Numerical simulations of a tapping-mode scanning force microscope operating in a liquid are presented. The simulations yield the time-dependent displacement of the tip, and the bursts of force pulses exerted by the vibrating tip on the sample that result in its indentation.
Atomic force microscopy, or its more general version, dubbed scanning force microscopy (SFM), has played a major role in characterizing surfaces of conducting as well as nonconducting samples (1,2). It is relatively easy to obtain the forces acting in an SFM operating in the contact or noncontact modes (3,4,5,6,7). In the contact mode, the deflection of a well-characterized cantilever is a direct measure of the force acting between tip and sample. In the noncontact mode, where the cantilever vibrates with a small amplitude relatively far away from the sample, one can use a first-order perturbation and obtain the force from either a shift in the resonance frequency of the cantilever or from the change in its amplitude of vibration (8). Recently, a tapping mode SFM has been developed to avoid the dragging action due to lateral forces as the tip raster scans across a sample (9,10,11). For this system, however, the tip-sample interaction is more complicated, because the vibrating tip traverses regions having a large variation in their force fields. The tip, at the highest point in its excursion, senses a small force field. As it descends toward the sample, the van der Waals attractive interaction increases, then decreases, and at a given distance falls to zero. For smaller distances, the repulsive forces take over and the tip deforms the sample. In this case one cannot use an approximation for the description of the forces exerted by the tip on the sample, and a numerical simulation is required. This is true in particular for the operation of a tapping-mode SFM under liquids (12,13), where the damping of the cantilever is very large and its vibration amplitude small, since the tip in this case is continuously acted upon by strong attractive and repulsive forces. The utility of operating an SFM in a liquid is that (a) one can control the Hamaker constant by a proper choice of a liquid, (b) biological samples can be imaged in their native solution, and (c) samples undergoing electrochemical processes can be imaged. It is therefore important to have an analytical tool that will shed light on the forces acting in a liquid-operated SFM. In this paper we present a numerical simulation that describes the operation of a tapping mode SFM with a critically-damped cantilever, using a range of parameters that are expected to cover most practical cases. The simulations, which use a sphere-plane geometry in a liquid medium, yield the displacement of the tip, and the bursts of force pulses exerted on the sample that result in its indentation.
For the attractive part of the force acting between tip and sample we consider a Lennard-Jones type interaction (4,14)
where H is Hamaker's constant. The distance at which the force drops to zero is z_0 = 30^-1/6 sigma. We will use Eq. 1 for tip-sample separations larger than z_0. For the repulsive part of the force, acting at z
where
Here, E_i is Young's modulus, and nu_i Poisson's ratio for the sphere and plane. A deformation of the plane by |z - z_0 | << R will therefore require a force, F(z), given by
We use the two force components in the equation of motion of our driven nonlinear oscillator system
Here, Q is the effective quality factor of the vibrating cantilever, and m its effective mass. Convenient initial conditions are z(0) = z_0 and v(0) = 0. Namely, at the beginning of the calculation, the cantilever is bent downward from its set-point at z_1 to the point z_0, where the tip-sample force is zero, and its velocity set to zero. Note that this model neglects dissipation associated with the deformation of the sample, assumes that the tip is nondeformable, and that the tip atoms do not bond to the surface atoms (16). Also note that both the repulsive and attractive parts of the force deform the sample, although the latter deformation is expected to be smaller.
For the water-immersed cantilever we chose a critically damped quality factor Q=0.5, with a spring constant k = 20 N/m, a radial frequency omega=2 \pi 10^5, effective mass m=k/omega^2, and a tip radius R=50nm. For the Hamaker constant of the tip-water-sample system we chose H=0.83 x 10^-20 J (quartz-water-quartz, (14) ). For the tip and sample we chose representative values sigma = 0.34 nm and nu = 0.5. For the samples we chose g_a=10^6[mks], g_b=10^7[mks], and g_c=10^8[mks]. This range of sample stiffness values is expected to cover most materials of interest. For example, E=9 GPa for an LB film (17), 74.5 GPa for gold (18), and 179 GPa for Si (4), corresponding to g=0.96 x 10^7 [mks], g=5.9 x 10^7 [mks], and g=10 x 10^7 [mks], respectively.
Figure 1 depicts the tip-sample force, Eqs. 1-4, where the attractive part is common to the three samples, while the repulsive part refers to the samples having (a) g=g_a, (b) g=g_b, and (c) g=g_c. The curve associated with the stiffest sample, g=g_c, is somewhat steeper than the repulsive part of the Lennard-Jones type force for small deformations. However, it increases only as (z - z_0)^3/2, whereas the repulsive part of the Lennard-Jones type force increases as z^-2.
Figure 2 shows the bottom part of the tip and a line representing the surface of the sample at their closest approach (Figs. 2a and 2c), and force F(t) (Figs. 2b and 2d), for the soft sample having g=g_a, for two set-points. Note that the force in the figures is depicted as positive for the attractive part, and negative for the repulsive part. In Fig. 2a the set-point is z=0.75 nm, and in Fig. 2c it is z=0.708 nm. The displacement z(t) for both resembles a sine wave, although, as will be discussed later, the acting forces will deform that shape. The force in Fig. 2b, for the set-point z=0.75 nm, is purely attractive, and grows as the tip approaches the sample. However, in Fig. 2d, where the set-point is decreased to z=0.75 nm, the tip approaches the zero-force line at z=z_0 on its down-swing. Here, the curve depicting the force develops a dimple, reflecting the passage of the tip through the dip of the attractive force curve.
Shown in Fig. 3 are the bottom part of the tip and a line representing the surface of the sample at their closest approach, Figs. 3a and 3c, and the force F(t), Figs. 3b and 3d. The simulations were performed for the soft sample with g=g_a and for the stiffer one with g=g_b. Here, the set-point z_1=0.65 nm is smaller than for that shown in Fig. 2. As in Fig. 2, the displacement resembles a sine wave, and the amplitude for the two samples is close to 0.5 nm. The attractive part of the force curves for these two samples is identical, while the repulsive part acts as a series of pulses that result in the indentation of the sample. However, the magnitude of the force pulses is markedly different in each case. For the soft sample, Fig. 3a, the maximum value of the repulsive part of the force is 0.4 nN. For the stiffer sample, Fig. 3d, it is 3.3 nN, The sample indentation radius is r=2.33 nm, and r=2.19 nm, for these two cases, respectively, and its depth is 0.0545 nm and 0.0481 nm, respectively. As expected, the tip dips into the soft sample somewhat deeper than into the stiffer one.
To investigate the effect that the forces have on the waveform describing the displacement of the tip, we calculated the velocity v(t), acceleration a(t), and force F(t) associated with a stiff sample having a g value of 10^8 [mks], as shown in Fig. 4a-4d. Fig. 4a depicts the bottom part of the tip and a line representing the surface of the sample at their closest approach. The maximum value of the repulsive force,Fig. 4b, is 19.9 nN, is almost two orders of magnitude larger than for the soft sample. The resultant radius and depth of indentation are r=1.78 nm and 0.0318 nm, respectively. The velocity and acceleration, Figs. 4c and 4d, clearly show the effect the forces have on their waveform. Note that the waveform describing the acceleration of the tip differs from the waveform describing the force. The reason for this is that the acceleration describes only the inertial part of the force. This can be readily observed from the equation of motion of the nonlinear driven oscillator, Eq. 5, which contains contributions to the force from the bending cantilever, tip-sample interactions, and the external force that drives the cantilever. Fig. 5 shows that even the shape of the displacement z(t), for the stiffest sample (dashed line), deviates from a pure sine wave (solid line). These simulations show, therefore, that the information regarding the forces acting between tip and sample are expressed in the waveform of the displacement and its first and second derivatives. These waveforms could conceivably be derived from the experimental waveforms of the displacement of the tip, z(t), obtained with the tapping mode scanning force microscope.
In conclusion, it is obvious that any modeling of the forces encountered in a tapping-mode scanning force microscope in general, and one operating in a liquid, in particular, have to take into account the elastic properties of the sample. Because the tapping amplitude of a tip immersed in a liquid is small, it is continuously under the influence of strong field variations. As such, it is not possible to use first-order approximations for modeling tip-sample forces, and one must resort to numerical simulations. The results presented in this paper should be helpful in understanding the interactions taking place in a tapping-mode scanning force microscope operating in a liquid for bio-molecular or electrochemical applications.
This work is partially supported by the Air Force Office of Scientific Research, and the Center for Microcontamination Control, University of Arizona. We would like to acknowledge helpful discussions with Dong Chen and Todd G. Ruskell.
References
Figure Captions
