Veloctity Dependent Friction Laws in Contact Mode Atomic-Force ...
PREPRINT *** to be published 2004 in ULTRAMICROSCOPY *** PREPRINT
Veloctity Dependent Friction Laws in Contact Mode Atomic-Force Microscopy Robert W. Stark 1 , Georg Schitter, Andreas Stemmer Nanotechnology Group, Swiss Federal Institute of Technology Zurich, ETH Zentrum/CLA, CH-8092 Zurich, Switzerland.
Abstract Friction forces in the tip-sample contact govern the dynamics of contact mode atomic-force microscopy. In ambient conditions typical contact radii between tip and sample are in the order of a few nanometers. In order to account for the large interaction area the dynamics of contact mode AFM is investigated under the assumption of a multi-asperity contact interface between tip and sample. Thus, the kinetic friction force between tip and sample is the product of the real contact area between both solids and the interfacial shear strength. The velocity strengthening of the lateral force is modeled assuming a logarithmic relationship between shear-strength and velocity. Numerical simulations of the system dynamics with this empirical model show the existence of two different regimes in contact mode AFM: steady sliding and stick-slip where the tip undergoes periodically stiction and kinetic friction. The state of the system depends on the scan velocity as well as on the velocity dependence of the interfacial friction force between tip and sample. Already small viscous damping contributions in the tip sample contact are sufficient to suppress stick-slip oscillations. keyword: atomic force microscopy PACS: 07.79.Lh, 68.37.Ps, 07.05.Tp
1
Introduction
In contact mode atomic-force microscopy frictional forces between tip and sample determine the dynamics of the system. On the atomic-scale a periodic ‘stick-slip’ motion of the tip occurs that is related to the periodicity of the lattice spacing of the specimen. This behavior was modeled for a one-dimensional 1
Corresponding author. Present address: Ludwig-Maximilians-Universit¨at Munich, Section Crystallography, Theresienstr. 41, 80333 Munich, Germany, Tel.: +49 89 2180 4329 Fax: +49 89 2180 4334 , e-mail: [email protected]
Preprint submitted to Elsevier Science
14 November 2003
+
L
-
FFM signal
er la s
R
bea m
photo diode
scan generator
vscan = const
Fig. 1. Scheme of friction-force microscopy (FFM). The sample is scanned with a triangular wave form leading to a constant velocity during each scan-line. Due to the friction between tip and sample the AFM cantilever is twisted. The twisting angle is detected by a split photodiode. The difference between the photo-currents in both parts of the diode is the friction-force microscopy signal.
system [1] and two-dimensional systems [2–4]. These models assume an ideal single asperity contact where a monoatomic tip interacts with the periodically modulated potential of an atomically flat surface. For a point-contact like configuration under ultra-high vacuum (UHV) conditions the frictional forces between a silicon tip and different carbon compounds were estimated to be roughly constant over a velocity range from nanometer per second to a few microns per second, whereas for larger velocities an increase of the friction with a velocity proportional friction force was predicted from simulations [5]. A logarithmic velocity dependence was reported later from experimental data obtained by atomic force microscope (AFM) in UHV [6,7]. The results were explained in terms of a modified Tomlinson [8] model that includes thermal activation. Experimental conditions in UHV differ from the standard situation in contact mode imaging under ambient conditions. For example, the typical loading force of the tip onto the sample is substantially higher than under ultrahigh vacuum conditions due to meniscus forces. This leads to contact radii of typically 4 nm between tip and sample. This implies that several atomic or molecular units are involved in the tip sample interaction. The situation is best described by a multi-asperity contact. A weak velocity dependence of the frictional force up to 50 µm/s and a following logarithmic increase for higher velocities was reported for friction force microscopy on silicon surfaces under ambient conditions [9]. Under similar conditions a logarithmic velocity dependence was reported for organosilanes grafted on silica [10]. The motion of a sliding block can range from steady sliding over stick-slip oscillations to chaotic behavior [11,12] depending on the velocity as well as on the details of the velocity dependency of the frictional forces. The velocity of the tip on to the sample can range from a few microns per second in typical 2
contact mode imaging over a few hundred microns per second in high speed AFM applications [13–15] to millimeters per second in dynamic friction mode [16–19]. Therefore one needs to know how the velocity dependence of the frictional forces determines the dynamics of contact mode AFM. For AFM imaging it is critical, whether the tip is steadily sliding on the sample surface or undergoing stick-slip cycles. To clarify the importance of a velocity dependency of the frictional forces for contact mode imaging three simple friction ’laws’ and their influence on the motion of the tip on the sample surface will be investigated.
2
Modeling
In the following the dynamics of ambient contact mode AFM will be investigated under the assumption of a multi-contact interface between tip and sample. Following the classical work of Bowden and Tabor [20] the kinetic friction force between tip and sample can be modelled as Fts = σs Σr which is the product of the real contact area between both solids Σr and the interfacial shear strength σs . State- and rate-dependency can be introduced, leading to a generalized Tabor equation Fts = σs (∆v)Σr (φ) [21,22]. Here, ∆v is the interfacial sliding velocity and the parameter φ can be understood as the age of the multi-contact interface. Several authors report on the modelling of friction forces in AFM experiments where a time independent contact area Σr is determined from contact-mechanical models [23–25]. Thus, for the analysis of contact mode AFM on hard specimen the age dependent contact strengthening Σr (φ) is approximated by the constant real contact area Σr of the Tabor equation. This assumes that the typical lattice dynamics of the tip and the sample are much faster as compared to the dynamics of the AFM cantilever and that boundary lubrication effects are negligible. The velocity strengthening of frictional forces is described by the empirical relation ∆v . (1) σs (∆v) = σs0 1 + α ln v0 The logarithmic relationship can be interpreted as the result of a thermally activated shear-induced creep process. There, α can be identified with α = kB T / (σs0 Vact ) [26]. The parameter σs0 is the shear strength in the low velocity limit and v0 a measure for the strength of the velocity dependence. The stress activation volume Vact is related to the elementary process and is e.g. about 12 nm3 for poly(methyl methacrylate) as well as for polystyrol [22]. Bouhacina et al. estimate this volume even smaller to about 0.2 nm3 [10]. In a typical AFM setup (Fig. 1) the base of the cantilever is fixed and the sample is scanned with respect to the laboratory frame of reference. The fric3
(a)
k, m, ω0
Vscan(t)
X(t)
(b)
x(t), x(t)
k c
m
m(Dv(t))
Vscan(t) Fig. 2. (a) An AFM tip in contact with a surface that is scanned with the velocity vscan . Due to friction the cantilever is twisted, which corresponds to a displacement x (t) of the tip apex. (b) Mechanical model of the system. The sample is replaced by a belt moving with the velocity vscan , the cantilever is replaced by an equivalent single-degree-of-freedom system with the mass m, spring constant k, and damping c.
tional forces lead to a torsion of the cantilever which is usually measured employing light lever detection. For the numerical modelling we assume that the friction force µ depends on the relative velocity between tip and sample ∆v (t) = vtip (t) − vscan (t) as illustrated in Fig. 2 (a). Here vtip and vscan are the velocities of the tip and sample in the laboratory frame of reference. The dynamics of this system depends on the mass m, torsional stiffness k and the damping c of the cantilever. The corresponding simplified mechanical model is shown in Fig. 2 (b). This single-degree-of-freedom approximation limits the use of the model to applications where other eigenmodes of the cantilever do not contribute to the response. For comprehensive reviews on the friction modeling of dynamic systems see [11,27]. The equation of motion is divided into two states (i) the sliding state, where the tip is sliding over the sample and (ii) the sticking state, where the relative velocity between tip and sample is zero. In the sliding state the equation of motion is given by m
d d2 x + c x + kx = Fts (∆v) , 2 dt dt 4
(2)
where x(t) is the displacement of the tip, and Fts (∆v) is the frictional force between tip and sample during sliding. The cantilever parameters are spring constant k, the effective mass of the oscillator m, resonant frequency ω0 = q k/m, and the damping coefficient c. The quality factor Q = mω0 /c is an alternate measure to describe the damping. The parameters of the harmonic oscillator are obtained from the first torsional mode of the cantilever. A dimensionless formulation of Eq. (2) is obtained with the transformations τ = ω0 t,
ξ=
k Fs,max
x,
(3)
where the maximum static friction is Fs,max . Thus, the normalized velocity is λ = v k/(Fs,max ω0 ). The normalized interfacial velocity is given by ∆λ(τ ) = λtip (τ ) − λscan (τ ). This leads to the equation of motion for the sliding state 1 ξ¨ + ξ˙ + ξ = µts (∆λ) . Q
(4)
The dot denotes the derivative d/dτ, scaling the time to the resonant frequency of the oscillator. The lateral tip-sample force is given by µts (∆λ) =
Fts (∆λ) . Fs,max
(5)
This means that the maximum static friction is normalized to µs, max = 1. The sliding phase ends if ∆λ(τ ) = 0 and the sum of inertia, damping, and spring force is smaller than the maximum stiction force, i.e. ξ¨ + Q1 ξ˙ + ξ ≤ 1. In the sticking state, the static friction force Fs is confined to the interval −Fs,max ≤ Fs ≤ Fs,max , which is in normalized coordinates −1 ≤ µs ≤ 1. All forces are balanced and the velocity between tip and sample equals zero which leads to a simple equation of motion λtip (τ ) = λscan (τ ).
(6)
The sticking phase ends if the forces overcome the static friction ξ¨ + Q1 ξ˙ + ξ > 1 and a new sliding phase begins.
In the following, the three different friction laws illustrated in Fig. 3 will be investigated: (a) Coulomb friction and friction laws with (b) linear and (c) logarithmic velocity dependence. For all three friction laws the dimensionless static friction coefficient is −1 ≤ µs ≤ 1. In the sliding regime (∆λ 6= 0) the velocity independent coulombic friction is given by µcoul (∆λ) = sign (∆λ) · µc , (7) 5
(a)
m
1
m = 0..5 c
0 -1
(b)
mv = 2
m
1
mv = 0..5
0 -1
(c)
l0-1= 4
m
1
l0-1= 1
0 -1 -0.5
0
difference velocity D l
0.5
Fig. 3. Illustration of the friction laws. The static friction at ∆λ = 0 is normalized to the interval −1 < µs < 1. (a) Coulombic friction with the friction coefficient 0 < µc ≤ 1. (b) The slope of this linear friction law is given by the viscous contribution µv . (c) The velocity dependence of the logarithmic force law is given by the parameter λ0 .
with 0 < µc ≤ 1. For the linear velocity dependence an additional viscous contribution is introduced µlin (∆λ) = sign (∆λ) · (µc + µv |∆λ|) . (8) In the limit µv → 0 again Eq. (7) is obtained. For a logarithmic velocity dependence the following force law is assumed "
µlog (∆λ) = sign (∆λ) · σs0
|∆λ| 1 + α ln + λc λ0
!#
.
(9)
The scaling parameter λ0 is a material parameter. The cutoff velocity λc is introduced in order to avoid negative friction forces at |∆λ| /λ0 < 1 and numerical divergence for very small velocities |∆λ| → 0. For the numerical simulations this parameter was set to λc = 1. The limit λ0 → ∞, leads to the 6
(a)
λtip
ξ
1
(c)
µc = 0.5 µv = 0.5
-1
λ0 = 1
0 0.2 0 -0.2 -0.4 µc = 0.95 µv = 0
1
ξ λtip
(b)
µc = 0.5 µv = 0
-1
µc = 0.5 µv = 2
λ0 = 4
0 0.2 0 -0.2 -0.4 0
100
200
300
time τ
400
0
100
200
300
time τ
400
0
100
200
300
time τ
400
Fig. 4. Normalized lateral tip displacement ξ and tip velocity λtip for the friction laws in Fig. 3 for a sample moving with the constant velocity λscan = 0.01. (a) Coulomb friction. The system is in a stick-slip state for µc = 0.5 (above) and µc = 0.95 (below). (b) With viscous damping in the tip sample contact the stick-slip does not vanish for µv = 0.5 (above). A damping of µv = 2 completely supresses the oscillation (below). (c) For the logarithmic force law stick-slip is present for λ0 = 1 (above) and vanishes for λ0 = 0.25 (below).
Coulomb friction law of Eq. (7). For small velocities |∆λ| /λ0 < λc Eq. (9) can be expanded to "
µ
log
(∆λ) = sign (∆λ) · σs0 1 + α ln (λc ) + α
∞ X (−1)n+1
n
n=1
|∆λ| λ0 λc
!n #
!
≈ sign (∆λ) · σs0 1 + α ln (λc ) + α
|∆λ| . λ0 λ c
(10)
Comparing Eqs. (8) and (10) we find for the linearized small velocity regime µv = σs0 α/ (λ0 λc ) and µc = σs0 (1 + α ln (λc )) .
3
Results and Discussion
In order to investigate basic features of contact mode AFM under ambient conditions a constant sample velocity was assumed for the numerical simulations. The friction was modeled employing the three friction laws in Eqs. (7, 8, 9) illustrated in Fig. 3. As initial condition a sticking state was selected. At t = 0 the displacement was ξ0 = 0, the relative tip sample velocity was ∆λ = 0 and the scan velocity was set to λscan = 0.01. The quality factor of the cantilevers torsional resonance was set to Q = 100. 7
Depending on the choice of the friction law different regimes could be distinguished. For a Coulomb friction law, the system remained in a stick-slip state for a dynamic friction force of µc = 0.5 (Fig. 4 (a), above) as well as for a value of µc = 0.95 (below). Initially the tip stuck to the sample and moved with the sample with a tip velocity of λtip = 0.01, i.e. ∆λ = 0. Thus, the spring was loaded until the spring force overcame the static friction. Then, the tip snapped back and a new stick-slip cycle began. The weak viscous damping Q−1 = 0.01 of the cantilever’s torsional resonance in the surrounding air was not sufficient to suppress the stick-slip oscillations. Adding viscous drag into the lateral tip sample force the stick-slip oscillations did not vanish for a viscous drag of µv = 0.5 (Fig. 4 (b), above). Increasing the the viscous drag to µv = 2 led to a steady sliding regime (below). After the initial stiction the tip slid across the sample with a constant relative velocity ∆λ = 0.01. For a logarithmic velocity dependence (parameters σs0 = 0.5, α = 1) similar results were obtained as shown in Fig. 4 (c). Another important parameter was the scan velocity λscan . The role of the combination of parameters for the final state (stick-slip or steady sliding) is visualized in the state diagram in Fig. 5. The state diagram for the linear viscous force law is given in Fig. 5 (a). For each combination of the parameters viscous contribution µv and scan velocity λscan a numerical simulation was carried out. The velocity independent contribution to the friction was set to µc = 0.5 for all simulations. The system was considered to be in a sliding (light gray) state if after τ = 100 no sticking was determined within a period of ∆τ = 100, otherwise the system behavior was marked as stick-slip (dark gray). In the limit of µv = 0 the damping of the cantilever in the surrounding medium was not sufficient to eliminate a stick-slip behavior at the scan velocities investigated here. The boundary between the stick-slip and the sliding state shifts towards smaller viscous damping values for increased scan velocity. Thus, for imaging in contact mode it is favorable to increase the scan velocity since this reduces the tendency to stick-slip oscillations. For the logarithmic force law a similar trend for the boundary between both states prevails (Fig. 5 (b)). This indicates that the dynamics of the system is mainly determined by the small velocity regime of the friction law that allows one to use the linear approximation of Eq. (10) in order to characterize the system state in contact mode imaging. For a further discussion of the theoretical results it is instructive to investigate the conversion of the normalized values of the forces, distances and velocities into typical values for both linear friction laws (Eqs. (7, 8)). The dynamic friction force in the Coulomb friction law is obtained with Fc = µc Fs,max . The scaling parameter is the static friction force which depends on the material. The viscous damping parameter µv in the friction law Eq. (8) is converted by cv = µv k/ω0 , i.e. the damping scales with parameters that characterize the torsional resonance of the cantilever. The tip displacement is given by x = 8
(a)
2
sliding
mv
1.5
1
0.5
(b)
stick-slip
0 4 3.5
sliding
3
l0
-1
2.5 2 1.5 1 0.5 0
stick-slip 0
0.1
0.2
0.3
0.4
scan velocity lscan
Fig. 5. State diagrams of the system. Light and dark gray indicate the sliding state and the stick-slip state, respectively. (a) (λscan , µv )-diagram for the viscous friction law in Eq. (8). (b) (λscan , λ−1 0 )-diagram for the logarithmic friction law in Eq. (9). For a better comparison with the viscous friction law, the plot is given for the inverse scaling velocity λ−1 0 .
ξ Fs,max /k and the relative velocity by ∆v = ∆λ ω0 Fs,max /k. Both, material properties and parameters of the cantilever determine the scaling behavior. For a given static friction force the ratio of the cantilever’s spring constant to the resonant frequency k/ω0 scales both axes in Fig. 5. The normalized viscous damping is increased for an increased ratio whereas the normalized velocity is decreased by the same factor. However, from Fig. 5 (a) it is clear, that a careful choice of the cantilever can determine the imaging state. For example, for the parameters (λscan ; µv ) = (0.1; 0.5) the system is in the stick-slip state. By doubling ω0 /k the normalized parameters are (0.05; 1) and the system is in the steady sliding state. The static friction Fs,max of the tip on the sample scales the horizontal axis of the state diagrams in Fig. 5. Obviously, a stronger stiction increases the tendency to stick-slip oscillations since it corresponds to a shift to lower velocities in the diagram. As a numerical example, the parameters for a typical contact mode cantilever 9
with a vertical spring constant of kver = 0.2 N/m were taken: lateral spring constant k = 10 N/m, torsional resonant frequency ω0 = 2π·200 kHz, and Q = 100. Under the assumption of a maximum static friction force Fs,max = 2 nN, the respective factors for the transformation from normalized displacement and velocity into to real-world displacement and velocity were Fs,max /k = 0.2 nm and ω0 Fs,max /k = 251 µm/s, respectively. Thus, a scan velocity of λscan = 0.3 corresponded to vscan = 75µm/s, which is a typical value for contact mode imaging. A viscous damping of the torsional cantilever vibration due to the ambient atmosphere of c = k/(ω0 Q) = 8.0 · 10−8 Ns/m was obtained. Assuming solely Coulomb friction (µc = 0.5, i.e. Fc = 1 nN), the system is in the stick-slip state as can be read directly from Fig. 5 (a) with µv = 0. An additional viscous damping of µv = 0.25 corresponding to cv = µv k/ω0 = 2.0 · 10−6 Ns/m damped the stick-slip oscillations. Thus, the velocity dependent friction force Fv = cv vscan = 0.15 nN contributed only 13% to the total lateral force Fts = Fc + Fv = 1.15 nN. The dependence of the dynamics on the initial conditions is another important question. The phase portraits in Fig. 6 show the transient response and the flow towards a stable state for the example discussed above with the viscous friction parameters (a) µv = 0.0, (b) µv = 0.1, and (c) µv = 0.25. The dashed lines indicate transient trajectories leading to a dynamic equilibrium with a stick-slip motion (thick black trajectory), whereas the gray solid lines are transients leading to the fixed point indicating steady sliding (black dot). The arrows indicate the direction of propagation. The fixed point can directly be calculated from Eq. (4) with λtip = 0.0; ξ = sign(λscan )µc + µv λscan ,
(11)
which means that in the case of steady sliding the tip acceleration and velocity are zero and the spring forces balance the tip-sample force. In the absence of viscous friction in the tip sample contact trajectories with initial values outside the stick-slip cycle finally led to the stick-slip limit cycle as shown in Fig. 6 (a). The inner area of the stick-slip cycle was divided into two basins of attraction. The outer set of initial conditions led to the stick-slip cycle, whereas in the inner area the trajectories slowly spiraled towards the fixed point (gray area). The extinction of a slip oscillation in this area was due to the viscous damping of the cantilever itself. For a simple Coulomb oscillator without a damper the slip trajectories are closed circles around the fixed point [28]. A moderate increase of the viscous damping to µv = 0.1 extended the basin of attraction for the fixed point (steady sliding) and reduced the diameter of the stick-slip cycle (Fig. 6 (b)). Thus, the area ratio of the basin of attraction for the stick-slip oscillation inside the limit cycle was decreased. After a further 10
Fig. 6. Transient phase space trajectories, limit cycles and fixed points for different viscous damping (Q = 100, µc = 0.5, λscan = 0.3). The arrows indicate the flow direction. The trajectories that lead to the stick-slip limit cycle (thick black line) are indicated by dashed lines. The trajectories towards the fixed point (steady sliding) are indicated by the continuous gray line. Viscous damping contribution (a) µv = 0, (b) µv = 0.1, (c) µv = 0.25.
11
increase of the viscous parameter to µv = 0.25 the stick-slip cycle vanished and steady sliding was the only stable state (Fig. 6 (c)). From Fig. 6 (b) it is clear that with increasing µv and the therefore growing basin of attraction for steady sliding smaller and smaller perturbations are sufficient to bring the system from stick-slip into the basin of attraction for steady sliding. The converse transition from steady sliding to the stick-slip state becomes more and more unlikely. In the presence of external perturbations this provides another mechanism that can suppress stick-slip oscillations in AFM experiments. Thus, even a relatively small velocity dependent component in the frictional forces can change the system dynamics from ‘stick-slip’ to ‘steady sliding’ behavior. This clearly illustrates the role of the velocity dependent friction forces for the dynamics of contact mode AFM.
4
Conclusions
Using numerical simulations we investigated the dynamics of contact mode AFM with respect to the velocity dependence of the frictional force between tip and sample. Assuming a multi asperity contact and a time and rate independent static friction together with a velocity dependent dynamic friction two different states of the dynamic system were found. In the stick-slip regime the tip oscillates whereas in the sliding state the tip continuously slides over the specimen. The latter is favorable for high resolution imaging. In the absence of a velocity dependent component of the frictional forces it turns out that the damping of the cantilever due to internal losses and damping by the surrounding atmosphere is not sufficient to suppress the stick-slip oscillations. However, already small viscous contributions to the lateral forces allow for a steady sliding of the tip on the sample.
References
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Veloctity Dependent Friction Laws in Contact Mode Atomic-Force Microscopy Robert W. Stark 1 , Georg Schitter, Andreas Stemmer Nanotechnology Group, Swiss Federal Institute of Technology Zurich, ETH Zentrum/CLA, CH-8092 Zurich, Switzerland.
Abstract Friction forces in the tip-sample contact govern the dynamics of contact mode atomic-force microscopy. In ambient conditions typical contact radii between tip and sample are in the order of a few nanometers. In order to account for the large interaction area the dynamics of contact mode AFM is investigated under the assumption of a multi-asperity contact interface between tip and sample. Thus, the kinetic friction force between tip and sample is the product of the real contact area between both solids and the interfacial shear strength. The velocity strengthening of the lateral force is modeled assuming a logarithmic relationship between shear-strength and velocity. Numerical simulations of the system dynamics with this empirical model show the existence of two different regimes in contact mode AFM: steady sliding and stick-slip where the tip undergoes periodically stiction and kinetic friction. The state of the system depends on the scan velocity as well as on the velocity dependence of the interfacial friction force between tip and sample. Already small viscous damping contributions in the tip sample contact are sufficient to suppress stick-slip oscillations. keyword: atomic force microscopy PACS: 07.79.Lh, 68.37.Ps, 07.05.Tp
1
Introduction
In contact mode atomic-force microscopy frictional forces between tip and sample determine the dynamics of the system. On the atomic-scale a periodic ‘stick-slip’ motion of the tip occurs that is related to the periodicity of the lattice spacing of the specimen. This behavior was modeled for a one-dimensional 1
Corresponding author. Present address: Ludwig-Maximilians-Universit¨at Munich, Section Crystallography, Theresienstr. 41, 80333 Munich, Germany, Tel.: +49 89 2180 4329 Fax: +49 89 2180 4334 , e-mail: [email protected]
Preprint submitted to Elsevier Science
14 November 2003
+
L
-
FFM signal
er la s
R
bea m
photo diode
scan generator
vscan = const
Fig. 1. Scheme of friction-force microscopy (FFM). The sample is scanned with a triangular wave form leading to a constant velocity during each scan-line. Due to the friction between tip and sample the AFM cantilever is twisted. The twisting angle is detected by a split photodiode. The difference between the photo-currents in both parts of the diode is the friction-force microscopy signal.
system [1] and two-dimensional systems [2–4]. These models assume an ideal single asperity contact where a monoatomic tip interacts with the periodically modulated potential of an atomically flat surface. For a point-contact like configuration under ultra-high vacuum (UHV) conditions the frictional forces between a silicon tip and different carbon compounds were estimated to be roughly constant over a velocity range from nanometer per second to a few microns per second, whereas for larger velocities an increase of the friction with a velocity proportional friction force was predicted from simulations [5]. A logarithmic velocity dependence was reported later from experimental data obtained by atomic force microscope (AFM) in UHV [6,7]. The results were explained in terms of a modified Tomlinson [8] model that includes thermal activation. Experimental conditions in UHV differ from the standard situation in contact mode imaging under ambient conditions. For example, the typical loading force of the tip onto the sample is substantially higher than under ultrahigh vacuum conditions due to meniscus forces. This leads to contact radii of typically 4 nm between tip and sample. This implies that several atomic or molecular units are involved in the tip sample interaction. The situation is best described by a multi-asperity contact. A weak velocity dependence of the frictional force up to 50 µm/s and a following logarithmic increase for higher velocities was reported for friction force microscopy on silicon surfaces under ambient conditions [9]. Under similar conditions a logarithmic velocity dependence was reported for organosilanes grafted on silica [10]. The motion of a sliding block can range from steady sliding over stick-slip oscillations to chaotic behavior [11,12] depending on the velocity as well as on the details of the velocity dependency of the frictional forces. The velocity of the tip on to the sample can range from a few microns per second in typical 2
contact mode imaging over a few hundred microns per second in high speed AFM applications [13–15] to millimeters per second in dynamic friction mode [16–19]. Therefore one needs to know how the velocity dependence of the frictional forces determines the dynamics of contact mode AFM. For AFM imaging it is critical, whether the tip is steadily sliding on the sample surface or undergoing stick-slip cycles. To clarify the importance of a velocity dependency of the frictional forces for contact mode imaging three simple friction ’laws’ and their influence on the motion of the tip on the sample surface will be investigated.
2
Modeling
In the following the dynamics of ambient contact mode AFM will be investigated under the assumption of a multi-contact interface between tip and sample. Following the classical work of Bowden and Tabor [20] the kinetic friction force between tip and sample can be modelled as Fts = σs Σr which is the product of the real contact area between both solids Σr and the interfacial shear strength σs . State- and rate-dependency can be introduced, leading to a generalized Tabor equation Fts = σs (∆v)Σr (φ) [21,22]. Here, ∆v is the interfacial sliding velocity and the parameter φ can be understood as the age of the multi-contact interface. Several authors report on the modelling of friction forces in AFM experiments where a time independent contact area Σr is determined from contact-mechanical models [23–25]. Thus, for the analysis of contact mode AFM on hard specimen the age dependent contact strengthening Σr (φ) is approximated by the constant real contact area Σr of the Tabor equation. This assumes that the typical lattice dynamics of the tip and the sample are much faster as compared to the dynamics of the AFM cantilever and that boundary lubrication effects are negligible. The velocity strengthening of frictional forces is described by the empirical relation ∆v . (1) σs (∆v) = σs0 1 + α ln v0 The logarithmic relationship can be interpreted as the result of a thermally activated shear-induced creep process. There, α can be identified with α = kB T / (σs0 Vact ) [26]. The parameter σs0 is the shear strength in the low velocity limit and v0 a measure for the strength of the velocity dependence. The stress activation volume Vact is related to the elementary process and is e.g. about 12 nm3 for poly(methyl methacrylate) as well as for polystyrol [22]. Bouhacina et al. estimate this volume even smaller to about 0.2 nm3 [10]. In a typical AFM setup (Fig. 1) the base of the cantilever is fixed and the sample is scanned with respect to the laboratory frame of reference. The fric3
(a)
k, m, ω0
Vscan(t)
X(t)
(b)
x(t), x(t)
k c
m
m(Dv(t))
Vscan(t) Fig. 2. (a) An AFM tip in contact with a surface that is scanned with the velocity vscan . Due to friction the cantilever is twisted, which corresponds to a displacement x (t) of the tip apex. (b) Mechanical model of the system. The sample is replaced by a belt moving with the velocity vscan , the cantilever is replaced by an equivalent single-degree-of-freedom system with the mass m, spring constant k, and damping c.
tional forces lead to a torsion of the cantilever which is usually measured employing light lever detection. For the numerical modelling we assume that the friction force µ depends on the relative velocity between tip and sample ∆v (t) = vtip (t) − vscan (t) as illustrated in Fig. 2 (a). Here vtip and vscan are the velocities of the tip and sample in the laboratory frame of reference. The dynamics of this system depends on the mass m, torsional stiffness k and the damping c of the cantilever. The corresponding simplified mechanical model is shown in Fig. 2 (b). This single-degree-of-freedom approximation limits the use of the model to applications where other eigenmodes of the cantilever do not contribute to the response. For comprehensive reviews on the friction modeling of dynamic systems see [11,27]. The equation of motion is divided into two states (i) the sliding state, where the tip is sliding over the sample and (ii) the sticking state, where the relative velocity between tip and sample is zero. In the sliding state the equation of motion is given by m
d d2 x + c x + kx = Fts (∆v) , 2 dt dt 4
(2)
where x(t) is the displacement of the tip, and Fts (∆v) is the frictional force between tip and sample during sliding. The cantilever parameters are spring constant k, the effective mass of the oscillator m, resonant frequency ω0 = q k/m, and the damping coefficient c. The quality factor Q = mω0 /c is an alternate measure to describe the damping. The parameters of the harmonic oscillator are obtained from the first torsional mode of the cantilever. A dimensionless formulation of Eq. (2) is obtained with the transformations τ = ω0 t,
ξ=
k Fs,max
x,
(3)
where the maximum static friction is Fs,max . Thus, the normalized velocity is λ = v k/(Fs,max ω0 ). The normalized interfacial velocity is given by ∆λ(τ ) = λtip (τ ) − λscan (τ ). This leads to the equation of motion for the sliding state 1 ξ¨ + ξ˙ + ξ = µts (∆λ) . Q
(4)
The dot denotes the derivative d/dτ, scaling the time to the resonant frequency of the oscillator. The lateral tip-sample force is given by µts (∆λ) =
Fts (∆λ) . Fs,max
(5)
This means that the maximum static friction is normalized to µs, max = 1. The sliding phase ends if ∆λ(τ ) = 0 and the sum of inertia, damping, and spring force is smaller than the maximum stiction force, i.e. ξ¨ + Q1 ξ˙ + ξ ≤ 1. In the sticking state, the static friction force Fs is confined to the interval −Fs,max ≤ Fs ≤ Fs,max , which is in normalized coordinates −1 ≤ µs ≤ 1. All forces are balanced and the velocity between tip and sample equals zero which leads to a simple equation of motion λtip (τ ) = λscan (τ ).
(6)
The sticking phase ends if the forces overcome the static friction ξ¨ + Q1 ξ˙ + ξ > 1 and a new sliding phase begins.
In the following, the three different friction laws illustrated in Fig. 3 will be investigated: (a) Coulomb friction and friction laws with (b) linear and (c) logarithmic velocity dependence. For all three friction laws the dimensionless static friction coefficient is −1 ≤ µs ≤ 1. In the sliding regime (∆λ 6= 0) the velocity independent coulombic friction is given by µcoul (∆λ) = sign (∆λ) · µc , (7) 5
(a)
m
1
m = 0..5 c
0 -1
(b)
mv = 2
m
1
mv = 0..5
0 -1
(c)
l0-1= 4
m
1
l0-1= 1
0 -1 -0.5
0
difference velocity D l
0.5
Fig. 3. Illustration of the friction laws. The static friction at ∆λ = 0 is normalized to the interval −1 < µs < 1. (a) Coulombic friction with the friction coefficient 0 < µc ≤ 1. (b) The slope of this linear friction law is given by the viscous contribution µv . (c) The velocity dependence of the logarithmic force law is given by the parameter λ0 .
with 0 < µc ≤ 1. For the linear velocity dependence an additional viscous contribution is introduced µlin (∆λ) = sign (∆λ) · (µc + µv |∆λ|) . (8) In the limit µv → 0 again Eq. (7) is obtained. For a logarithmic velocity dependence the following force law is assumed "
µlog (∆λ) = sign (∆λ) · σs0
|∆λ| 1 + α ln + λc λ0
!#
.
(9)
The scaling parameter λ0 is a material parameter. The cutoff velocity λc is introduced in order to avoid negative friction forces at |∆λ| /λ0 < 1 and numerical divergence for very small velocities |∆λ| → 0. For the numerical simulations this parameter was set to λc = 1. The limit λ0 → ∞, leads to the 6
(a)
λtip
ξ
1
(c)
µc = 0.5 µv = 0.5
-1
λ0 = 1
0 0.2 0 -0.2 -0.4 µc = 0.95 µv = 0
1
ξ λtip
(b)
µc = 0.5 µv = 0
-1
µc = 0.5 µv = 2
λ0 = 4
0 0.2 0 -0.2 -0.4 0
100
200
300
time τ
400
0
100
200
300
time τ
400
0
100
200
300
time τ
400
Fig. 4. Normalized lateral tip displacement ξ and tip velocity λtip for the friction laws in Fig. 3 for a sample moving with the constant velocity λscan = 0.01. (a) Coulomb friction. The system is in a stick-slip state for µc = 0.5 (above) and µc = 0.95 (below). (b) With viscous damping in the tip sample contact the stick-slip does not vanish for µv = 0.5 (above). A damping of µv = 2 completely supresses the oscillation (below). (c) For the logarithmic force law stick-slip is present for λ0 = 1 (above) and vanishes for λ0 = 0.25 (below).
Coulomb friction law of Eq. (7). For small velocities |∆λ| /λ0 < λc Eq. (9) can be expanded to "
µ
log
(∆λ) = sign (∆λ) · σs0 1 + α ln (λc ) + α
∞ X (−1)n+1
n
n=1
|∆λ| λ0 λc
!n #
!
≈ sign (∆λ) · σs0 1 + α ln (λc ) + α
|∆λ| . λ0 λ c
(10)
Comparing Eqs. (8) and (10) we find for the linearized small velocity regime µv = σs0 α/ (λ0 λc ) and µc = σs0 (1 + α ln (λc )) .
3
Results and Discussion
In order to investigate basic features of contact mode AFM under ambient conditions a constant sample velocity was assumed for the numerical simulations. The friction was modeled employing the three friction laws in Eqs. (7, 8, 9) illustrated in Fig. 3. As initial condition a sticking state was selected. At t = 0 the displacement was ξ0 = 0, the relative tip sample velocity was ∆λ = 0 and the scan velocity was set to λscan = 0.01. The quality factor of the cantilevers torsional resonance was set to Q = 100. 7
Depending on the choice of the friction law different regimes could be distinguished. For a Coulomb friction law, the system remained in a stick-slip state for a dynamic friction force of µc = 0.5 (Fig. 4 (a), above) as well as for a value of µc = 0.95 (below). Initially the tip stuck to the sample and moved with the sample with a tip velocity of λtip = 0.01, i.e. ∆λ = 0. Thus, the spring was loaded until the spring force overcame the static friction. Then, the tip snapped back and a new stick-slip cycle began. The weak viscous damping Q−1 = 0.01 of the cantilever’s torsional resonance in the surrounding air was not sufficient to suppress the stick-slip oscillations. Adding viscous drag into the lateral tip sample force the stick-slip oscillations did not vanish for a viscous drag of µv = 0.5 (Fig. 4 (b), above). Increasing the the viscous drag to µv = 2 led to a steady sliding regime (below). After the initial stiction the tip slid across the sample with a constant relative velocity ∆λ = 0.01. For a logarithmic velocity dependence (parameters σs0 = 0.5, α = 1) similar results were obtained as shown in Fig. 4 (c). Another important parameter was the scan velocity λscan . The role of the combination of parameters for the final state (stick-slip or steady sliding) is visualized in the state diagram in Fig. 5. The state diagram for the linear viscous force law is given in Fig. 5 (a). For each combination of the parameters viscous contribution µv and scan velocity λscan a numerical simulation was carried out. The velocity independent contribution to the friction was set to µc = 0.5 for all simulations. The system was considered to be in a sliding (light gray) state if after τ = 100 no sticking was determined within a period of ∆τ = 100, otherwise the system behavior was marked as stick-slip (dark gray). In the limit of µv = 0 the damping of the cantilever in the surrounding medium was not sufficient to eliminate a stick-slip behavior at the scan velocities investigated here. The boundary between the stick-slip and the sliding state shifts towards smaller viscous damping values for increased scan velocity. Thus, for imaging in contact mode it is favorable to increase the scan velocity since this reduces the tendency to stick-slip oscillations. For the logarithmic force law a similar trend for the boundary between both states prevails (Fig. 5 (b)). This indicates that the dynamics of the system is mainly determined by the small velocity regime of the friction law that allows one to use the linear approximation of Eq. (10) in order to characterize the system state in contact mode imaging. For a further discussion of the theoretical results it is instructive to investigate the conversion of the normalized values of the forces, distances and velocities into typical values for both linear friction laws (Eqs. (7, 8)). The dynamic friction force in the Coulomb friction law is obtained with Fc = µc Fs,max . The scaling parameter is the static friction force which depends on the material. The viscous damping parameter µv in the friction law Eq. (8) is converted by cv = µv k/ω0 , i.e. the damping scales with parameters that characterize the torsional resonance of the cantilever. The tip displacement is given by x = 8
(a)
2
sliding
mv
1.5
1
0.5
(b)
stick-slip
0 4 3.5
sliding
3
l0
-1
2.5 2 1.5 1 0.5 0
stick-slip 0
0.1
0.2
0.3
0.4
scan velocity lscan
Fig. 5. State diagrams of the system. Light and dark gray indicate the sliding state and the stick-slip state, respectively. (a) (λscan , µv )-diagram for the viscous friction law in Eq. (8). (b) (λscan , λ−1 0 )-diagram for the logarithmic friction law in Eq. (9). For a better comparison with the viscous friction law, the plot is given for the inverse scaling velocity λ−1 0 .
ξ Fs,max /k and the relative velocity by ∆v = ∆λ ω0 Fs,max /k. Both, material properties and parameters of the cantilever determine the scaling behavior. For a given static friction force the ratio of the cantilever’s spring constant to the resonant frequency k/ω0 scales both axes in Fig. 5. The normalized viscous damping is increased for an increased ratio whereas the normalized velocity is decreased by the same factor. However, from Fig. 5 (a) it is clear, that a careful choice of the cantilever can determine the imaging state. For example, for the parameters (λscan ; µv ) = (0.1; 0.5) the system is in the stick-slip state. By doubling ω0 /k the normalized parameters are (0.05; 1) and the system is in the steady sliding state. The static friction Fs,max of the tip on the sample scales the horizontal axis of the state diagrams in Fig. 5. Obviously, a stronger stiction increases the tendency to stick-slip oscillations since it corresponds to a shift to lower velocities in the diagram. As a numerical example, the parameters for a typical contact mode cantilever 9
with a vertical spring constant of kver = 0.2 N/m were taken: lateral spring constant k = 10 N/m, torsional resonant frequency ω0 = 2π·200 kHz, and Q = 100. Under the assumption of a maximum static friction force Fs,max = 2 nN, the respective factors for the transformation from normalized displacement and velocity into to real-world displacement and velocity were Fs,max /k = 0.2 nm and ω0 Fs,max /k = 251 µm/s, respectively. Thus, a scan velocity of λscan = 0.3 corresponded to vscan = 75µm/s, which is a typical value for contact mode imaging. A viscous damping of the torsional cantilever vibration due to the ambient atmosphere of c = k/(ω0 Q) = 8.0 · 10−8 Ns/m was obtained. Assuming solely Coulomb friction (µc = 0.5, i.e. Fc = 1 nN), the system is in the stick-slip state as can be read directly from Fig. 5 (a) with µv = 0. An additional viscous damping of µv = 0.25 corresponding to cv = µv k/ω0 = 2.0 · 10−6 Ns/m damped the stick-slip oscillations. Thus, the velocity dependent friction force Fv = cv vscan = 0.15 nN contributed only 13% to the total lateral force Fts = Fc + Fv = 1.15 nN. The dependence of the dynamics on the initial conditions is another important question. The phase portraits in Fig. 6 show the transient response and the flow towards a stable state for the example discussed above with the viscous friction parameters (a) µv = 0.0, (b) µv = 0.1, and (c) µv = 0.25. The dashed lines indicate transient trajectories leading to a dynamic equilibrium with a stick-slip motion (thick black trajectory), whereas the gray solid lines are transients leading to the fixed point indicating steady sliding (black dot). The arrows indicate the direction of propagation. The fixed point can directly be calculated from Eq. (4) with λtip = 0.0; ξ = sign(λscan )µc + µv λscan ,
(11)
which means that in the case of steady sliding the tip acceleration and velocity are zero and the spring forces balance the tip-sample force. In the absence of viscous friction in the tip sample contact trajectories with initial values outside the stick-slip cycle finally led to the stick-slip limit cycle as shown in Fig. 6 (a). The inner area of the stick-slip cycle was divided into two basins of attraction. The outer set of initial conditions led to the stick-slip cycle, whereas in the inner area the trajectories slowly spiraled towards the fixed point (gray area). The extinction of a slip oscillation in this area was due to the viscous damping of the cantilever itself. For a simple Coulomb oscillator without a damper the slip trajectories are closed circles around the fixed point [28]. A moderate increase of the viscous damping to µv = 0.1 extended the basin of attraction for the fixed point (steady sliding) and reduced the diameter of the stick-slip cycle (Fig. 6 (b)). Thus, the area ratio of the basin of attraction for the stick-slip oscillation inside the limit cycle was decreased. After a further 10
Fig. 6. Transient phase space trajectories, limit cycles and fixed points for different viscous damping (Q = 100, µc = 0.5, λscan = 0.3). The arrows indicate the flow direction. The trajectories that lead to the stick-slip limit cycle (thick black line) are indicated by dashed lines. The trajectories towards the fixed point (steady sliding) are indicated by the continuous gray line. Viscous damping contribution (a) µv = 0, (b) µv = 0.1, (c) µv = 0.25.
11
increase of the viscous parameter to µv = 0.25 the stick-slip cycle vanished and steady sliding was the only stable state (Fig. 6 (c)). From Fig. 6 (b) it is clear that with increasing µv and the therefore growing basin of attraction for steady sliding smaller and smaller perturbations are sufficient to bring the system from stick-slip into the basin of attraction for steady sliding. The converse transition from steady sliding to the stick-slip state becomes more and more unlikely. In the presence of external perturbations this provides another mechanism that can suppress stick-slip oscillations in AFM experiments. Thus, even a relatively small velocity dependent component in the frictional forces can change the system dynamics from ‘stick-slip’ to ‘steady sliding’ behavior. This clearly illustrates the role of the velocity dependent friction forces for the dynamics of contact mode AFM.
4
Conclusions
Using numerical simulations we investigated the dynamics of contact mode AFM with respect to the velocity dependence of the frictional force between tip and sample. Assuming a multi asperity contact and a time and rate independent static friction together with a velocity dependent dynamic friction two different states of the dynamic system were found. In the stick-slip regime the tip oscillates whereas in the sliding state the tip continuously slides over the specimen. The latter is favorable for high resolution imaging. In the absence of a velocity dependent component of the frictional forces it turns out that the damping of the cantilever due to internal losses and damping by the surrounding atmosphere is not sufficient to suppress the stick-slip oscillations. However, already small viscous contributions to the lateral forces allow for a steady sliding of the tip on the sample.
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