency response in tapping mode AFM ((a) amplitude, (b) phase). The open circles show the experimental result measured on polyethylene. The lines show the theoretical results. Driving amplitude u =0.62 nm, free vibration damping 0 a =0.002, free resonance frequency 148.24 kHz and cantilever 0 spring constant 40 N/m. The arrows show the frequency scanning directions and jumps. The dotted lines represent unstable solutions.
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are obtained by minimizing the difference between experimental and theoretical results using the least square algorithm. We obtained K=1.35 GPa, w= 0.37 N/m and a =0.014. If one assumes a Poisson 1 ratio of 0.35, the determined Young’s modulus of the specimen (E ) is 0.89 GPa. This value is less s than the Young’s modulus (1.5 GPa) of this sample in the handbook. The determined attractive force f is 69.7 nN. It agrees with direct force measure0 ments [10,15] that show the maximum attractive force is very large in ambient measurements (>50 nN ). The typical frequency response shown by the solid line includes three branches, i.e. free vibration (D∞A and CB∞), attractive force dominated branches (DA∞ and CC∞) and repulsive force dominated branch (A∞B). These branches are linked by unstable solution shown by the dash lines. The vibration responses (amplitude and phase) jump among these branches as driving frequency sweeps. As the driving frequency sweeps from low frequency to high (D∞ to B∞), the vibration amplitude increases as the driving frequency approaches the free resonance frequency (R). The tip reaches the sample surface at point A and the amplitude jumps from A to A∞. The amplitude follows A∞B as the driving frequency increases until it drops suddenly to free vibration at point B. As the driving frequency sweeps from high to low (B∞ to D∞), the amplitude and phase slowly increase along CC◊ when the tip reaches the sample surface at point C. At point C∞, the amplitude jumps from C∞ to C◊. Similar to branch C◊B at the higher frequency side, there is a branch at the lower frequency side A∞D and the amplitude jumps to free vibration at D instead of A∞. As described by Eq. (2), the interaction force is attractive (adhesion force) when the tip begins to touch the surface. The tip reaches the sample surface when the driving frequency approaches the free resonance frequency on both sides (point A and C ). Because attractive force shifts the resonance to lower frequency according to Eq. (8), the vibration amplitude becomes larger than that of the free vibration at lower frequency side (DA∞) and smaller at higher frequency side (CC∞) in the attractive force dominated regimes. The interaction becomes repulsive when deformation increases (A∞B). This repulsive
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force shifts the resonance to higher frequency and leads to larger amplitude at higher frequency side (C◊B). Fig. 4a and b shows the results under different interactions. The experimental result shows the frequency response at high damping condition. The two curves representing without elastic force (K=0) or adhesion force ( f =0) clearly show the 0 effects of these forces: (1) the attractive force leads to decreasing the resonance frequency and bending
Fig. 4. Frequency response under three interaction conditions ((a) amplitude, (b) phase). The three interaction conditions are without adhesion force (w=0), without elastic force (K=0) and without sample damping (a =0). The open circles represent the s experimental result. The parameters for the vibration system are driving amplitude u =0.152 nm, free vibration damping 0 a =0.0006, free resonance frequency 148.51 kHz and cantilever 0 spring constant 40 N/m.
the resonance curve to lower frequency and (2) the repulsive force leads to increasing the resonance frequency and bending the resonance curve to higher frequency. Without damping, the transitions among the attractive force dominated vibration, repulsive force dominated vibration and free vibration have very large hysteresis; the amplitude jumps from one vibration state to another vibration state and the phase changes slowly with driving frequency and jumps between negative and positive values. The damping effect decreases the unstable ranges and smooths the transition between the two states dominated by adhesion or elastic force. According to Eq. (7), the amplitude increase leads to effective damp a increasing. But e the effective damping increase will lead to amplitude decrease as described by Eq. (5). Therefore the damping can smooth the amplitude variation and significantly decrease the unstable range. The sample also becomes effectively ‘stiff ’ due to the damping and the deformation becomes small as shown by the solid line which corresponds to high damping. Therefore the interaction force is limited within the attractive force dominated range at high damping, and damping increase leads to phase decrease. Fig. 5 shows the frequency of the jump positions versus the damping constant. The frequency has been normalized by the free resonance
Fig. 5. The frequency of the jump positions (A, B, C∞, D) shown in Fig. 3 versus material damping. The frequency has been normalized by the free resonance frequency. The jumps at high frequency side (B, C∞) disappear at high damping.
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frequency. When damping is small, there are four bifurcation positions (A, B, C∞, D) as shown in Fig. 3a. The frequencies of A and D are less than the free resonance frequency and the frequency of B is much larger than the free resonance frequency. As the damping increases, the frequency of B decreases quickly and the frequency of D increases. A and C∞ are not much influenced by damping. C∞ and B disappear as the frequency of B reaches free resonance frequency (B=1). Fig. 5 clearly shows that the responses are more significantly influenced by the damping in the repulsive dominated regime (B) than in the attractive force dominated regimes (C∞ and D). This phenomenon has been experimentally observed by Cleveland et al. [16 ] and can be explained by Eqs. (2) and (7). In order to reach repulsive force regime, the deformation d has to be larger as shown in Eq. (2). This large deformation leads to larger contact phase h and effective 0 damping a . e Fig. 6 shows the frequency responses (phase) calculated by Hertzian, DMT, JKR, and Maugis models. The corresponding interaction forces described by these models are shown in Fig. 2. At slight tapping regimes (AD and CC∞), the phase is determined by the long range force. At regime (A∞B), the phase calculated by the DMT model is less than that calculated by the JKR model because the DMT model predicts larger attractive force. The phase calculated by the Maugis model is
Fig. 6. The influence of different interaction models on frequency response.
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between the values calculated by the DMT and JKR models.
4. Setpoint amplitude ratio response under difference damping Another important adjustable parameter in tapping mode AFM is the setpoint amplitude ratio that is defined as the ratio of tapping vibration amplitude and free vibration amplitude. In tapping mode AFM, the vibration amplitude is maintained constant by feedback. Then the movement of the Z scanner represents the height image and the phase difference between driving signal and output signal shows the phase image. Fig. 7 shows the relation between phase and setpoint amplitude ratio under different damping. The solid lines are calculated by sixth order Runge–Kutta method and the dot line is calculated by the approximated analytical solution as described above. The analytical solution has good agreement with the numerical method when setpoint amplitude ratio is larger than 0.4. In smaller setpoint amplitude ratio, the contact phase h becomes large because the vibra0
Fig. 7. Phase versus setpoint amplitude ratio under different interaction conditions. Driving frequency is equal to the free resonance frequency. The relation of phase and setpoint amplitude ratio in the absence of damping (a =0) and adhesion force s (w=0) is represented by F . F shows the relation in the + − absence of damping and elastic force (K=0). The other curves show the influence of material damping. The unit for force f 0 is nN and for modulus K is GPa.
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tion amplitude decreases and the vibration system becomes a strong nonlinear vibration system. Therefore the solution cannot be expressed as a simple harmonic function. As shown in Fig. 7, without damping the phase setpoint amplitude ratio relation simply becomes two curves F and F which are independent of + − the magnitude of the interactions. F corresponds + to the repulsive force control and F corresponds − to the attractive force control. As the setpoint decreases, because the initial interaction is attractive, the phase first follows the F branch. Then − at point A the phase jumps to F branch as the + setpoint amplitude ratio decreases. The phase increases as setpoint amplitude ratio decreases after the phase jumps to the repulsive force control branch F . The phase reaches maximum (C ) and + then decreases as setpoint amplitude ratio decreases and may jump again to attractive control branch (BB∞). At the end, the tapping mode vibration changes to force modulation vibration and the phase approaches 90°. According to Eq. (5), if the sample damping is zero (a =a ), the effective e 0 resonance v can be determined by the amplitude e which is adjusted by the setpoint amplitude ratio. Then the phase Q can also be obtained by Eq. (6). Therefore the magnitude of attractive and repulsive force only affects the transition position (A) without sample damping. In the presence of damping, the amplitude variation affects not only the effective resonance frequency v but also the effece tive damping a . Therefore the phase cannot be e absolutely determined by the vibration amplitude and it becomes dependent on the magnitude of repulsive or attractive forces, i.e. adhesion and elasticity. As shown in Fig. 7, the phase is significantly dependent on the damping. As damping increases, the phase decreases and shifts from repulsive force control F to attractive force con+ trol F . As discussed in the frequency response − section, the damping increase leads to an effective ‘stiff ’ surface and small deformation. As the deformation decreases, the average interaction force in one vibration cycle is shifted from positive to negative. It leads to the phase decreasing and negative regimes increasing. At high damping, the phase becomes negative in all setpoint amplitude
ratio. These phenomena agree with the experimental observations shown by Magonov et al. [4].
5. The influence of damping on mechanical properties mapping It has been shown that the phase response is more sensitive to the sample mechanical properties than amplitude or height response. Winkler et al. [5] showed the phase and modulus relation under different damping without the attractive force. As described above, the repulsive, attractive force and damping influence each other and make the responses in tapping mode AFM complicated. Figs. 8 and 9 show the influences of damping on elasticity and adhesion mapping. As shown by the solid line in these two figures, without damping, the phase only jumps between about 45° and –45°. These two states correspond to repulsive or attractive force control and the values are determined by the setpoint amplitude ratio as described in Section 4. As the modulus K increases, the phase stays on the attractive force control branch until the modulus K reaches a critical value (A). The repulsive and attractive force control regimes have
Fig. 8. The phase versus Young’s modulus for three different damping values (a =0, 0.005, 0.1). The adhesion force is con1 stant ( f =40 nN ). The setpoint amplitude ratio is 0.7 and the 0 driving frequency is equal to the free resonance frequency. The solid line shows that the phase jumps between two values without damping. In the presence of damping, the phase increases as the Young’s modulus increases.
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in tapping mode AFM have been analyzed by an analytical solution. All features can be divided into two regimes which are dominated by attractive or repulsive forces. The transition between these two regimes is significantly dependent on the damping. The damping increase leads to the attractive force control regime increasing and the magnitude of hysteresis decreasing. The sensitivity of phase imaging to surface properties is significantly dependent on the damping. To map the material surface stiffness and adhesion using phase imaging, the vibration conditions (driving frequency, amplitude and setpoint amplitude ratio) should be optimized to obtain high sensitivity. Fig. 9. The phase versus adhesion force modulus for three different damping values (a =0, 0.005, 0.1). The Young’s mod1 ulus is 5 GPa. Other conditions are the same as those in Fig. 7. The solid line shows that the phase jumps between two values without damping. In the presence of damping, the phase decreases as adhesion force decreases.
an overlapping range AB. Within this range, the phase can be on either of these branches. In the presence of damping, the transition between these states becomes smoother and the overlap regime decreases. As shown in Fig. 8, the phase increases as the modulus increases. However, unstable regimes exist for small damping. The same phenomenon happens as the adhesion force changes. The phase decreases as attractive force increases. As shown in these figures, the best sensitivity is around the transition range where the phase changes around 0°. Therefore, for given sample, we can adjust the vibration parameter (setpoint amplitude ratio, drive amplitude and frequency) to make the work point around the transition ranges. The phase is insensitive to modulus or attractive force in the attractive force control regimes.
6. Conclusion By using a model which includes the attractive, repulsive force and damping, the complex features
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