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FORCE MEASUREMENTS AT THE NANOSCALE 7
In classical SFA, the confining surfaces are typically back-silvered atomically smooth mica surfaces of centimetric radius of curvature (Fig. 1.1B). The deflection of the spring and the position between the two surfaces is obtained via white light interferometry, with angstrom resolution. Typical spring constants are of the order of 103 N.m 1 [8, 12, 13, 26].
For a given displacement sensitivity l, the force sensitivity of the instrument in static mode is inversely proportional to the spring constant K. High force sensitivity thus requires the use of small spring constant K.
However, small value of the spring constant K are also detrimental as they lead to in-stabilities when measuring strong attractive forces. If the attractive force gradient j@F=@xj of the measured force field becomes larger than the stiffness K of the cantilever (e.g. due to capillary forces or electrostatic forces…), mechanical instabilities can occur, leading to « jump to contact » or « snap to contact » of the cantilever. Those instabilities preclude the use of cantilevers with small spring constants in presence of strong attractive force gradients.
Another intrinsic limitation associated with static force measurements relates to the 1=f noise in cantilever fluctuations. This flicker noise can be also a strong limiting factor for static force resolution [5].
Dynamic force measurements
One way to overcome those limitations is to operate the cantilever or spring in dynamic mode. In dynamic mode, the cantilever or spring is deliberately vibrated by an external force. When the vibrating probe interacts with the external force field, the oscillation amplitude and phase shift between the forcing and the cantilever oscillation is modified, allowing the detection of the interaction.
Figure 1.2: Log-log plot for the sensitivity or transfer function s = a=F for a mass-spring resonator (inset) as a function of excitation frequency f. F is the driving force, a the oscillation amplitude, K [N.m 1] the spring constant, M the equivalent mass, f0 [Hz] the resonance frequency, f [Hz] the half width of the resonance, and Q = f0= f [-] the quality factor at resonance.
A crucial parameter in dynamic mode is the sensitivity s, which quantifies the effect of an additional force F on the amplitude change a or phase change of the oscillator, via a = s F . The sensitivity can be simply taken as the transfer function of the resonator s = a=F . For a simple 2nd order mass spring resonator, the sensitivity has the typical shape shown in Fig. 1.2, with a resonance at a frequency f0 K=M. The resonance is p characterized by the quality factor Q, which can be estimated from the resonance half-width as Q = f0= f.
Far before the resonance, for small oscillation frequencies, we have s = 1=K, as in the static case. Interestingly, at resonance, the sensibility is given by s = Q=K, and can thus increase by orders of magnitudes compared to the low frequency or static case, depending on the value of Q. The quality factor Q is thus a crucial parameter, as it sets the maximal sensitivity in the dynamic case. The quantity 1=s = K=Q is referred to as the dynamic stiffness, characterizing force sensitivity at the resonance in dynamic mode.
Dynamic force measurements are obtained via two basic operation modes, known as amplitude modulation and frequency modulation.
In amplitude modulation techniques, the cantilever is driven at a fixed frequency f. Elastic and dissipative interactions will cause a change in both the amplitude and the phase shift (relative to the driving signal) of the cantilever. Such amplitude modulation techniques are for example used in dynamic SFA [2, 22].
In dynamic AFM, the driving frequency is generally chosen close to the resonance frequency f0, in order to maximize the response of the oscillator for high quality factor. However, changes in amplitude or phase due to variation of the force field will not occur instantaneously, but on a timescale Q=f0. A high Q maximizes the sensitivity but is detrimental in term of response time.
Frequency modulation techniques were introduced to combine the benefit of high dy-namic sensitivity through high quality factor Q with low response time. With frequency modulation techniques, the oscillator is systematically excited at its resonant frequency, and interactions are measured via measurement of the changes of the resonant frequency and amplitude at resonance. The response time is then given by the time to measure the change in resonant frequency, which scales as the inverse of the oscillator resonant fre-quency 1=f0 (set by the phase detection bandwidth) [18], and is not related to the intrinsic quality factor Q.
The quartz tuning fork based AFM
The main characteristic of quartz tuning forks is precisely their high quality factor Q, which make them ideal candidates for their use in frequency modulation modes. Their excellent behavior as mechanical resonators has lead to their commercial production and use as oscillators in wristwatch (Fig. 1.3).
FORCE MEASUREMENTS AT THE NANOSCALE 9
Figure 1.3: Quartz tuning forks are commercially produced for their use as excellent me-chanical resonators in wristwatch.
Microscopy [6]. Since then, they have been successfully used in a variety of home-made Atomic Force Microscope set-up for hard condensed matter or surface science measurements [7, 15, 25], for example in the groups of F. Giessibl [3, 4] and E. Meyer [10, 11].
They were introduced in the group by A. Nigues in 2014 in the context of dissipation measurement during interlayer sliding of nanotubes [19].
Interestingly, they were only scarcely used in the context of soft matter [21].
When used with frequency modulation techniques in dynamic mode, quartz-tuning forks combine several advantages.
In comparison to other force measuring techniques, the main characteristic of the quartz-tuning forks are there extremely large static stiffness, of the order of tens of kN/m. This large stiffness gives the tuning fork a high mechanical stability, allowing perfect control of the tip position even under strong attractive forces (e.g. with the tip immersed in liquids) or strong repulsive forces (e.g. during indentation experiments).
The second characteristic of tuning forks is their excellent resonator characteristics, characterized by large quality factors Q of up to 10,000 in air, 50,000 in vacuum and 2,000-5,000 in liquids. These large quality factors confer the tuning fork a low effective dynamic stiffness K=Q, corresponding to an increased dynamic force sensitivity. For a quality factor of 10,000, the effective dynamic stiffness is thus of the order of a few N.m 1, equivalent to standard AFM cantilevers.
As we will show in the following, working in frequency-modulation AFM mode allows direct quantitative measurements of the conservative and dissipative mechanical impedance of the force field with the tuning fork.
One disadvantage of frequency modulation techniques is that the excitation frequency for the dynamic impedance measurement is fixed by the resonant frequency of the system (of the order of tens of kHz) and cannot be tuned.
The tuning fork as a mechanical resonator
The tuning fork and its resonant frequencies
Figure 1.4: Schematic of the quartz-tuning fork used throughout this study, with the principal geometrical and physical parameters [24].
Tuning forks have several resonance frequencies, corresponding to symmetric and anti-symmetric motion of there prongs. In this thesis, we used the two fundamental resonances corresponding to antisymmetric motion of the prongs (see Fig. 1.5). Because those antisym-metric resonance lead to negligible displacements of the center of mass, the corresponding quality factors are very large, providing an excellent dynamic force sensitivity. When a tip is attached to one prong of the tuning fork, those resonances lead respectively to normal (N) and tangential (T) motion of the attached tip with respect to the substrate, and are thus important in the context of friction studies (see Chapter 4).
Figure 1.5: Schematic of the two principal antisymmetric oscillation modes for the tuning fork, corresponding to the excitation of (A) normal (N) and (B) tangential (T) oscillation of the tuning fork with respect to the substrate.
Quartz-based sensing
The prong and body of the tuning fork are made of a single quartz crystal, which has piezoelectric properties. When the tuning fork vibrates, stresses lead to polarization and charge dissociation inside the quartz material. An alternating stress field, caused by an oscillatory motion, will thus lead to an alternating current. This current is read out using the electrodes deposited on the quartz-tuning fork, making the tuning fork a self-sensing device. In industrially made quartz-tuning forks, as the ones we used, those contacts are positioned at strategic places so as to cancel out any electrical signal stemming from non-antisymmetric deformation of the tuning fork, maximizing the signal to noise ratio for antisymmetric deformations.
One thus has a simple relation between the alternating current i(t) = i0 exp(i!t) and the amplitude of oscillation a(t) = a0 exp(i!t) of the tuning fork, as:
The factor has been calibrated using interferometry [24] and its calibration confirmed over the course of this PhD (see Table 1.1).
Mechanical excitation
We excite the tuning fork mechanically, using a piezo-dither glued close to the tuning fork. The piezo-dither induces mechanical vibration of the tuning fork through its holding support. Mechanical excitation allows for a perfect decoupling between the excitation signal sent to the piezo-dither and the electric signal generated by the oscillation of the quartz prongs of the tuning fork. This decoupling contrasts with the standard electrical excitation method, which uses the electrodes and the piezoelectric properties of the quartz to simultaneously excite and detect the oscillation of the tuning fork.
One can model the effect of the piezo-dither as that of an oscillatory force F = F0 exp(i!t) acting on the tuning fork. We have a perfectly linear relation between this excitation force and the oscillatory voltage E(t) = E0 exp(i!t) exciting the piezo-dither, with:
The transduction factor C depends on the clamping and on the added mass on the tuning fork’s prong, and has to be calibrated at the beginning of each experiment (see following section).
Resonanc
Close to the resonance, one can in a very good approximation model the tuning fork as an effective one-dimensional mechanical oscillator [1, 24]. The validity of this assumption can be verified by probing the Lorentzian-like shape of the resonance.
In the absence of interactions, the dynamics of the tuning fork close to its resonant frequency can be simply modeled as a second order mass-spring resonator, with the tuning fork oscillation amplitude a(t) solution of:
with Meff [kg] the equivalent mass, eff [N.s.m 1] a viscous damping coefficient, Keff [N.m 1] the equivalent dynamic spring constant of the tuning fork and Fext [N] the external exci-tation forcing due to the piezo dither.
Figure 1.6: Measured amplitude (A) and phase shift (B) of the tuning fork as a function of excitation frequency f, along with theoretical fits (Eqs. 1.8 and 1.9), for a center frequency f0 = 32;756 Hz and quality factor Q = 13;725.
We can solve Eq. 1.5 under the oscillatory forcing F ext(t) = Fext exp(i!t), leading to the oscillation a(t) = a exp(i!t + ) of the prongs of the tuning fork. The amplitude a around the resonance as a function of the excitation frequency ! is given by:
We show in Fig. 1.6 the measured amplitude and phase shift for the resonance of a free tuning fork, along with fits of the corresponding Eqs. 1.8 and 1.9.
Additionally, we show in Fig. 1.7 the resonance for the normal and tangential oscillation modes of the tuning fork.
Parameters calibration
To obtain quantitative information on the force field, we need to calibrate the equivalent parameters describing the tuning fork’s motion.
Among the equivalent parameters characterizing the tuning fork, the equivalent stiffness Keff is calibrated once and for all by pressing on cantilevers of known spring constant, and measuring the resulting frequency shift for the resonance [16, 20, 24] (see following Section). The equivalent stiffness for normal and tangential oscillatory modes are given in Table 1.1.
The quality factor Q0 in the absence of interactions and resonant frequency f0 (Eq. 1.3) are measured for each tuning forks at the beginning of the experiments, by fitting the resonance (Fig. 1.6).
The transduction factor C between the excitation voltage and the force F is also obtained from the quality factor and the amplitude at resonance (Eq. 1.10) for a given excitation voltage E0, with:
C = Fext = Keffa0 (1.13).
We summarize in the following table the main parameters for the two oscillatory modes of the tuning fork.
Quantitative measurements of dissipative and con-servative response
Conservative and dissipative force field
When submitted to an external force field, the shape of the resonance of the tuning fork is modified. We show in this section that quantitative informations about the conservative and dissipative part of the interacting force field can be obtained, via measurements of the shift of the resonance frequency, amplitude at resonance and external excitation force due to the piezo-dither.
In particular, we show that non-linear dissipative forces such as solid friction forces can also be extracted from such measurements.
Figure 1.8: Tuning fork’s tip oscillating at frequency ! and amplitude a. The tip is interacting with a force field, modeled by a conservative part corresponding to a stiffness ki or real mechanical impedance Z0, and dissipative part FD, modeled by the sum of viscous like damping force ix and solid-like friction force FSx=jxj, and characterized by the dissipative mechanical impedance Z00 = FD=a.
We model the external force field by the sum of conservative and dissipative contribu-tions (Fig. 1.8).
We can also characterize the external force field by its mechanical impedance Z = F =a [N/m], with F the complex force felt by the tuning fork. This notation is especially convenient when measuring materials’ properties, for which one need to compare both the elastic and the dissipative mechanical impedance.
We thus obtain the following relation between the amplitude a at resonance, the quality factor Q at resonance, the external excitation force Fext, the solid-like dissipative force FS and the stiffness Keff with: a0 (Fext 4FS= ) = (1.21) Q Keff We first consider the limit where FS = 0.
In the presence of non-linear solid friction force, one can still measure the dissipative forces FD by measuring the additional force Fext Fext0 necessary to provide to the tun-ing fork to maintain a constant oscillation amplitude a0 and compensate for additional dissipation, with 0 = i v + 4FS FD (1.23).
with Fext0 = Keffa0=Q0 the external excitation force in the absence of interactions (charac-terizing internal damping of the tuning fork).
The factor 4= 1:27, corresponds to an error of 27 % and characterizes the largest error on the measurement of non-linear dissipative forces due to the loss of informations in the harmonics.
One can thus directly measure the sum of all dissipative forces (viscous or not) via the measurement of the external excitation force necessary to apply on the tuning fork to keep a constant oscillation amplitude a0. This excitation force Fext is directly proportional to the excitation volt-age E of the piezo-dither, via:
The factor C = Keffa0=Q0E0 characterizes the transduction of the piezo-dither, and one can equivalently express the dissipative forces as :
Ring-down experiments
The nature and amplitude of the dissipative forces can be obtained alternatively from « ring-down » experiments, where we monitor the relaxation of the oscillator after the in-terruption of the excitation. This is a classical way to measure damping of oscillators, and has been recently applied in the context of tuning forks [27].
The dynamics of the oscillator under both sliding friction and viscous-like damping is not trivial. We follow here Ricchiuto [23], and compute the amplitude at the resonance frequency, given by: !0t=2Q) exp( =2Q).
For an oscillator damped only by sliding friction, Q = 1 and we recover a linear decay of the motion, with x(t) a0(1 !0t) and x(t) = a0!0 .
Tuning-Fork based AFM set-up
Commercially available tuning forks have to be prepared before each experiment, to become usable AFM probes. In this Section, we present briefly the workflow necessary to prepare the tuning forks.
Tuning Fork preparation
The tuning forks are commercially available (Radiospare). They are stored under con-trolled atmosphere in a shell. The first step is to get the tuning fork out of its shell.
The second step is the gluing of the tip to one of the prong of the tuning fork (typically tungsten or gold). The wire of 200 m diameter is cut to a few centimeters and cleaned with ethanol. A dot of conductive epoxy glue is deposited on one prong of the tuning fork (preferentially touching one of the tuning fork’s electrodes in order to ground the tip). The wire is put in contact to the glue, which can be cured using a hair-dryer.
Table of contents :
1 The Tuning Fork based Atomic Force Microscope
1.1 Force measurements at the nanoscale
1.1.1 Static force measurements
1.1.2 Dynamic force measurements
1.1.3 The quartz tuning fork based AFM
1.2 The tuning fork as a mechanical resonator
1.2.1 The tuning fork and its resonant frequencies
1.2.2 Quartz-based sensing
1.2.3 Mechanical excitation
1.2.4 Resonance
1.2.5 Quality factor and force sensitivity
1.2.6 Parameters calibration
1.3 Dissipative and conservative response
1.3.1 Conservative and dissipative force field
1.3.2 Tuning fork in interaction
1.3.3 Ring-down experiments
1.4 Tuning-Fork based AFM set-up
1.4.1 Tuning Fork preparation
1.4.2 Integrated AFM Set-up
1.4.3 Signal acquisition
1.4.4 Signal processing and control
1.5 Limitation
1.5.1 Fundamental limitations
1.5.2 Experimental limitations
1.6 Conclusion
2 Capillary Freezing in Ionic Liquids
2.1 General Context
2.2 Experimental Set-up
2.2.1 General Set-up
2.2.2 Ionic Liquids
2.2.3 Substrates
2.2.4 Tip
2.3 Solid-like response and prewetting
2.3.1 Dissipation of an AFM tip oscillating in a viscous fluid
2.3.2 Approach curve in the ionic liquid
2.3.3 Prewetting
2.4 Confinement-induced freezing transition
2.4.1 Gibbs-Thompson effect
2.4.2 Dependence on the metallicity of the substrate
2.5 Role of electronic screening
2.5.1 Electronic screening by a Thomas-Fermi metal
2.5.2 Influence of the electronic screening on charges in the ionic liquid
2.5.3 Effect of metallicity on surface tensions
2.5.4 Effect of metallicity on the freezing transition
2.5.5 Comparison with the experimental data
2.6 Effect of tension and bulk melting temperature
2.6.1 Effect of tension
2.6.2 Effect of bulk melting temperature
2.7 Conclusion
3 Molecular Rheology of Atomic Gold Junctions
3.1 General Context
3.2 Experimental set-up
3.2.1 Experimental set-up
3.2.2 Static mechanical properties of the junction
3.3 Rheology of a gold nanojunction
3.3.1 Viscoelastic junction properties
3.3.2 Typical rheological curves and reversibility
3.4 Yield stress and yield force for plastic flow
3.4.1 Yield force, yield stress and yield strain
3.4.2 Interpretation of the deformation mechanism
3.5 Dissipative response in the plastic regime
3.5.1 Friction coefficient
3.5.2 Liquid-like dissipative response
3.5.3 Frequency dependence of the plastic transition
3.5.4 Solid-like dissipation regime at large oscillation amplitude
3.6 Conservative force response and capillary attraction
3.6.1 Capillary attraction
3.6.2 Shear induced melting of the junction
3.6.3 Jump to contact at large oscillation amplitude
3.7 Prandtl-Tomlinson model
3.7.1 Equations and non-dimensionalization
3.7.2 Simulation procedure
3.7.3 Simulation results and limiting cases
3.7.4 Discussion
3.8 Conclusion
4 Non-Newtonian Rheology of Suspensions
4.1 General context
4.1.1 Rheology of non-brownian suspensions
4.1.2 Shear thickening
4.1.3 Shear thinning
4.2 Experimental Set-up
4.2.1 Measuring normal and tangential force profiles between two approaching beads with the AFM
4.2.2 Particles, substrate and solvent
4.2.3 Rheology of macroscopic suspensions
4.3 Nanoscale force profile
4.3.1 Typical approach curve
4.3.2 Normal dissipative force
4.3.3 Normal force gradient
4.3.4 Tangential dissipative force
4.3.5 Approach in presence of a surface asperity
4.3.6 Approach between cornstarch particles
4.4 Frictional force profile
4.4.1 Characterization of the frictional regime
4.4.2 Distribution of friction coefficient and normal critical load
4.4.3 Ring-down and characterization of non-linearity
4.4.4 Measurements under moderate and high normal load
4.5 Results and Discussions
4.5.1 The shear thickening transition in PVC and Cornstarch
4.5.2 Shear thinning at low shear rate in PVC suspensions
4.5.3 Shear thinning at high shear rate in PVC suspensions
4.6 Conclusion
5 Conclusion and Perspectives
5.1 General Conclusion and Perspectives
5.1.1 Nanoscale Capillary Freezing in Ionic Liquids
5.1.2 Molecular Rheology of Gold Nanojunctions
5.1.3 Non Newtonian Rheology of Suspensions
5.1.4 Instrumental Perspectives
5.2 On-Going Perspectives on Reactive Lubrication
5.2.1 The Tuning Fork based dynamic Surface Force Apparatus
5.2.2 Reactive Lubrication in Skiing
5.2.3 Reactive Lubrication in Ionic Liquids
